On the Pluri-Lagrangian Structure of the Kdv Hierarchy

Master’s Thesis

On the pluri-Lagrangian structure of the KdV hierarchy

Mats Vermeeren

Technische UniversitätBerlin FakultätII Institut fürMathematik AG Geometrie und Mathematische Physik

Supervisor: Prof. Dr. Yuri B. Suris Second reader: Prof. Dr. Alexander I. Bobenko

October 31, 2014 (Updated January 20, 2015) Erkl¨arung

Hiermit erkläre ich, dass ich die vorliegende Arbeit selbstständig und ei- genhändig sowie ohne unerlaubte fremde Hilfe und ausschlielich unter Ver- wendung der aufgefuhrten¨ Quellen und Hilfsmittel angefertigt habe. Die selbständige und eigenständige Anfertigung versichert an Eides statt:

Berlin, den 31. Oktober 2014

Mats Vermeeren Acknowledgments

It’s better to stand by someone’s side than by yourself. Jack London There are many people who have contributed to the creation this thesis. Whether that was directly or indirectly, knowingly or not, I would like to thank them all. I feel a few of them should be mentioned explicitly. First and foremost I would like to thank my supervisor, Yuri Suris. Without his helpful comments, corrections and suggestions I would have never been able to finish this work. Furthermore, his supportive attitude helped me to stay motivated and to enjoy working on this project. I am very grateful to the Berlin Mathematical School (BMS). Not only for providing me with a scholarship and an office, but also for the invaluable help concerning German bureaucracy. I would also like to thank my fellow BMS students for making the BMS lounge and offices a very productive working environment and for providing some pleasant dis- tractions when needed. A separate thank you goes out to my office mate Stanley for correcting a fair amount of language errors in the summary in German. It goes without saying that any mistakes that are still in there are to be blamed on me. Finally I would like to thank my family for encouraging me to follow my interests and for not complaining too much when I decided to move abroad. Special thanks goes out to my brother Stijn for introducing me to beautiful bits of mathematics around the time high school math classes were boring me to death, and thus sparking my interest in mathematical studies.

Contents

1. Introduction 1 1.1. Variational methods ...... 1 1.2. Integrable systems ...... 4 1.3. Variational structure of integrable hierarchies ...... 6

2. Pluri-Lagrangian systems 9 2.1. Approximation by stepped surfaces ...... 10 2.2. Multi-time Euler-Lagrange equations for curves ...... 12 2.3. Multi-time Euler-Lagrange equations for two-dimensional surfaces . . . 14

3. Pluri-Lagrangian structure of the sine-Gordon equation 21

4. The KdV Hierarchy 25 4.1. Lax Pairs ...... 25 4.2. Variational relations ...... 28 4.3. Resolvents ...... 31

5. Pluri-Lagrangian structure of the PKdV hierarchy 35 5.1. Time derivatives of the Lagrangians ...... 35 5.2. Construction of the Lagrangian two-form ...... 37 5.3. The multi-time Euler-Lagrange equations ...... 41

6. Relation to Hamiltonian formalism 47

7. Conclusion 49

A. Appendix 51 A.1. A very short introduction to the variational bicomplex ...... 53 A.2. Traditional Euler-Lagrange equations are not ﬁrst order evolution equations ...... 55 A.3. Construction of the KdV hierarchy from the resolvent equation . . . . . 56 A.4. Table of explicitly computed quantities for the KdV hierarchy ...... 60

B. Zusammenfassung in deutscher Sprache 63

1. Introduction

This thesis explores the concept of pluri-Lagrangian systems and investigates if it is a reasonable characterization of integrability for a hierarchy of variational diﬀerential equations. This notion was suggested by Suris [16] and already investigated in detail for discrete variational systems by Boll, Petrera and Suris [5], building on work of Lobb and Nijhoﬀ [10]. Before discussing pluri-Lagrangian systems, let us quickly review traditional variational methods and spend a few words on the notion of integrable systems.

1.1. Variational methods

Nature is thrifty in all its actions.

Pierre Louis Maupertuis1

The solutions of most differential equations that occur in physics have a remarkable property. They minimize an abstract quantity called the action. A necessary condition for this to happen is that for any infinitesimal variation of the solution the corresponding variation of the action is zero. This condition can be used to find solutions.

Variations

N M N n Consider a vector bundle X = R × R over R and its n-th jet bundle J X. Let n N N L : J X → Ω (R ) be smooth a form-valued function. In other words, L is a N-form N M that depends on a function u : R → R and its partial derivatives up to order n. We sometimes emphasize this dependence by writing L[u]. We call u = (u1, . . . , uM ) the N field and L the Lagrangian N-form. The action on an open submanifold S ⊂ R is defined by Z S := L[u]dV. S One way to formalize the idea of infinitesimal variations would be to introduce the

1Common, though far from exact, translation of “la Nature dans la production de ses eﬀets agit toˆujourspar les voies les plus simples”.

1 1. Introduction

n n M variation as a function δL : J X × J X → R that satisﬁes

∞ N M ∀δu ∈ C0 (R , R ): L[u + εδu] = L[u] + εδL[u, δu] + O (ε).

This has the advantage of being immediately understandable, but the theory will turn out nicer if we use the so-called variational bicomplex, which is introduced in Appendix A.1. In this context, variations V of the ﬁeld u are “vertical” vector ﬁelds, and δL is the “vertical” exterior derivative of L. The corresponding variation of the action is given by Z δS = ιpr V δL, S where ι denotes the interior product and pr V the n-th jet prolongation of V . We want N δS to be zero for any open submanifold S ⊂ R and any V with compact support in S. If this is the case we say that the action is critical or stationary.

The Euler-Lagrange equations

N Consider coordinates (t1, . . . , tN ) on R . We will use the multi-index notation for partial derivatives: for any multi-index I = (i1, . . . , iN ) we set

|I| ∂ uk (uk)I := i i , (∂t1) 1 ... (∂tN ) N

α where |I| = i1 + ... + iN . The notations Ik and Ik will represent the multi-indices (i1, . . . , ik + 1, . . . iN ) and (i1, . . . , ik + α, . . . iN ) respectively. When convenient we will α also use the notations Itk and Itk for these multi-indices. We will write k 6∈ I (or tk 6∈ I) if ik = 0 and k ∈ I (or tk ∈ I) if ik > 0. We will denote by Di or Dti the total derivative with respect to coordinate direction ti,

N X X ∂ Di := Dti := (uk)Ii , ∂(uk)I k=1 I and by D := Di1 DiN the corresponding higher order derivatives. I t1 ... t1 Write L = L dV , where dV is a volume form that does not depend on the ﬁeld u or its derivatives. Then we have δ(dV ) = 0, hence there holds

δL = δL ∧ dV + L δ(dV ) = δL ∧ dV.

2 1.1. Variational methods

If the coeﬃcient L of L = L dV is polynomial in u and its derivatives, we have

N Z Z X X ∂L δS = ιpr V δL = δ(uk)I (V ) dV, ∂(uk)I S S k=1 I where the second sum is taken over all multi-indices, including the empty one, and δ(uk)I (V ) := ιpr V δ(uk)I is the vertical one-form δ(uk)I evaluated on the prolonged variation pr V . Integration by parts gives us

Z N X X |I| ∂L δS = (−1) DI δuk(V ) dV. ∂(uk)I S k=1 I

There is no boundary term because V vanishes on the boundary of S. We see that δS is zero for any variation V if and only if δL X |I| ∂L := (−1) DI = 0 for k = 1,...,M. (1.1) δuk ∂(uk)I I These are the Euler-Lagrange equations. The quantity on the left hand side of Equa- tion (1.1) is called the variational derivative of L with respect to uk. The variational R R P δL derivative is characterized by the property that δL ∧ dV = δuk ∧ dV . k δuk The theory outlined above is called Lagrangian ﬁeld theory. The most famous part of it is Lagrangian mechanics. In this case we have N = 1, so X is a vector bundle over the real line R, which represents time and has coordinate t. Furthermore, the Lagrangian depends only on the ﬁrst jet bundle, so the Euler-Lagrange equations (1.1) reduce to ∂L ∂L − Dt = 0 for k = 1,...,M. (1.2) ∂uk ∂(uk)t

In the rest of this text, we will generally have N > 1. On the other hand, we will restrict to M = 1, so the ﬁeld u will be a real-valued function.

Literature

An excellent introduction to Lagrangian mechanics, along with other topics in classical mechanics, can be found in the well-known book by Arnold [2]. For more about Lagrangian ﬁeld theory and the variational bicomplex, see e.g. Dickey [6, Chapter 19].

3 1. Introduction

1.2. Integrable systems

An integrable system is a system that I can solve but you cannot.

Unknown origin2 The field of integrable systems is vast and diverse. There is not even a general definition of the concept of integrability. Roughly speaking, an integrable system is a system of differential equations (or a single one) that possesses some non-obvious structure which allows one either to find an exact solution, or to derive properties of solutions. Without knowledge of this structure, the problem is usually very inaccessible.

Hamiltonian systems In many cases, the hidden structure of an integrable system is uncovered using the language of Hamiltonian mechanics.A canonical Hamiltonian system consists of the diﬀerential equations

∂H ∂H q˙k = , p˙k = − , for k = 1,...,N (1.3) ∂pk ∂qk

2N where H : R → R :(q1, . . . , qN , p1, . . . , pN ) 7→ H(q1, . . . , qN , p1, . . . , pN ) is the Hamil- ton function or Hamiltonian, and the dot represents the time derivative. These equations are closely related to the equations (1.2) of Lagrangian mechanics. The Hamilto- nian can be obtained from the corresponding Lagrangian through the so-called Legendre transform. The equations of motion (1.3) can be written in a coordinate-free way by introducing the Poisson bracket N X ∂F ∂G ∂F ∂G {F,G} := − . (1.4) ∂pk ∂qk ∂qk ∂pk k=1 2N Then the time evolution of any function F : R → R :(q1, . . . , qN , p1, . . . , pN ) 7→ F (q1, . . . , qN , p1, . . . , pN ) is given by

F˙ = {H,F }. (1.5)

In particular, F is a conserved quantity if and only if {H,F } = 0. If this is the case we say that F and H are in involution. Note that this also implies that H is a conserved quantity of the Hamiltonian ﬂow of F , since the Poisson bracket is anti-symmetric.

2I heard this quote from Prof. Dr. Thomas Kriecherbauer during a summer school at TU Munich in July 2014. He did not know its origin and I have not been able to ﬁnd out.

4 1.2. Integrable systems

It is in this framework that the best-known deﬁnition of integrability is stated. A ﬂow 2N with Hamiltonian H : R → R is completely integrable in the sense of Liouville-Arnold 2N if there exist functions H2,...,HN : R → R such that

• H1 := H, H2,..., HN are functionally independent, and

• H1,..., HN are (pairwise) in involution. The Liouville-Arnold theorem states two important facts about completely integrable systems. On the computational side it says that any such system is solvable in quadra- tures, which means that a solution can by obtained by algebraic manipulations and integration. On the abstract side it determines the topology of the symplectic leaves of a completely integrable system. These are the sets of all points that can be reached from a given point by following the flows of H1,..., HN . If the symplectic leaves are compact, the Liouville-Arnold theorem states that they must be tori. Otherwise, they are cylinders. A more general description of Hamiltonian mechanics is obtained by defining a Pois- son bracket as a bilinear operation that is anti-symmetric and satisfies the Jacobi identity {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0. Then, after fixing a Poisson bracket, generally different from the one given in Equation (1.4), we can define the flow of a Hamiltonian H by Equation (1.5). Such generalized Poisson brackets can also be defined in infinite dimensional spaces. The study of those systems is called Hamiltonian field theory. Integrable equations rarely occur by themselves. Indeed, if a system is completely integrable in the sense of Liouville-Arnold, then not only H is the Hamiltonian of an integrable flow, but so are H2,..., HN . Infinite dimensional integrable equations are usually part of infinite hierarchies. A well-known example of such an integrable hierarchy is the Korteweg-de Vries (KdV) hierarchy, which will be the main example in this work.

Lax equations A second property that is often used to characterize integrable systems is the existence of a Lax pair. This is a pair of operators (L, A) such that the Lax equation

L˙ = [A, L] := AL − LA (1.6) is equivalent to the given system. The fundamental property of Lax equations is that the time-evolution given by Equation (1.6) of an operator L is isospectral. As a very simple example, consider the harmonic oscillator described by the equations

q˙ = p andp ˙ = −ω2q. (1.7)

5 1. Introduction

A possible Lax pair for this system is formed by the (2 × 2)-matrices

p ωq 1 0 −ω L := and A := . ωq −p 2 ω 0

Indeed, we have

1 −ω2q ωp 1 ω2q −ωp −ω2q ωp [A, L] = − = , 2 ωp ω2q 2 −ωp −ω2q ωp ω2q so the Lax equation (1.6) is equivalent to the equations of motion (1.7). The eigenvalues p of L are λ = ± p2 + ω2q2, so in this case the isospectrality of the ﬂow is equivalent to 1 2 1 2 2 the conservation of the energy 2 p + 2 ω q .

Literature More about Hamiltonian mechanics can be found in Arnold’s book [2]. For a thorough introduction to the classical theory of integrable systems, see e.g. Babelon, Bernard and Talon [3].

1.3. Variational structure of integrable hierarchies

This idea is, doubtless, rather inven- tive (not to say exotic) in the framework of the classical calculus of variations.

From the introduction of [5]

Even though Hamiltonian and Lagrangian mechanics are very closely related, there is no well-developed Lagrangian theory of integrable hierarchies. This thesis explores a candidate for such a notion. Suppose we have N − 1 two-dimensional Lagrangian problems for a ﬁeld u(x, t). We think of x as a space coordinate and of t as a time coordinate. We let the Lagrangian problems share the space coordinate x, but introduce separate time coordinates ti, i = 2,...,N, for each of them. Simultaneous solutions can then be considered as N a function u : R → R :(x, t2, . . . , tN ) 7→ u(x, t2, . . . , tN ). The usual Lagrangian R formulation of the individual equations requires the action S L to be critical on any (x, ti)-plane S. In the pluri-Lagrangian formulation, we will impose a much stronger N condition: the action must be critical on any two-dimensional surface S in R . This approach might seem unmotivated, but closely related ideas are already established in other areas of mathematics. Consider for example the theory of pluriharmonic

6 1.3. Variational structure of integrable hierarchies

N functions [9, Section 2.2]. A function of several complex variables f : C → R is pluriharmonic if ∂2f = 0 ∂zi∂zj for all i, j. This is equivalent to saying that f ◦ Γ is harmonic,

∂2(f ◦ Γ) = 0, ∂z∂z

N for any holomorphic curve Γ : C → C [12]. Another way to put this is to say that f minimizes the Dirichlet functional Z 2 ∂(f ◦ Γ) dz ∧ dz¯ Γ ∂z

m along any holomorphic curve Γ : C → C . This property is similar to the pluri- Lagrangian condition. Other precursors of the pluri-Lagrangian approach are Baxter’s Z-invariance in statistical mechanics and the classical notion of variational symmetries, as explained in [16]. The aim of this thesis is to develop the theory of pluri-Lagrangian systems. There are three main goals. First we will derive the multi-time Euler-Lagrange equations for one- and two-dimensional pluri-Lagrangian systems. Then we will construct a pluri- Lagrangian form for the KdV hierarchy. Finally, we provide a brief discussion of the relation between pluri-Lagrangian systems and the established Hamiltonian theory of integrable hierarchies.

Literature The pluri-Lagrangian approach for lattice systems is discussed by Boll, Petrera, and Suris [5]. First steps towards a pluri-Lagrangian theory for continuous integrable hierarchies were made by Suris [15, 16].

2. Pluri-Lagrangian systems

N M N n Consider a vector bundle X = R × R over R and its n-th jet bundle J X. Let n d N N L : J X → Ω (R ) be a smooth a form-valued function (d < N). We call R the multi- time, u the field, and L the Lagrangian d-form. We will use coordinates (t1, . . . , tN ) on N R . Definition 2.1. We say that the field u solves the pluri-Lagrangian problem for L if u is R N a critical point of the action S L for all d-dimensional surfaces S in R simultaneously. The differential equations describing this condition are called the multi-time Euler- Lagrange equations. We say that they form a pluri-Lagrangian system and that L is a pluri-Lagrangian structure for for these equations. To derive the multi-time Euler-Lagrange equations, we consider the vertical derivative δL of L in the variational bicomplex, and a generalized vertical vector field V . The criticality condition is described by the equation Z ιpr V δL = 0, (2.1) S for every d-dimensional surface S and any variation V on the interior of S. Proposition 2.2. The exterior derivative dL of the Lagrangian is constant on solutions of the multi-time Euler-Lagrange equations.

Proof. Consider a solution u and a small (d + 1)-dimensional ball B. Because S := ∂B has no boundary Equation (2.1) is satisﬁed for any variation V . Using the properties that δd + dδ = 0 and ιpr V d + d ιpr V = 0 (Propositions A.1 and A.3 in Appendix A.1), and Stokes’ theorem, we ﬁnd that Z Z Z Z ιpr V δ(dL) = − ιpr V d(δL) = d(ιpr V δL) = ιpr V δL = 0. B B B S

Since this holds for any ball B it follows that ιpr V δ(dL) = 0 for any variation V of a solution u. Therefore dL is constant on solutions.

We will take a closer look at this property in Chapter 6, when we discuss the link with Hamiltonian theory. The rest of the current chapter is devoted to the derivation of the multi-time Euler-Lagrange equations for one- and two-dimensional systems in full generality. Previously, they where only known in some special cases [15, 16].

9 2. Pluri-Lagrangian systems

2.1. Approximation by stepped surfaces

The key to deriving the multi-time Euler-Lagrange equations is to observe that it suffices to consider a very specific type of surface. Definition 2.3. A d-dimensional coordinate surface is a surface S such that for d ∂ ∂ distinct i1, . . . , id and for all x ∈ S we have TxS = span ,..., , i.e. a surface of ∂ti1 ∂tid the form {(t1, . . . , tN ) | ∀j 6= i1, . . . , id : tj = cj}. A stepped surface is a finite union of coordinate surfaces. Lemma 2.4. If the action is stationary on every stepped surface, then it is stationary on every smooth surface. The proof of this lemma is quite long and technical. The reader will not be judged for skipping it on the first reading.

Proof. Assume that the action is stationary on all d-dimensional stepped surfaces in N N R . Let S be a smooth d-dimensional surface. Partition the space R into hypercubes Ci of edge length ε. We can choose this partitioning in such a way that the surface N S does not contain the center of any of the hypercubes. Denote by Si := S ∩ Ci the piece of S that lies in the cube Ci. N We give each hypercube its own Euclidean coordinate system [−1, 1] → Ci and identify the hypercube with its coordinates. In each punctured hypercube [−1, 1]N \{0} we deﬁne a family of balloon maps  αx  if kxkmax < α N N N k k Bα :[−1, 1] \{0} → [−1, 1] \{0} : x 7→ x max x if kxkmax ≥ α for α ∈ [0, 1]. Here, kxkmax := max(|x1|,... |xN |) denotes the maximum norm with respect to the local coordinates. The idea is that from the center of each hypercube we inﬂate a “square” balloon which pushes the curve away from the center until it lies on the boundary of the hypercube. N−1 N N Indeed, the deformed curve Si := B1 (Si ) lies on the boundary of the hypercube, i.e. within the (N − 1)-faces of the hypercube. We want it to lie within the d-faces of the hypercube, which would imply that it is a stepped surface. To achieve this, we introduce a balloon map  αx  if kxkmax < α N−1,j N−1 N−1 k k Bα :[−1, 1] \{0} → [−1, 1] \{0} : x 7→ x max x if kxkmax ≥ α

j in each of the (N − 1)-faces Ci of the hypercube Ci, which pushes the surface into the N−2 N−1 (N − 2)-faces. We denote the surface we obtain this way by Si . If the surface Si happens to contain the center of a (N − 1)-face, we can slightly perturb the surface

10 2.1. Approximation by stepped surfaces

2 Figure 2.1.: Balloon maps in nine adjacent squares deforming a curve in R . From left to right: α = 0.2, α = 0.7, and α = 1.

3 Figure 2.2.: The second and last iteration for a curve in R . From left to right: α = 0.1, α = 0.6, and α = 1.

k,j without aﬀecting the argument. By iterating this procedure, using balloon maps Bα j d in each k-face Ci (N ≥ k ≥ d + 1), we obtain a surface Si that lies in the d-faces. Consider the (d + 1)-dimensional surface

N [ [ [ k,j k j Mi := Bα (Si ∩ Ci ) k=d+1 j α∈[0,1] j: Ci is a k-face of Ci

N that is swept out by the consecutive application of the balloon maps to Si = S ∩ Ci. Assuming that ε is small compared to the curvature of S, the (d + 1)-dimensional S k,j k j d+1 volume of each of the α∈[0,1] Bα (Si ∩ Ci ) is of the order ε . The number of such volumes making up Mi only depends on the dimensions N and d, not on ε, so the

11 2. Pluri-Lagrangian systems

d+1 (d + 1)-dimensional volume |Mi| of Mi is of the order |Mi| = O(ε ). Now consider a variation V with compact support and restrict the surface S to this S d support. Denote by Sb := i Si the stepped surface obtained from S by repeated S application of balloon maps in all the hypercubes, and by M := i Mi the (d + 1)- dimensional surface swept out by these balloon maps. The boundary of M consists of S, Sb, and a small strip of area O(ε) connecting the boundaries of S and Sb (the dotted line in Figure 2.1). The number of hypercubes intersecting S is of order ε−d, so |M| = O(ε−d)O(εd+1) = O(ε). It follows that Z Z Z Z

ιpr V δL − ιpr V δL = ιpr V δL + O(ε) = d(ιpr V δL) + O(ε) → 0 Sb S ∂M M as ε → 0. By assumption R ι δL = 0 for all ε, so the action on S will be stationary Sb pr V as well.

2.2. Multi-time Euler-Lagrange equations for curves

Our main result is the derivation of the multi-time Euler-Lagrange equations for two- dimensional surfaces (d = 2). That will allow us to study the KdV hierarchy as a pluri-Lagrangian system. However, it is instructive to ﬁrst derive the multi-time Euler- Lagrange equations for curves (d = 1). P Theorem 2.5. Consider a Lagrangian one-form L = i Li dti. The corresponding multi-time Euler-Lagrange equations are

δiLi = 0 ∀I 63 i, (2.2) δuI δiLi δjLj = ∀I, (2.3) δuIi δuIj where i and j are distinct, and

δiLi X α α ∂Li ∂Li ∂Li 2 ∂Li := (−1) Di = − Di + Di − ... δuI ∂uIiα ∂uI ∂uIi ∂uIi2 α∈N is the variational derivative of Li with respect to uI . Remark. Note that δi diﬀers from the traditional variational derivative in that the δuI additional derivatives are with respect to the coordinate ti only. If it is clear in which direction the extra derivatives are taken, we will also use the notation δ . δuI Remark. In the special case that L only depends on the ﬁrst jet bundle, the system (2.2)–(2.3) reduces to the equations found in [15].

Proof of Theorem 2.5. It is suﬃcient to look at a general L-shaped curve S = Si ∪ Sj,

12 2.2. Multi-time Euler-Lagrange equations for curves t t t t j j p j j p Si Si Sj Sj Sj Sj S Si i p p

ti ti ti ti

Figure 2.3.: The four rotations of an L-shaped curve.

where T S = span ∂ and T S = span ∂ . Denote the vertex by p := S ∩ S . x i ∂ti x j ∂tj i j We orient the curve such that Si induces the positive orientation on the point p and Sj the negative orientation. Figure 2.3 shows this for the four possible rotations of the L-shape in the (ti, tj)-plane. The variation of the action is Z Z Z ιpr V δL = (ιpr V δLi) dti + (ιpr V δLj) dtj S Si Sj Z Z X ∂Li X ∂Lj = δuI (V ) dti + δuI (V ) dtj, ∂uI ∂uI Si I Sj I where the sums are taken over all multi-indeces, including the empty one. Note that these sums are actually ﬁnite. Indeed, since L depends on the n-th jet bundle, all terms with |I| > n vanish. Now we expand the sum in the ﬁrst of the integrals and perform integration by parts. Z (ιpr V δLi) dti Si Z X ∂Li ∂Li ∂Li ∂Li = δuI (V ) + δuIi(V ) + δuIi2 (V ) + δuIi3 (V ) + ... dti ∂uI ∂uIi ∂u 2 ∂u 3 Si I63i Ii Ii Z X ∂Li ∂Li 2 ∂Li 3 ∂Li = − Di + Di − Di + ... δuI (V ) dti ∂uI ∂uIi ∂u 2 ∂u 3 Si I63i Ii Ii X ∂Li ∂Li ∂Li + δuI (V ) + δuIi(V ) − Di δuI (V ) ∂uIi ∂u 2 ∂u 2 I63i Ii Ii ! ∂Li ∂Li 2 ∂Li + δuIi2 (V ) − Di (V )δuIi(V ) + Di δuI (V ) + ... . ∂uIi3 ∂uIi3 ∂uIi3 p

13 2. Pluri-Lagrangian systems

Using the language of variational derivatives, this reads Z Z X δiLi X δiLi δiLi (ιpr V δLi) dti = δuI (V ) dti + δuI (V ) + δuIi(V ) + ... δuI δuIi δu 2 Si Si I63i I63i Ii p Z X δiLi X δiLi = δuI (V ) dti + δuI (V ) . δuI δuIi Si I63i I p

The other piece, Sj, contributes Z Z X δjLj X δjLj (ιpr V δLj) dtj = δuI (V ) dtj − δuI (V ) , δuI δuIj Sj Sj I63j I p where the minus sign comes from the fact that Sj induces negative orientation on the point p. Summing the two contributions we ﬁnd Z Z Z X δiLi X δjLj ιpr V δL = δuI (V ) dti + δuI (V ) dtj δuI δuI S Si I63i Sj I63j X δiLi δjLj + − δuI (V ) . (2.4) δuIi δuIj I p

Now require that the variation (2.4) of the action is zero for any variation V . If we consider variations that are zero on Sj, then we find for every I which does not contain i that δiLi = 0. δuI Given this equation, and its analogue for the index j, only the last term remains in the right hand side of Equation (2.4). Considering variations around the vertex p we then find δiLi δjLj = . δuIi δuIj It is clear these two equations combined are also sufficient for the action to be critical.

2.3. Euler-Lagrange equations for two-dimensional surfaces

The two-dimensional case (d = 2) covers many known integrable hierarchies, including the potential KdV hierarchy which we will discuss in detail later on. We consider a P Lagrangian two-form L = i

14 2.3. Multi-time Euler-Lagrange equations for two-dimensional surfaces

Theorem 2.6. The multi-time Euler-Lagrange equations for two-dimensional surfaces are

δijLij = 0 ∀I 63 i, j, (2.5) δuI δijLij δ L = ik ik ∀I 63 i, (2.6) δuIj δuIk δijLij δjkLjk δ L + + ki ki = 0 ∀I, (2.7) δuIij δuIjk δuIki where i, j, and k are distinct, and

δijLij X α+β α β ∂Lij := (−1) Di Dj δuI ∂uIiαjβ α,β∈N is the variational derivative. Remark. In the special case that L depends on the second jet bundle and N = 3, the system (2.5)–(2.7) reduces to the equations given in [16]. Before proceeding with the proof of Theorem 2.6, we introduce some terminology and prove a lemma. A two-dimensional stepped surface consisting of q flat pieces intersecting at some point p is called a q-flower around p. The flat pieces are called its petals. If the action is stationary on every q-flower, then it is stationary on any stepped surface. By Lemma 2.4 the action will then be stationary on any surface. The following Lemma shows that it is sufficient to consider 3-flowers. Lemma 2.7. Take any q > 3. If the action is stationary on every 3-flower, it is stationary on every q-flower.

Proof. Let F be a q-flower around a point p. Denote its petals corresponding to coordinate directions (ti1 , ti2 ), (ti2 , ti3 ), . . . , (tiq , ti1 ) by S12, S23,..., Sq1 respectively. Consider the 3-flower F123 = S12 ∪ S23 ∪ S31, where S31 is a petal in the coordinate direction (ti3 , ti1 ) such that F123 is a flower around the same point p. Similarly, define F134,...,F1 q−1 q. Then (for any integrand) Z Z Z + + ... + F123 F134 F1 q−1 q Z Z Z Z Z Z Z Z Z = + + + + + + ... + + + . S12 S23 S31 S13 S34 S41 S1 q−1 Sq−1 q Sq1

Here, S21, S32, . . . are the petals S12, S23, . . . but with opposite orientation (see Figure 2.4), hence all terms where the index of S contains 1 cancel, except for the ﬁrst and

15 2. Pluri-Lagrangian systems

tj tj

Sij

∂Sj ti Sjk

ti p ∂S tk i ∂Sk

tl Ski tk Figure 2.4.: Two 3-flowers composed to form a 4-flower. The common petal does Figure 2.5.: The 3-flower considered in not contribute to the integral because it oc- the proof of Theorem 2.6. curs twice with opposite orientation. last, leaving Z Z Z Z Z Z Z Z + ... + = + + + ... + + = . F123 F1 q−1 q S12 S23 S34 Sq−1 q Sq1 F By assumption the action is stationary on every 3-flower, so Z Z Z ιpr V δL = ιpr V δL + ... + ιpr V δL = 0 F F123 F1 q−1 q

Proof of Theorem 2.6. Consider a 3-ﬂower S = Sij ∪ Sjk ∪ Ski around the point p. Denote its interior edges by

∂Si := Sij ∩ Ski, ∂Sj := Sjk ∩ Sij, ∂Sk := Ski ∩ Sjk.

On Si, Sj and Sk we choose the orientations that induce negative orientation on p. We choose the orientation on the petals in such a way that the orientations of Si, Sj and Sk are induced by Sij, Sjk and Ski respectively. Then the orientations of Si, Sj and Sk are the opposite of those induced by Ski, Sij and Sjk respectively (see Figure 2.5). We will calculate Z Z Z Z ιpr V δL = ιpr V δL + ιpr V δL + ιpr V δL (2.8) S Sij Sjk Ski

16 2.3. Multi-time Euler-Lagrange equations for two-dimensional surfaces and require it to be zero for any variation V which vanishes on the (outer) boundary of S. This will give us the multi-time Euler-Lagrange equations.

For the ﬁrst term of the right hand side of Equation (2.8) we ﬁnd Z Z ιpr V δL = (ιpr V δLij) dti ∧ dtj Sij Sij Z X ∂Lij = δuI (V ) dti ∧ dtj ∂uI Sij I Z X X ∂Lij = δuIiλjµ (V ) dti ∧ dtj. S ∂uIiλjµ ij I63i,j λ,µ∈N

First we perform integration by parts with respect to ti as many times as possible. Z Z X X λ λ ∂Lij ιpr V δL = (−1) Di δuIjµ (V ) dti ∧ dtj. S S ∂uIiλjµ ij ij I63i,j λ,µ∈N Z λ−1 X X X π π ∂Lij − (−1) Di δuIiλ−π−1jµ (V ) dtj, ∂S ∂uIiλjµ j I63i,j λ,µ∈N π=0 where the minus sign is due to the choice of orientations. Next integrate by parts with respect to tj as many times as possible. Z Z X X λ+µ λ µ ∂Lij ιpr V δL = (−1) Di Dj δuI (V ) dti ∧ dtj. (2.9) S S ∂uIiλjµ ij ij I63i,j λ,µ∈N Z λ−1 X X X π π ∂Lij − (−1) Di δuIiλ−π−1jµ (V ) dtj (2.10) ∂S ∂uIiλjµ j I63i,j λ,µ∈N π=0 Z µ−1 X X X λ+ρ λ ρ ∂Lij − (−1) Di Dj δuIjµ−ρ−1 (V ) dti. (2.11) ∂S ∂uIiλjµ i I63i,j λ,µ∈N ρ=0 The minus sign of (2.11) is not due to orientation, but due to the anti-symmetry of two-forms: for any function f there holds d(fdti) = (Dj f) dtj ∧dti = −(Dj f) dti ∧dtj.

We can rewrite the ﬁrst integral (2.9) as Z X δijLij δuI (V ) dti ∧ dtj. δuI Sij I63i,j

17 2. Pluri-Lagrangian systems

The last integral (2.11) takes a similar form if we replace the index µ by β = µ − ρ − 1.

Z µ−1 X X X λ+ρ λ ρ ∂Lij − (−1) Di Dj δuIjµ−ρ−1 (V ) dti ∂S ∂uIiλjµ i I63i,j λ,µ∈N ρ=0 Z X X λ+ρ λ ρ ∂Lij = − (−1) Di Dj δuIjβ (V ) dti ∂S ∂uIiλjβ+ρ+1 i I63i,j β,λ,ρ∈N Z X X δijLij = − δuIjβ (V ) dti ∂S δuIjβ+1 i I63i,j β∈N Z X δijLij = − δuI (V ) dti. δuIj ∂Si I63i

To write the integral (2.10) in this form we ﬁrst perform integration by parts.

Z λ−1 X X X π π ∂Lij − (−1) Di δuIiλ−π−1jµ (V ) dtj ∂S ∂uIiλjµ j I63i,j λ,µ∈N π=0 Z λ−1 X X X π+µ π µ ∂Lij = − (−1) Di Dj δuIiλ−π−1 (V ) dtj ∂S ∂uIiλjµ j I63i,j λ,µ∈N π=0 λ−1 µ−1 X X X X π+ρ π ρ ∂Lij + (−1) Di Dj δuIiλ−π−1jµ−ρ−1 (V ) . ∂uIiλjµ I63i,j λ,µ∈N π=0 ρ=0 p Then we replace λ by α = λ − π − 1 and in the last term µ by β = µ − ρ − 1.

Z λ−1 X X X π π ∂Lij − (−1) Di δuIiλ−π−1jµ (V ) dtj ∂S ∂uIiλjµ j I63i,j λ,µ∈N π=0 Z X X π+µ π µ ∂Lij = − (−1) Di Dj δuIiα (V ) dtj ∂S ∂uIiα+π+1jµ j I63i,j α,µ,π∈N X X π+ρ π ρ ∂Lij + (−1) Di Dj δuIiαjβ (V ) ∂uIiα+π+1jβ+ρ+1 I63i,j α,β,π,ρ∈N p Z X X δijLij X X δijLij = − δuIiα (V ) dtj + δuIiαjβ (V ) ∂S δuIiα+1 δuIiα+1jβ+1 j I63i,j α∈N I63i,j α,β∈N p Z X δijLij X δijLij = − δuI (V ) dtj + δuI (V ) . δuIi δuIij ∂Sj I63j I p

18 2.3. Multi-time Euler-Lagrange equations for two-dimensional surfaces

Putting everything together we ﬁnd Z Z Z X δijLij X δijLij ιpr V δL = δuI (V ) dti ∧ dtj − δuI (V ) dti δuI δuIj Sij Sij I63i,j ∂Si I63i Z X δijLij X δijLij − δuI (V ) dtj + δuI (V ) . δuIi δuIij ∂Sj I63j I p

Expressions for the integrals over Sjk and Ski are found by cyclic permutation of the indices. Finally, keeping in mind that Lki = −Lik, we obtain Z Z Z X δijLij X δijLij δkiLik ιpr V δL = δuI (V ) dti ∧ dtj − − δuI (V ) dti δuI δuIj δuIk S Sij I63i,j ∂Si I63i X δijLij + δuI (V ) + cyclic permutations in i, j, k. (2.12) δuIij I p From this we can read oﬀ the multi-time Euler-Lagrange equations.

3. Pluri-Lagrangian structure of the sine-Gordon equation

The ﬁrst example of a pluri-Lagrangian system in our sense was discussed by Suris [16] at a time when the multi-time Euler-Lagrange equations were only known in some special cases. It consists of the sine-Gordon equation and the modiﬁed KdV equation. The presentation in this chapter closely follows [16].

The sine-Gordon equation uxy = sin u is the Euler-Lagrange equation for the La- grangian 1 L := u u − cos u. 2 x y ∂ Consider the vector ﬁeld ϕ ∂u with 1 ϕ := u + u3 xxx 2 x P ∂ and its prolongation Dϕ := (DI ϕ) . It is known that Dϕ is a variational symmetry I ∂uI for the sine-Gordon equation [13, p. 336]. In particular, we have that

Dϕ L = − Dx N + Dy M (3.1) with 1 1 1 M := ϕu − u4 + u2 , 2 x 8 x 2 xx 1 1 N := − ϕu + u2 cos u + u (u − sin u). 2 y 2 x xx xy

Now we introduce a new independent variable z corresponding to the “flow” of the generalized vector field Dϕ, i.e. uz = ϕ. Consider simultaneous solutions of the Euler- δL Lagrange equation δu = 0 and of the flow uz = ϕ as functions of 3 independent variables x, y, and z. Then Equation (3.1) expresses the closedness of the two-form

L := L dx ∧ dy + M dz ∧ dx + N dy ∧ dz.

In other words, dL = 0 on simultaneous solutions, so is consistent with Proposition 2.2. Therefore L is a reasonable candidate for a Lagrangian two-form.

21 3. Pluri-Lagrangian structure of the sine-Gordon equation

Theorem 3.1. The multi-time Euler-Lagrange equations for the Lagrangian two-form

L = L12 dx ∧ dy + L13 dz ∧ dx + L23 dy ∧ dz with the components 1 L = u u − cos u, (3.2) 12 2 x y 1 1 1 L = u u − u4 + u2 , (3.3) 13 2 x z 8 x 2 xx 1 1 L = − u u + u2 cos u + u (u − sin u), (3.4) 23 2 y z 2 x xx xy consist of the sine-Gordon equation

uxy = sin u, the modiﬁed KdV equation 1 u = u + u3, z xxx 2 x and corollaries thereof. On solutions of either of these equations the two-form L is closed.

Proof. Let us calculate the multi-time Euler-Lagrange equations (2.5)–(2.7) one by one:

δ12L12 • The equation = 0 yields uxy = sin u. δu δ12L12 For any α > 0 the equation = 0 yields 0 = 0. δuzα

δ13L13 3 2 • The equation = 0 yields uxz = u uxx + uxxxx. δu 2 x δ13L13 For any α > 0 the equation = 0 yields 0 = 0. δuyα

δ23L23 1 2 • The equation = 0 yields uyz = u sin u + uxx cos u. δu 2 x δ23L23 The equation = 0 yields uyxx = ux cos u. δux δ23L23 The equation = 0 yields uxy = sin u. δuxx δ23L23 For any α > 2 , the equation = 0 yields 0 = 0. δuxα

22 δ13L13 δ23L23 1 3 • The equation = yields uz = uxxx + ux. δux δuy 2

δ13L13 δ23L23 The equation = yields uxx = uxx. δuxx δuxy

δ13L13 δ23L23 For any other I the equation = yields 0 = 0. δuIx δuIy

δ12L12 δ13L13 1 1 • The equation = yields 2 ux = 2 ux. δuy δuz

δ12L12 δ13L13 For any nonempty I, the equation = yields 0 = 0. δuIy δuIz

δ12L12 δ23L32 1 1 • The equation = yields 2 uy = 2 uy. δux δuz δ12L12 δ23L32 For any nonempty I, the equation = yields 0 = 0. δuIx δuIz

δ12L12 δ23L23 δ13L31 • For any I the equation + + = 0 yields 0 = 0. δvIxy δvIyz δvIzx

It remains to notice that all nontrivial equations in this list are corollaries of the equa- 1 3 tions uxy = sin u and uz = uxxx + 2 ux, derived by diﬀerentiation. The closedness of L can be veriﬁed by direct calculation: 1 D L − D L + D L = (u u + u u ) + u sin u z 12 y 13 x 23 2 yz x zx y z 1 1 1 − u u − u u + u3u − u u 2 yz x 2 z xy 2 x xy xx xxy 1 1 1 − u u − u u + u u cos u − u3 sin u 2 xz y 2 z xy x xx 2 x + uxxx(uxy − sin u) + uxx(uxxy − ux cos u) 1 = − u − u3 − u (u − sin u). z 2 x xxx xy

Remark. The sine-Gordon equation and the modiﬁed KdV equation are the simplest equations of their respective hierarchies. Furthermore, those hierarchies can be seen as the positive and negative parts of one single hierarchy that is inﬁnite in both directions [11, Sections 3c and 5k]. It seems likely that this whole hierarchy possesses a pluri- Lagrangian structure.

4. The KdV Hierarchy

Our second and most important example will be the Korteweg-de Vries (KdV) hierarchy. The KdV equation played a central role in the development of the modern theory of integrable systems. That does not only make for an interesting history lesson (see e.g. Kasman [8] for a very accessible history, or Palais [14] for a more technical text), but it also has the consequence that many diﬀerent approaches are available to introduce and study this equation (see e.g. Dickey [6]). We will use Lax pairs and pseudodiﬀerential operators because they provide an elegant way to construct the whole KdV hierarchy at once. This chapter borrows heavily from [6] and to a lesser extent from [4].

4.1. Lax Pairs

Consider the Schr¨odingeroperator

L := ∂2 + u. (4.1)

We look for diﬀerential operators P such that the commutator [P, L ] is a zeroth-order operator, i.e a function. Then it is possible to consider the Lax equation Lt = [P, L ], where the subscript t denotes the time derivative taken coeﬃcient-wise. This equation is equivalent to the evolution equation

ut = [P, L ]. (4.2)

The equations obtained by plugging different operators P into Equation (4.2) form the KdV hierarchy. To construct suitable operators P we need a generalization of the concept of differential operator. Definition 4.1. A pseudodifferential operator (PDO) A is a formal series of the form

m X i A = Ai∂ ,Am 6= 0, i=−∞ where the coeﬃcients Ai are diﬀerential polynomials in u, ux, uxx, . . . . The highest index, m, is called the order of A .

25 4. The KdV Hierarchy

The diﬀerential part A+ and the integral part A− of A are deﬁned as

m −1 X i X i A+ := Ai∂ and A− := Ai∂ . i=0 i=−∞

The coefficient of ∂−1 in a PDO A is called the residue of A and denoted by res(A ). We call A a pure function if Ai = 0 for all i 6= 0. The set of PDOs has a ring structure. The addition of PDOs is defined coefficient- wise, m m m X i X i X i Ai∂ + Bi∂ = (Ai + Bi)∂ . i=−∞ i=−∞ i=−∞ The multiplication of PDOs is defined by the distributivity property and the rule

k k ∂k(A∂j) = A∂j+k + (D A)∂j+k−1 + (D2 A)∂j+k−2 + ..., 1 x 2 x

k k(k − 1) ... (k − i + 1) where := for negative values of k as well as for positive i i! ones. −1 1 Proposition 4.2. The PDOs L and L 2 exist and are unique. They commute with each other and with L . Remark. In fact these properties hold for any PDO A with essentially the same proof. However, we will only need them for the Schr¨odinger operator L = ∂2 + u.

−1 −1 −2 −3 −4 Proof. If L exists, it must be of the form L = ∂ + X3∂ + X4∂ + .... The identity LL −1 = 1 gives

−1 −2 −3 −4 1 + X3∂ + (X4 + u) ∂ + (X5 + uX3) ∂ + (X6 + uX4) ∂ + ... = 1, from which we obtain X3 = 0, X4 = −u, and a recurrence relation Xk+2 = −uXk. This determines L −1 to be

L −1 = ∂−2 − u∂−4 + u2∂−6 − ....

1 1 1 −1 −2 Similarly, if L 2 exists, it must be of the form L 2 = ∂ + X0 + X1∂ + X2∂ + .... 1 1 The identity L 2 L 2 = L gives us another recurrence relation. We ﬁnd that 1 1 −1 1 −2 1 1 2 −3 3 1 −4 L 2 = ∂ + u∂ − ux∂ + uxx − u ∂ + uux − uxxx ∂ + .... 2 4 8 8 8 16

− 1 1 By the same procedure we can also uniquely determine the inverse L 2 of L 2 . It − 1 − 1 − 1 − 1 1 1 −1 satisﬁes L 2 L 2 L = L 2 L 2 L 2 L 2 = 1, so it is also the square root of L .

26 4.1. Lax Pairs

We need this operator to prove commutativity, which is now elementary:

h −1 1 i − 1 − 1 1 1 − 1 − 1 − 1 − 1 L , L 2 = L 2 L 2 L 2 − L 2 L 2 L 2 = L 2 − L 2 = 0, −1 L , L = 1 − 1 = 0, h 1 i 1 3 1 3 L , L 2 = L 2 − L 2 = 0.

k Proposition 4.2 allows us to define the operator L 2 for every integer k. These 1 3 5 operators commute with L as well. The differential parts of L 2 , L 2 , L 2 , . . . are the operators we are looking for. Define

2k−2 2k−1 Pk := 2 L 2 . (4.3) +

The normalization factor 22k−2 is chosen such that the highest order terms of the KdV equations will have coeﬃcient 1. Proposition 4.3. The commutator [Pk, L ] is a pure function.

2k−1 Proof. Since L 2 commutes with L we have

h 2k−1 i h 2k−1 2k−1 i 2k−2 2 2k−2 2 2 [Pk, L ] = 2 L +, L = 2 L − L −, L h 2k−1 i 2k−2 2 = −2 L −, L .

2k−1 2 The operator L − is of order −1 and L is of order 2, so the order of the commu- 2k−1 2 tator L −, L is at most ((−1) + 2) − 1 = 0. On the other hand, since both Pk and L are diﬀerential operators, [Pk, L ] has no terms of negative order. We conclude that [Pk, L ] only contains a zeroth-order term, hence it is a pure function.

Definition 4.4. The k-th KdV equation (k ≥ 1) is the evolution equation for u given by Lt = [Pk, L ], where L is the Schrödingeroperator (4.1) and Pk is given by Equation (4.3). The second KdV equation Lt = [P2, L ] is known as the KdV equation. The first few KdV equations are

(k = 1) ut = ux,

(k = 2) ut = uxxx + 6uux, 2 (k = 3) ut = ux5 + 20uxuxx + 10uuxxx + 30u ux.

Theorem 4.5. The right hand sides of the KdV equations are derivatives,

2k−1 2k−1 [Pk, L ] = 2 Dx res L 2 .

27 4. The KdV Hierarchy

Proof. We have

h 2k−1 i h 2k−1 i h 2 2k−1 i L 2 , L = − L 2 , L = ∂ + u, L 2 . + − + − +

2k−1 2 Since u, L − has no diﬀerential part, it follows that

h 2k−1 i h 2 2k−1 i 2k−1 L 2 , L = ∂ , L 2 = 2 Dx res L 2 . + − +

Motivated by Theorem 4.5, we deﬁne

2k−1 2k−1 rk := 2 res L 2 . (4.4)

Note that this also makes sense for nonpositive k, in particular

1 −1 1 −1 1 r0 = res L 2 = res ∂ + ... = . 2 2 2

The derivatives Dx rk (k ≥ 1) are the right hand sides of the KdV equations. The equations of the KdV hierarchy are not Lagrangian because the variational δ derivative δu cannot produce terms of the form ut (see Appendix A.2). However, if we introduce the potential v that satisfies vx = u and the functions gk[v] := rk[vx], the corresponding equations vxt = Dx gk (4.5) are in fact Lagrangian. Definition 4.6. The set of Equations (4.5) for k ≥ 1 is called the differentiated potential Korteweg-de Vries (DPKdV) hierarchy. The evolutionary version of this hierarchy,

vt = gk (k ≥ 1), (4.6) is called the potential Korteweg-de Vries (PKdV) hierarchy

4.2. Variational relations

In this section we show that the right hand sides of the KdV hierarchy are related to each other through variational derivatives. Lemma 4.7. Let the vertical exterior derivative δ act coeﬃcient-wise on PDOs. For any PDOs A and B the Leibniz rule holds, δ(AB) = (δA )B + A (δB).

Proof. It is suﬃcient to show this for monomial PDOs. Since δ is an exterior derivative,

28 4.2. Variational relations the Leibniz rule holds when it is applied to diﬀerential polynomials, so ! X i δ(A∂i B∂j) = δ A Dk B ∂j+i−k k x k X i X i = (δA) Dk B ∂j+i−k + A δ(Dk B) ∂j+i−k. k x k x k k

k k By Proposition A.2 in Appendix A.1 there holds that δ(Dx B) = Dx(δB), so we ﬁnd

X i X i δ(A∂i B∂j) = (δA) Dk B ∂j+i−k + A Dk(δB) ∂j+i−k k x k x k k = (δA ∂i)B∂j + A∂i(δB ∂j).

Lemma 4.8. For any PDOs A and B we have Z Z res(AB) = res(BA ),

R where the integrals are formal, i.e. denotes the equivalence class modulo Dx.

Proof. It is suﬃcient to prove this for monomials A = A∂i and B = B∂j. Without loss of generality we can assume that i ≥ 0, j < 0, and i + j ≥ −1. We will show that the residue res[A , B] is a derivative. We have

i ∞ X i X j [A , B] = A (Dk B)∂j+i−k − B (Dk A)∂i+j−k, k x k x k=0 k=0 hence i j res[A , B] = A Dj+i+1 B − B Dj+i+1 A. j + i + 1 x j + i + 1 x From the deﬁnition of the binomial coeﬃcient we see that i i(i − 1) ... (−j) j(j − 1) ... (−i) j = = (−1)i+j+1 = (−1)i+j+1 , j + i + 1 (j + i + 1)! (j + i + 1)! j + i + 1 so i j+i+1 i+j j+i+1 res[A , B] = A D B + (−1) B D A j + i + 1 x x i+j ! i X k k i+j−k = Dx (−1) D A D B . i + j + 1 x x k=0

29 4. The KdV Hierarchy

Proposition 4.9. The quantities rk deﬁned by Equation (4.4) satisfy Z Z δrk = (4k − 2)rk−1δu, hence they are variational derivatives,

δr k = (4k − 2)r . δu k−1

Proof. By virtue of Lemma 4.7 we have Z Z 2k−1 2k−1 δrk = 2 res δL 2 Z 2k−1 1 2k−2 1 1 2k−3 2k−2 1 = 2 res δL 2 L 2 + L 2 δL 2 L 2 + ... + L 2 δL 2 .

Using Lemma 4.8 we can write this as Z Z 2k−2 1 2k−2 1 1 2k−3 δrk = 2 (2k − 1) δ(L 2 )L 2 + L 2 δ(L 2 )L 2 .

Applying Lemma 4.7 once more we ﬁnd Z Z 2k−2 2k−3 δrk = 2 (2k − 1) res δLL 2 Z 2k−2 2k−3 = 2 (2k − 1) res δu L 2 Z = 2(2k − 1)rk−1δu.

Remark. Two other proofs of this relation can be found in [7] and a third one in [18] (see also Appendix A.3). However, all these proofs start from the resolvent equation (4.7) rather than directly from the Lax representation. 1 Corollary 4.10. Set hk := 4k+2 gk+1, then gk and hk satisfy

δgk δhk = (4k − 2) gk−1 and = gk. δvx δvx

To show that the DPKdV equations are Lagrangian we need one more little lemma. Lemma 4.11. For any diﬀerential polynomial f in v, vx, vxx, . . . we have δf ∂f δf Dx = − . δvx ∂v δv

30 4.3. Resolvents

Proof. By direct calculation: δf ∂f ∂f 2 ∂f Dx = Dx − Dx + Dx + ... δvx ∂vx ∂vxx ∂vxxx ∂f 2 ∂f 3 ∂f ∂f δf = Dx − Dx + Dx + ... = − . ∂vx ∂vxx ∂vxxx ∂v δv If f does not depend on v itself, this result becomes even simpler. Corollary 4.12. For any diﬀerential polynomial f in vx, vxx, vxxx, . . . there holds δf δf Dx = − . δvx δv 1 1 Theorem 4.13. The functions Lk := 2 vxvt −hk, with hk = 4k+2 gk+1, are Lagrangians for the DPKdV Equations (4.5).

1 Proof. Since hk = 4k+2 gk+1 does not depend on v directly, it follows from Corollary 4.12 that δLk δhk δhk = −vtx − = −vtx + Dx . δv δv δvx Then from Corollary 4.10 it follows that

δLk = −vtx + Dx g . δv k 4.3. Resolvents

We now sketch an alternative way to introduce the KdV hierarchy, and derive an equation that will prove useful later on. Consider the the formal power series

X r X r + r −r−2 2 − r −r−2 2 T := 2 z L − and T := 2 (−z) L −. r∈Z r∈Z

2−1 They are called basic resolvents because their average equals the resolvent 4L − z . Indeed, + − −1 T + T X 2k −2k−2 k X 2k −2k−2 k = 2 z L = 2 z L , 2 − k∈Z k=−∞ so

+ − −1 −1 T + T 2 X 2k+2 −2k−2 k+1 X 2k −2k k 4L − z = 2 z L − 2 z L = 1. 2 k=−∞ k=−∞

2 ± ± 2 Proposition 4.14. The PDOs 4L − z T and T 4L − z are diﬀerential operators.

31 4. The KdV Hierarchy

Proof. There holds ! X r 2 ± 2 r −r−2 2 4L − z T − = 4L − z 2 (±z) L − r∈Z − ! 2 X r −r−2 r = 4L − z 2 (±z) L 2 r∈Z − ! X r+2 −r−2 r+2 X r −r r = 2 (±z) L 2 − 2 (±z) L 2 = 0. r∈Z r∈Z −

± 2 Analogously, T 4L − z − = 0.

Corollary 4.15. The basic resolvents T ± lie in the Kernel of the Adler mapping, deﬁned as

2 2 2 2 H : T 7→ 4L − z T + 4L − z − 4L − z T 4L − z + Proof. We have

± 2 ± 2 2 ± 2 H(T ) = − 4L − z T − 4L − z + 4L − z T 4L − z − = 0.

This corollary leads us to a useful equation that the basic resolvents T ± satisfy. −1 −2 Theorem 4.16. If T = T1∂ + T2∂ + ... satisﬁes the condition H(T ) = 0, then

1 1 u T + 2 u − z2 D T + D3T = 0. (4.7) x 1 4 x 1 2 x 1

Proof. In this proof we denote total x-derivatives by a prime, 0. The equation H(T ) = 0 is equivalent to

2 1 2 −1 −2 2 1 2 0 = ∂ + u − z T1∂ + T2∂ + ... ∂ + u − z 4 + 4 2 1 2 −1 −2 2 1 2 − ∂ + u − z T1∂ + T2∂ + ... ∂ + u − z 4 4 + 0 2 1 2 2 1 2 = T1∂ + 2T1 + T2 ∂ + u − 4 z − ∂ + u − 4 z (T1∂ + T2) 3 1 2 0 0 2 1 2 0 2 1 2 = T1∂ + u − 4 z T1∂ + u T1 + 2T1∂ + 2 u − 4 z T1 + T2∂ + u − 4 z T2 3 0 2 00 2 0 00 1 2 1 2 − T1∂ + 2T1∂ + T1 ∂ + T2∂ + 2T2∂ + T2 + u − 4 z T1∂ + u − 4 z T2 00 0 0 1 2 0 00 = − T1 + 2T2 ∂ + u T1 + 2 u − 4 z T1 − T2 .

0 −1 00 0 1 2 0 1 000 It follows that T2 = 2 T1 and u T1 + 2 u − 4 z T1 + 2 T1 = 0.

32 4.3. Resolvents

We call Equation (4.7) the resolvent equation. It can be used as the starting point to deﬁne the KdV hierarchy. To see this, consider the formal power series

X h r i X h r i r−1 2 −r−2 r−1 2 −r−2 2 L +, L (±z) = 2 L +, L (±z) r∈N r∈Z X r−1 h r i −r−2 = 2 L , (L 2 )− (±z) r∈Z 1 ± = L , T 2

r 2 Since each of the commutators L +, L is a pure function, we know that the ± 2 ± only nonzero term of [L , T ] = ∂ + u, T is the zeroth-order term, which equals ± 2 Dx(res T ). Therefore

X h r i r−1 2 −r−2 ± 2 L +, L (±z) = Dx(res T ). r∈N

± On the other hand, T1 := res T satisﬁes the resolvent equation (4.7). This gives us the following recipe to cook up the KdV hierarchy. Take an “appropriate” solution of Equation (4.7) and expand its derivative as a power series in z. The coeﬃcients of the h 2k−1 i 2k−2 2 odd powers in this series will be the right hand sides [Pk, L ] = 2 L +, L of the KdV hierarchy. More details can be found in Appendix A.3 or in [6, Section 3.7].

5. Pluri-Lagrangian structure of the PKdV hierarchy

In the last chapter we have merely reviewed known facts about the KdV hierarchy. In this chapter we will use those facts to give a pluri-Lagrangian form of the PKdV hierarchy.

5.1. Time derivatives of the Lagrangians

In the pluri-Lagrangian formulation dL is constant on solutions. In order to find a pluri- Lagrangian structure for the PKdV hierarchy, we will impose the stronger condition that the Lagrangian two-form is closed, dL = 0. Some of the coefficients of L will be the Lagrangians we computed in the last chapter. The closedness condition involves time derivatives of these coefficients and the following lemma will help in handling them. Lemma 5.1. (a) The polynomials

δ1hi δ1hi δ1hi aij := vtj + vxtj + vxxtj + ... δvx δvxx δvxxx satisfy

Dx aij = Dj hi + vtj Dx gi, 1 where gi are the right hand sides of the PKdV equations and hi = 4i+2 gi+1.

(b) There are polynomials bij in v, vx, vxx,... such that

Dx bij = gj Dx gi.

Proof. (a) We have δ1hi δ1hi δ1hi Dx aij = Dx vtj + vxtj + vxxtj + ... δvx δvxx δvxxx δ1hi δ1hi δ1hi = vxtj + vxxtj + vxxxtj + ... δvx δvxx δvxxx δ1hi δ1hi δ1hi + vtj Dx + vxtj Dx + vxxtj Dx + .... δvx δvxx δvxxx

35 5. Pluri-Lagrangian structure of the PKdV hierarchy

By virtue of Corollary 4.12 it follows that

δ1hi δ1hi δ1hi Dx aij = vxtj + vxxtj + vxxxtj + ... δvx δvxx δvxxx δ1hi ∂hi δ1hi ∂hi δ1hi ∂hi − vtj + vtj − vxtj + vxtj − vxxtj + vxxtj − ... δv ∂v δvx ∂vx δvxx ∂vxx δ1hi ∂hi ∂hi ∂hi = −vtj + vtj + vxtj + vxxtj + ... δv ∂v ∂vx ∂vxx δ1hi = Dj hi − vt j δv Finally, by Corollary 4.10 it follows that

Dx aij = Dj hi + vtj Dx gi.

1 (b) We use induction with respect to j. For j = 0 we ﬁnd g0 Dx gi = 2 Dx gi, so 1 bi0 := 2 gi does the job.

Now assume we have found all bij for some ﬁxed j. Write the resolvent equa- + P∞ −2k−1 tion (4.7) for the power series T1 := res T = k=0 rkz in terms of its coeﬃcients: 3 Dx rk + 4u Dx rk + 2uxrk = Dx rk+1. It follows that 3 Dx gk + 4vx Dx gk + 2vxxgk = Dx gk+1. Now

gj+1 Dx gi = Dx(gj+1gi) − Dx gj+1 gi 3 = Dx(gj+1gi) − Dx gj + 4vx Dx gj + 2vxxgj gi 2 2 = Dx gj+1gi − Dx gj gi + Dx gj Dx gi − gj Dx gi − 4vxgjgi 3 + gj Dx gi + 4vx Dx gi + 2vxxgi 2 2 = gj Dx gi+1 + Dx gj+1gi − Dx gj gi + Dx gj Dx gi − gj Dx gi − 4vxgjgi ,

so 2 2 bi j+1 := bi+1 j + gj+1gi − Dx gj gi + Dx gj Dx gi − gj Dx gi − 4vxgjgi does the job.

Observe that aij and bij do not depend on v, only on its derivatives.

36 5.2. Construction of the Lagrangian two-form

Lemma 5.2. The identity bij + bji = gigj holds for all i, j.

Proof. We know that

Dx(bij + bji) = gj Dx gi + gi Dx gj = Dx(gigj).

Since neither bij + bji nor gigj contain any constant terms, the claim follows.

5.2. Construction of the Lagrangian two-form

Now we are in a position to construct a pluri-Lagrangian structure for the PKdV N hierarchy. We restrict to a finite number of equations, so that our multi-time R remains finite-dimensional. As justified by the first KdV equation, ut1 = ux, we identify t1 with x. We look for a Lagrangian two-form X L = Lij dti ∧ dtj, i

X ∂ Dgj := (DI gj) ∂vI I and the polynomial

(gj ) δ1hi δ1hi 2 δ1hi aij := gj + Dx gj + Dx gj + ... δvx δvxx δvxxx obtained by identiﬁcation of vtj with gj from Dj and aij respectively.

37 5. Pluri-Lagrangian structure of the PKdV hierarchy

Ansatz 5.3. We want Dgj to be a variational symmetry of L1i. In particular, we look for Lij such that

Dx Lij − Di L1j + Dgj L1i = 0.

We have 1 1 D L = v g + v D g − D h , i 1j 2 tix j 2 x i j i j 1 1 D L = D g v + v D g − D h , gj 1i 2 x j ti 2 x i j gj i so 1 1 D L − D L = v g − v D g − D h + D h i 1j gj 1i 2 tix j 2 ti x j i j gj i 1 1 (g ) = v g − v D g − D a + v D g + D a j − g D g 2 tix j 2 ti x j x ji ti x j x ij j x i 1 1 (g ) = v g + v D g − D a − a j − g D g 2 tix j 2 ti x j x ji ij j x i 1 (g ) = D (v g ) + D a j − a − D b . 2 x ti j x ij ji x ij We denote the antiderivative with respect to x of this quantity by

(i) 1 (g ) L := v g + a j − a − b . (5.2) ij 2 ti j ij ji ij

The analogous calculation in the (x, tj)-plane yields

1 (g ) D L − D L = − D (v g ) + D a − a i + D b . gi 1j j 1i 2 x tj i x ij ji x ji We denote its antiderivative by

(j) 1 (g ) L := − v g + a − a i + b . (5.3) ij 2 tj i ij ji ji

We look for a function Lij on the three-dimensional (x, ti, tj)-space that reduces to (i) (j) Lij and Lij after the substitution vtj = gj and vti = gi respectively. Ansatz 5.4. The function Lij is of the form

∞ ∞ X X α α Lij = cvti vtj + vx ti pα + vx tj qα + r, (5.4) α=0 α=0 where c ∈ R is a constant and pα, qα, r are diﬀerential polynomials in v, vx, vxx,....

38 5.2. Construction of the Lagrangian two-form

Theorem 5.5. For every choice of c ∈ R there is exactly one such Lij. It is given by

1 1 1 Lij = cvti vtj + (aij − aji) + 2 − c vti gj − 2 + c vtj gi − 2 (bij − bji) + cgigj.

Proof. After the substitution vtj = gj, Equation (5.4) becomes

∞ ∞ (i) X X α α Lij = cvti gj + vx ti pα + qα Dx gj + r, α=0 α=0

δ1hj 1 δ1hj Comparing with Equation (5.2), we ﬁnd that p0 = − + − c gj and pα = − δvx 2 δvxα+1 for α > 0, i.e. ∞ X 1 α vx ti pα = −aji + ( 2 − c)vti gj. α=0

By substituting vti = gi instead in Equation (5.4), and comparing to Equation (5.3), δ1hi 1 δ1hi we ﬁnd that q0 = − + c gi and qα = for α > 0, i.e. δvx 2 δvxα+1

∞ X 1 α vx tj qα = aij − 2 + c vtj gi. α=0 This leaves us with

1 1 Lij = cvti vtj − aji + aij + 2 − c vti gj − 2 + c vtj gi + r. (5.5) Now we again do the substitutions and compare with Equations (5.2) and (5.3). We ﬁnd that 1 1 r = −bij + 2 + c gigj = bji − 2 − c gigj (5.6) modulo terms that vanish after the substitution vti = gi or vtj = gj. However, there are no such (nonzero) terms that only depend on v, vx, vxx,..., so Equation (5.6) holds exactly. The two expressions for r in Equation (5.6) are equal by virtue of Lemma 5.2. In fact, using the same Lemma, we ﬁnd a more aesthetically pleasing way to express r:

1 r = 2 (bji − bij) + cgigj. Plugging this into Equation (5.5) yields the desired result.

Our theory does not depend in any essential way on the choice of Lij among this family. For simplicity we choose c = 0, i.e.

1 1 Lij = 2 (vti gj − vtj gi) + (aij − aji) − 2 (bij − bji). (5.7)

39 5. Pluri-Lagrangian structure of the PKdV hierarchy

Another interesting choice is to formally put c = ∞, or, in other words, to take only the c-linear part. This gives us

Le1j = 0 and Leij = (vti − gi)(vtj − gj). (5.8)

We will call the form with coefficients (5.1) and (5.7) the first form and the one with coefficients (5.8) the second form. The second form can be considered for any family of evolution equations vtj = gj. However, it does not have any connection to the classical

Lagrangian formulation of the individual diﬀerentiated equations vxtj = Dx gj. The reason we assumed Ansatz 5.3 is that it gives us the following closedness property. P Proposition 5.6. (a) The two-form L = i

(b) The same property holds when the coeﬃcients are given by (5.8).

Proof. Consider the first form, i.e. the one with coefficients given by (5.1) and (5.7). The coefficients of dL are given by Dk Lij − Dj Lik + Di Ljk for all i < j < k. Consider the case that i = 1. By construction (Ansatz 5.3), Dk L1j −Dj L1k +Dx Ljk vanishes as soon as either vtj = gj or vtk = gk is satisfied. Indeed, we have that

DkL1j − Dj L1k + Dx Ljk 1 1 1 1 = v v + v v − D h − v v − v v + D h 2 tj tk x 2 tj xtk k j 2 tj tk x 2 tk xtj j k 1 + v g + v D g − v g − v D g 2 xtj k tj x k xtk j tk x j 1 + D h + v D g − D h − v D g − (g D g − g D g ) k j tk x j j k tj x k 2 k x j j x k 1 = v v − v v + v g − v D g − v g + v D g − g D g + g D g 2 tj xtk tk xtj xtj k tj x k xtk j tk x j k x j j x k 1 1 = (v − g )D (v − g ) − (v − g )D (v − g ). (5.9) 2 tj j x tk k 2 tk k x tj j

If i, j, k > 1 we can assume without loss of generality that vti = gi and vtj = gj are satisfied. Regardless of whether the corresponding equation for vtk holds, we do not make any identification involving vtk , vxtk , . . . . Using Equation (5.9) we find

Dx (Dk Lij − Dj Lik + Di Ljk) = Dk (Di L1j − Dj L1i)

− Dj (Di L1k − Dk L1i)

+ Di (Dj L1k − Dk L1j) = 0.

Since these polynomials do not contain constant terms, it follows that

Dk Lij − Dj Lik + Di Ljk = 0.

40 5.3. The multi-time Euler-Lagrange equations

For the second form, where the coeﬃcients are given by Equation (5.8), the claim is trivial.

5.3. The multi-time Euler-Lagrange equations

Theorem 5.7. (a) The multi-time Euler-Lagrange equations for the ﬁrst Lagrangian P two-form L = i

vt2 = g2, vt3 = g3, . . . vtN = gN ,

and equations that follow from these.

(b) The same result holds for the second form given by (5.8). The proof of Theorem 5.7 consists of checking all multi-time Euler-Lagrange equations (2.5)–(2.7) individually for both forms, and will take up the rest of this section. If N > 3 we ﬁx k > j > i > 1. If N = 3 we take j = 3, i = 2 and in the following ignore all equations containing k. We use the convention Lji = −Lij, etc.

First form, coeﬃcients (5.1) and (5.7)

Equations (2.7)

• The equations δ L δ L δ L 1i 1i + ij ij + 1j j1 = 0 δvIxti δvItitj δvItj x and δ L δ L δ L ij ij + jk jk + ki ki = 0 δvItitj δvItj tk δvItkti are trivial because all terms vanish.

Equations (2.6)

• The equation δ L δ L 1i 1i = ij ji δvx δvtj

41 5. Pluri-Lagrangian structure of the PKdV hierarchy

yields

1 δ1ihi 1 δijaij vti − = gi − 2 δvx 2 δvtj 1 δij δ1hi δ1hi δ1hi = gi − vtj + vtj x + vtj xx + ... 2 δvtj δvx δvxx δvxxx 1 δ1hi = gi − . 2 δvx This simpliﬁes to the PKdV equation

vti = gi. (5.10)

• For α > 0, the equation δ L δ L 1i 1i = ij ji α δvxα+1 δvtj x yields δ1ihi δij δ1hi δ1hi δ1hi − = − vtj + vtj x + vtj xx + ... α δvxα+1 δvtj x δvx δvxx δvxxx δ1hi = − , δvxα+1 which is trivial.

• Similarly, the equation δ L δ L 1j 1j = ij ij δvx δvti yields PKdV equation

vtj = gj, (5.11) and for α > 0, the equation δ L δ L 1j 1j = ij ij α δvxα+1 δvtix is trivial.

• All equations of the form

δ1iL1i δijLji δ1jL1j δijLij = (ti 6∈ I) and = (tj 6∈ I) δvxI δvtj I δvxI δvtiI

where I contains any tl (l > 1) are trivial because each term is zero.

42 5.3. The multi-time Euler-Lagrange equations

• The equations δ1iL1i δ1jL1j = (x 6∈ I) δvIti δvItj 1 are trivial because both sides are zero for nonempty I and both are equal to 2 vx for empty I.

Equations (2.5) δ L • By construction, the equations 1i 1i = 0 are the DPKdV equations δv

vxti = Dx gi. (5.12)

δ1iL1i For I containing any tl, l > 1, l 6= i, the equations = 0 are trivial. δvI • We discuss the last family of equations as a lemma because its calculation is far from trivial. δ L Lemma 5.8. The equations ij ij = 0 are corollaries of the PKdV equations. δvxα At first sight there is nothing in our construction that forces this claim to be true. So before proceeding with the proof, we present a heuristic argument why the claim should hold. N Consider a small three-dimensional cube C in R , and a solution v to the PKdV equations. As we have already calculated, v satisfies all multi-time Euler-Lagrange δ L equations except possibly ij ij = 0 for all α. Hence for any variation V , the δvxα variation of the action (Equation (2.12)) on the boundary of the cube simplifies to Z Z X δijLij ιpr V δL = δvxα (V ) dti ∧ dtj. ∂C ∂C δvxα α∈N If we choose the variations along simultaneous solutions of all but one of the PKdV equations, L will be closed along these variations, i.e. ιpr V δL will be closed. There- fore Z Z Z X δijLij δvxα (V ) dti ∧ dtj = ιpr V δL = d(ιpr V δL) = 0, ∂C δvxα ∂C C α∈N

δijLij so the equations = 0 for k ≥ 0 are satisﬁed. δvxα The reason this argument is only a heuristic one, is that the set of variations that we consider might be too small.

43 5. Pluri-Lagrangian structure of the PKdV hierarchy

Proof of Lemma 5.8. From Equation (5.7) we see that the variational derivative of Lij contains only three nonzero terms, δijLij ∂Lij ∂Lij ∂Lij = − Di − D j . (5.13) α α α α δvx ∂vx ∂vx ti ∂vx tj To determine the ﬁrst term we use an indirect method. Assume that the dimension of multi-time N is at least 4 and ﬁx k > 1 distinct from i and j. Let v be a solution

of all PKdV equations except vtk = gk. By Proposition 5.6 we have

X ∂Lij vItk = Dk Lij = Dj Lik − Di Ljk. (5.14) ∂vI I

∂L Since ij does not contain any derivatives with respect to t , we can determine ∂vI k ∂Lij by looking at the terms in the right hand side of Equation (5.14) containing ∂vxα α vx tk . These are 1 δ1hi δ1hi Dj − 2 givtk + vtk + vxtk + ... δvx δvxx 1 δ1hj δ1hj − Di − 2 gjvtk + vtk + vxtk + ... . δvx δvxx Now we expand the brackets. By again throwing out all terms that do not contain

α any vx tk , and those that cancel modulo vti = gi or vtj = gj, we get δ1hi δ1hi δ1hi + vtk Dj + vxtk Dj + vxxtk Dj + ... δvx δvxx δvxxx δ1hj δ1hj δ1hj − vtk Di − vxtk Di − vxxtk Di − .... δvx δvxx δvxxx

Comparing this to Equation (5.14), we ﬁnd that ∂Lij δ1hj δ1hi = − Di + Dj . ∂vxα δvxα+1 δvxα+1

On the other hand we have ∂Lij ∂Lij δ1hj δ1hi − Di − Dj = Di − Dj , α α ∂vx ti ∂vx tj δvxα+1 δvxα+1

δ L so Equation (5.13) implies that ij ij = 0. δvxα

44 5.3. The multi-time Euler-Lagrange equations

δ23L23 Since = 0 does not depend on the dimension N ≥ 3, the result for N ≥ 4 δvxα implies the claim for N = 3.

Second form, coeﬃcients (5.8)

Equations (2.7)

• The equations δ1iLe1i δijLeij δ1jLej1 + + = 0 δvIxti δvItitj δvItj x and δijLeij δjkLejk δkiLeki + + = 0 δvItitj δvItj tk δvItkti are trivial because all terms vanish.

Equations (2.6)

• The equations δ1iLe1i δijLeji δ1jLe1j δijLeij = and = δvx δvtj δvx δvti yield the PKdV equations

0 = gi − vti and 0 = vtj − gj (5.15)

respectively.

• All equations of the form

δ1iLe1i δijLeji δ1jLe1j δijLeij = (ti 6∈ I) and = (tj 6∈ I) δvxI δvtj I δvxI δvtiI

where I is nonempty are trivial because each term is zero.

δ1iLe1i δ1jLe1j • The equations = , with x 6∈ I, are trivial because both sides are zero. δvIti δvItj

Equations (2.5)

δ1iLe1i • The equations = 0, with x 6∈ I and ti 6∈ I, are trivial for every such I. δvI

45 5. Pluri-Lagrangian structure of the PKdV hierarchy

• The equations δijLeij = 0 δvxα yield ∂gi ∂gj − (vtj − gj) + (vti − gi) = 0, ∂vxα ∂vxα which are corollaries of the PKdV equations (5.15).

δijLeij For I containing any tl with l > 1 and l 6= i, the equations = 0 are trivial. δvI This concludes the proof of Theorem 5.7. It is remarkable that multi-time Euler-Lagrange equations are capable of producing first order evolution equations. This is a striking difference from both the traditional variational theories and the discrete case of the pluri-Lagrangian theory. In the traditional Lagrangian formalism, the variational derivative does not produce terms of the form vt (see Appendix A.2), hence the Euler-Lagrange equations cannot be first order evolution equations. In pluri-Lagrangian lattice systems the evolution equations (quad equations) imply the multi-time Euler-Lagrange equations (corner equations), but they themselves are not variational [5].

46 6. Relation to Hamiltonian formalism

In Proposition 2.2 we saw that dL is constant on solutions. For the one-dimensional case (d = 1) with L depending on the first jet bundle only, Suris [15] has shown that this is equivalent to the commutativity of the corresponding Hamiltonian flows. Moreover, if the constant is zero, then the Hamiltonians are in involution. Now we will prove a similar result for the two-dimensional case. We will use a Poisson bracket on formal integrals, i.e. equivalence classes of functions modulo x-derivatives [6, Chapter 1–2]. It is defined by Z R R δ1F δ1G F, G := Dx . δu δu Using integration by parts we see that this bracket is anti-symmetric. Less obvious is the fact that it satisfies the Jacobi identity [13, Chapter 7]. In this section, the integral sign R will always denote an equivalence class, not an integration operator. As we did when studying the KdV hierarchy, we introduce a potential v that satisfies vx = u and we identify the space-coordinate x with the first coordinate t1 of multi-time. We can now re-write the Poisson bracket as Z Z R R δ1F δ1G δ1F δ1G F, G = Dx = − , (6.1) δvx δvx δv δvx for functions F and G that depend on the x-derivatives of v but not on v itself.

Assume that the coefficients L1j of the Lagrangian two-from L have the form 1 L = v v − h , 1j 2 x tj j where hj is a differential polynomial in vx, vxx,.... This is consistent with the first Lagrangian two-form of the PKdV hierarchy. The L1j are Lagrangians of the equations

vxtj = Dx gj or utj = Dx gj,

δ h δ h where g := 1 j , hence 1 j = − D g . It turns out that the formal integral R h is the j δvx δv x j j Hamilton functional for the equation utj = Dx gj with respect to the Poisson bracket

47 6. Relation to Hamiltonian formalism

(6.1). Formally: Z R R R δ1hj hj, u(y) = hj, vx δ(· − y) = − δ(· − y) = Dx gj(y), δv where δ denotes the Dirac delta.

Theorem 6.1. If dL = 0 on solutions of the evolution equations vtj = gj, then the Hamiltonians are in involution, R R hi, hj = 0. P Proof. Write dL = i

Using Lemma 5.1(a) (which, as opposed to Lemma 5.1(b), is independent of the form of hi and gi), the evolution equations vtj = gj, and integration by parts, we ﬁnd that

Z Z 1 M dx = (v v − v v ) − D a + v D g + D a − v D g 1jk 2 xtk tj xtj tk x jk tk x j x kj tj x k Z 1 = − (g D g − g D g ) − D a + D a 2 j x k k x j x jk x kj Z = gk Dx gj Z δ h δ h = − 1 j 1 k δv δvx R R = hj, hk .

Hence if dL = 0 on solutions of the evolution equations vtj = gj, then the Hamilton functionals are in involution.

48 7. Conclusion

The truth is rarely pure and never simple Oscar Wilde In this thesis I have formulated the pluri-Lagrangian theory of integrable hierarchies, and together with Yuri Suris I propose it as a definition of integrability [17]. The motivation for this definition comes from the discrete case [5, 10, 15] and the fact that there is a relation with the Hamiltonian side of the theory. For the Hamiltonians to be in involution, we need the additional property that the Lagrangian two-form is closed on solutions. However, I am not aware of any examples were this is not the case. Indeed, in all examples considered in this text, the two-form is closed as soon as all but one of the equations are satisfied. Since the (P)KdV hierarchy is one of the most important examples of an integrable hierarchy, the the fact that it possesses a pluri-Lagrangian structure is an additional indication that the existence of a pluri-Lagrangian structure is a reasonable definition of integrability. On the other hand, the second Lagrangian two-form we found for the KdV hierarchy is too general to be a characterization of integrability. It can be considered for any family of first order evolution equations. Furthermore, the link to Hamiltonian theory does not apply to this form. Therefore it would make sense to exclude such a form from the definition of a pluri-Lagrangian system. This could be done by introducing a weak extra criterion, e.g. that none of the coefficients of the Lagrangian two-form are identically zero, or that the two-form does not vanish on solutions. In contrast to the discrete case, continuous pluri-Lagrangian systems are capable of producing evolutionary equations. This discrepancy invites additional research. Another question that is left for future to decide, is whether the pluri-Lagrangian formulation provides any new insight in the theory integrable systems. However, it seems unlikely to me that such a nice theory would remain without applications.

Appendix

A.1. A very short introduction to the variational bicomplex

Here we introduce the variational bicomplex and state the basic results that are used in the text. We follow Dickey, who provides a more complete discussion in [6, Chapter 19]. Another good source on a (subtly different) variational bicomplex is Anderson’s N unfinished manuscript [1]. For ease of notation we restrict to real fields u : R → R, rather than vector-valued fields. The space of (p, q)-forms A(p,q) consists of all formal sums

p,q X ω = f δuI1 ∧ ... ∧ δuIp ∧ dtj1 ∧ ... ∧ dtjq , where f is a polynomial in t1,..., tN , v, and partial derivatives of v with respect to any coordinates. We call (0, q)-forms horizontal and (p, 0)-forms vertical. The vertical ∂ one-forms δuI are dual to the vector ﬁelds . The action of the derivative Di on the ∂uI (p, q)-form ωp,q is

p,q X Di ω := (Di f) δuI1 ∧ ... ∧ δuIp ∧ dtj1 ∧ ... ∧ dtjq

+ f δuI1i ∧ ... ∧ δuIp ∧ dtj1 ∧ ... ∧ dtjq + ...

+ f δuI1 ∧ ... ∧ δuIpi ∧ dtj1 ∧ ... ∧ dtjq .

The integral of ωp,q over an q-dimensional manifold is the (p, 0)-form deﬁned by Z Z p,q X ω := f dtj1 ∧ ... ∧ dtjq δuI1 ∧ ... ∧ δuIp .

The horizontal exterior derivative d : A(p,q) → A(p,q+1) and the vertical exterior derivative δ : A(p,q) → A(p+1,q) are deﬁned by the anti-derivation property

p1,q1 p2,q2 p1,q1 p2,q2 p1+q1 p1,q1 p2,q2 (a) d (ω1 ∧ ω2 ) = dω1 ∧ ω2 + (−1) ω1 ∧ dω2 ,

p1,q1 p2,q2 p1,q1 p2,q2 p1+q1 p1,q1 p2,q2 δ (ω1 ∧ ω2 ) = δω1 ∧ ω2 + (−1) ω1 ∧ δω2 , and by the way they act on (0, 0)-, (1, 0)-, and (0, 1)-forms:

! X X ∂f X ∂f (b) df = Dj f dtj = + uIj dtj and ∂tj ∂uI j j I X ∂f δf = δuI for polynomials f in tj, v, and partial derivatives of v, ∂uI I

53 X (c) dδuI = − δuIj ∧ dtj and δ(δuI ) = 0, j

(d) d(dtj) = 0 and δ(dtj) = 0.

Properties (a)–(d) determine the action of d and δ on any form. In particular, we have

X X δduI = δ uIj dtj = δuIj ∧ dtj = −dδuI . j j

The corresponding mapping diagram is known as the variational bicomplex...... ↑ δ ↑ δ ↑ δ ↑ δ d d d d A(1,0) −→ A(1,1) −→ ... −→ A(1,n−1) −→ A(1,n) ↑ δ ↑ δ ↑ δ ↑ δ d d d d A(0,0) −→ A(0,1) −→ ... −→ A(0,n−1) −→ A(0,n)

The following claims follow by elementary calculations from properties (a)–(d). Proposition A.1. We have d2 = δ2 = 0 and dδ + δd = 0. k k+1 k Sk (i,k−i) Remark. This implies that d+δ : A → A , where A := i=0 A , is an exterior derivative as well. Proposition A.2. We have Di δ = δ Di P ∂ Proposition A.3. For a diﬀerential polynomial h, deﬁne ∂ := (DI h) . We h I ∂uI have d ι∂h + ι∂h d = 0.

Proof. It suﬃces to show this for every polynomial f in tj, v, and partial derivatives of v, for dtj, and for δuI . For the ﬁrst two, both terms of the claimed identity are zero,

X d(ι∂h f) = 0, ι∂h (df) = ι∂h Dj f dtj = 0, d(ι∂h dtj) = 0, ι∂h (ddtj) = 0. j

For vertical one-forms we ﬁnd

X X ι∂h (dδuI ) = ι∂h − δuIj ∧ dtj = − DIj h dtj = −d(DI h) = −d(ι∂h δuI ). j j

54 A.2. Traditional Euler-Lagrange equations are not ﬁrst order evolution equations

A.2. Traditional Euler-Lagrange equations are not ﬁrst order evolution equations

The traditional Lagrangian formalism does not produce ﬁrst order evolution equations ut = .... The following proposition explains why. N Proposition A.4. Let f be a diﬀerential polynomial on R (t1, . . . , tN ). The varia- δ1...N f tional derivative δu does not contain terms of the form uti .

Proof. Consider a monomial m := uI1 . . . uIn . We call n its degree and |I1| + ... + |In| its diﬀerential order. All terms of the variational derivative δ1...N m X |J| ∂m = (−1) DJ δu ∂uJ J

δf have degree n − 1 and diﬀerential order |I1| + ... + |In|. Hence, the only way δu could possibly contain a term uti is if f contains a second degree term of ﬁrst order. Such a term must be of the form uuti , but that is a full ti-derivative, so its variational derivative vanishes,

δ1...N uuti = ut − Di u = 0. δu i As we have seen in Chapters 3 and 5, the pluri-Lagrangian formalism does in fact produce ﬁrst order evolution equations. It is easy to see where the above proof fails in these cases. If i 6∈ I, then the variational derivative

δI uuti δu does not vanish. Instead it yields exactly uti .

55 A.3. Construction of the KdV hierarchy from the resolvent equation

In this section we mainly follow Dickey [6, Section 3.7]. One way to introduce the Korteweg-de Vries Hierarchy is to consider a formal power series

∞ X −2k−1 R = rkz k=0 satisfying the resolvent equation

3 2 Dx R + 4u Dx R + 2uxR − z Dx R = 0. (A.1)

Multiplying this equation by R and integrating with respect to x we ﬁnd

1 1 R D2 R − (D R)2 + 2 u − z2 R2 = C(z), (A.2) x 2 x 4

P∞ −2k −2 where C(z) = k=0 ckz is a formal power series in z , with constant real coeﬃ- cients ck. Proposition A.5. For every C with c0 ≥ 0 there exists a real solution to Equation (A.2).

Proof. Equation (A.2) provides a recurrence relation for rk. Indeed, we have

∞ l ! 2 X X 2 −2l−2 R Dx R = rl−m Dx rm z , l=0 m=0 ∞ l ! 1 X X 1 (D R)2 = D r D r z−2l−2, 2 x 2 x l−m x m l=0 m=0 ∞ l ! 2 X X −2l−2 2uR = 2rl−mrmu z , l=0 m=0 ∞ l ! ∞ l+1 ! 1 X X 1 X X 1 z2R2 = r r z−2l = r r z−2l−2. 2 2 l−m m 2 l+1−m m l=0 m=0 l=−1 m=0

Therefore Equation (A.2) is equivalent to the system of equations consisting of 1 r2 = c (A.3) 2 0 0

56 A.3. Construction of the KdV hierarchy from the resolvent equation and

l X 1 1 1 r D2 r − D r D r − r r + 2r r u − r r = c l−m x m 2 x l−m x m 2 l+1−m m l−m m 2 0 l+1 l+1 m=0 for every l ≥ 0. Writing the term for m = 0 separately we get

l X 1 1 r D2 r − D r D r − r r + 2r r u l−m x m 2 x l−m x m 2 l+1−m m l−m m m=1 1 1 1 + r D2 r − D r D r − r r + 2r r u − r r = c . l x 0 2 x l x 0 2 l+1 0 l 0 2 0 l+1 l+1

Solving this equation for rl+1 we get an explicit recurrence relation.

l 1 X 2 1 1 r = r D rm − Dx r Dx rm − r rm + 2r rmu l+1 r l−m x 2 l−m 2 l+1−m l−m 0 m=1 2 rl Dx r0 Dx rl Dx r0 cl+1 + − + 2rlu − . r0 2r0 r0

Hence once r0 is ﬁxed, all rl are uniquely determined. From Equation (A.3) we see that for every C with positive c0 there exist two solutions to Equation (A.2). If c0 = 0 there exists exactly one solution.

1 1 We consider Equation (A.2) for C = 8 , i.e. c0 = 8 and ck = 0 for k 6= 0. Let us −1 −3 −5 calculate the ﬁrst few coeﬃcients of the power series R = r0z + r1z + r2z + ....

√ 1 1 • The first coefficient is r0 = ± 2c0 = ± 2 . We choose the positive sign: r0 = 2 . Remark. The choice of sign, and previously the choice of C, are motivated by the fact that with this normalization the highest order term in each KdV equation will have coefficient 1.

−2 • The coeﬃcient of the z -term is r1 = 2r0u = u.

• The coeﬃcient of the z−4-term is 1 r = 2 r D2 r − r r + 2r r u + 2r u 2 0 x 1 2 1 1 0 1 1 1 1 = 2 u − u2 + u2 + 2u2 2 xx 2 2 = uxx + 3u .

57 • The coeﬃcient of the z−6-term is 1 1 1 r = 2 r D2 r − (D r )2 − r r + 2r2u + r D2 r − r r + 2r r u + 2r u 3 1 x 1 2 x 1 2 2 1 1 0 x 2 2 1 2 0 2 2 1 1 1 1 = 2 uu − u2 − ur + 2u3 + D2 r − ur + ur + 2ur xx 2 x 2 2 2 x 2 2 2 2 2 2 3 2 = 2uuxx − ux + 4u + Dx r2 + 2ur2 2 3 2 3 = 2uuxx − ux + 4u + uxxxx + 6uuxx + 6ux + 2uuxx + 6u 2 3 = uxxxx + 10uuxx + 5ux + 10u .

The Korteweg-de Vries hierarchy can be deﬁned as follows. Deﬁnition A.6. • The KdV hierarchy is the family of equations

ut = Dx rk[u](k ≥ 1).

• Write gk[v] := rk[vx]. The diﬀerentiated potential KdV (DPKdV) hierarchy is the family of equations vxt = Dx gk[v](k ≥ 1).

• The potential KdV (PKdV) hierarchy is the family of equations

vt = gk[v](k ≥ 1).

δr Proposition A.7. The r satisfy k = (4k − 2) r . k δu k−1

0 Proof. In this proof we use a prime, , as well as Dx, to denote the total x-derivative. We apply the vertical exterior derivative δ to Equation (A.2):

1 R00δR + RδR00 − R0δR0 + 4 u − z2 RδR + 2R2δu = 0. 4

2 ∂R Rζ Write ζ := z and Rζ := ∂ζ . Multiply the last equation by 2R2 to obtain

00 00 0 0 Rζ R δR Rζ δR Rζ R δR 1 Rζ δR 0 = + − + 2 u − z2 + R δu 2R2 2R 2R2 4 2R ζ 00 0 0 0 Rζ R 1 2 Rζ Rζ δR Rζ δR = + 2 u − z δR + Dx − + R δu. (A.4) 2R2 4 R 2R 2R ζ

58 A.3. Construction of the KdV hierarchy from the resolvent equation

Deriving Equation (A.2) with respect to ζ we get

1 1 R R00 + RR00 − R R0 + 4 u − z2 RR − R2 = 0, ζ ζ x ζ 4 ζ 2 and thus 00 00 0 0 1 2 Rζ R Rζ R Rζ 2 u − z Rζ 1 + − + 4 = . (A.5) 2R2 2R 2R2 R 4 Combining Equations (A.4) and (A.5) we arrive at

0 0 0 00 0 0 1 Rζ δR Rζ δR Rζ δR R Rζ δR δR = −R δu − Dx + + − 4 ζ 2R 2R 2R 2R2 0 0 ! Rζ δR Rζ δR = −R δu − Dx + . ζ 2R 2R

After (formally) integrating this becomes Z Z δR = −4Rζ δu.

δR It follows that δu = −4Rζ , so ∞ ∞ ∞ X δrk −k− 1 X 1 −k− 3 X −k− 1 ζ 2 = 4 k + r ζ 2 = (4k − 2) r ζ 2 . δu 2 k k−1 k=0 k=0 k=1 Therefore δr k = (4k − 2) r . δu k−1 1 Corollary A.8. Set hk := 4k+2 gk+1, then the gk and hk satisfy

δgk δhk = (4k − 2) gk−1 and = gk. δvx δvx Theorem A.9. The functions 1 L := v v − h k 2 x t k are Lagrangians for the DPKdV equations.

Proof. Literally the same as the proof of Theorem 4.13.

59 A.4. Table of explicitly computed quantities for the KdV hierarchy

i = 1 i = 2 i = 3

2 2 3 ri u uxx + 3u ux4 + 10uuxx + 5ux + 10u

2 Dx ri ux uxxx + 6uux ux5 + 20uxuxx + 10uuxxx + 30u ux

2 2 3 gi vx vxxx + 3vx vx5 + 10vxvxxx + 5vxx + 10vx

2 Dx gi vxx vx4 + 6vxvxx vx6 + 20vxxvxxx + 10vxvx4 + 30vxvxx

vxxx 1 2 1 1 2 1 3 2 hi 6 + 2 vx 10 vx5 + 2 vxx + 14 vx7 + vxvx5 + 2vxxvx4 + 2 vxxx + 3 2 2 5 4 vxvxxx + vx 5vxvxxx + 5vxvxx + 2 vx

2 1 4 a23 3vt3 vx + vxvt3xx + vt3 vxxx + 10 vt3x 3 2 2 a32 10vt2 vx + 5vt2 vxx + 5vxvt2xx + 10vt2 vxvxxx + 1 4 5 6 2vt2xxvxxx + vxxvt2xxx + vxvt2x + vt2 vx + 14 vt2x 5 2 2 3 2 2 b23 201vx + 960vxvxx + 640vxvxxx + 692vxxvxxx + 569vxvxxx + 762vxvxxvx4 + 2 2 3 104vx4 + 189vxvx5 + 171vxxxvx5 + 81vxxvx6 + 27vxvx7 + 2 vx9 5 2 2 3 2 2 b32 207vx + 945vxvxx + 660vxvxxx + 699vxxvxxx + 575vxvxxx + 750vxvxxvx4 + 2 2 3 103vx4 + 192vxvx5 + 172vxxxvx5 + 81vxxvx6 + 27vxvx7 + 2 vx9

60 A.4. Table of explicitly computed quantities for the KdV hierarchy

Deﬁnitions and usage • Schr¨odingeroperator L := ∂2 + u

2k−1 2k−1 • rk := 2 res L 2

• KdV: uti = Dx ri[u]

• g[v] := r[vx]

• DPKdV: vxti = Dx gi[v]

• PKdV: vti = gi[v] 1 1 • Lagrangians for DPKdV: Li := L1i := vxvt − hi, where hi := gi+1 2 i 4i + 2

δ1hi δ1hi δ1hi • aij := vtj + vxtj + vxxtj + ... δvx δvxx δvxxx 1 • b := g , b := b + g g − D2 g g + D g D g − g D2 g − 4v g g i0 2 i i j+1 i+1 j j+1 i x j i x j x i j x i x j i X • Lagrangian two-form: L = Lij dti ∧ dtj, where, for i, j > 1, i

B. Zusammenfassung in deutscher Sprache

Ich glaube nicht, dass es irgendetwas auf der ganzen Welt gibt, was man in Berlin nicht lernen k¨onnte- außer der deutschen Sprache!

Mark Twain1 In der klassischen Mechanik sind die Hamiltonsche und Lagrangesche Formulierung äquivalent. Im Bereich der integrierbaren Systeme ist die Lage ganz anders. Manche der wichtigsten Definitionen von Integrierbarkeit benutzen einen Hamiltonschen Ge- sichtspunkt, aber eine allgemein akzeptierte Lagrangesche Theorie von integrierbaren Systemen gibt es im Moment nicht. Einen Vorschlag fur¨ solch ein Konzept wird in dieser Arbeit entwickelt. Gegeben N − 1 kommutierende Lagrangesche Flusse,¨ können wir unterschiedliche Zeit- koordinaten ti fur¨ jeden einzelnen Fluss einfuhren¨ und gleichzeitige Lösungen der Euler- N Lagrange-Gleichungen als Funktionen u : R → R :(x, t2, . . . , tN ) 7→ u(x, t2, . . . , tN ) betrachten. Das ubliche¨ Lagrangesche Kriterium ist in diesem Kontext, dass die Wir- R kung S L auf jeder (x, ti)-Ebene S stationär ist. In einem Pluri-Lagrangeschen System gilt ein viel stärkeres Kriterium: die Wirkung muss auf jeder zwei-dimensionalen Man- N nigfaltigkeit S in R stationär sein. Dieser Ansatz scheint vielleicht unmotiviert, aber ähnliche Ideen sind in anderen Ge- bieten der Mathematik gängig. Ein Beispiel dafur¨ ist die Theorie der pluriharmoni- schen Funktionen. Eine Funktion f heißt pluriharmonisch, wenn das Dirichletfunktio- R ∂(f◦Γ) 2 m nal Γ ∂z dz ∧ dz¯ fur¨ jede holomorphe Kurve Γ : C → C minimal ist. Andere Beispiele fur¨ verwandte Ideen sind Baxters Z-Invariante in der statistischen Mechanik und das klassische Konzept von Variationssymmetrien. Die Absicht dieser Arbeit ist eine Pluri-Lagrangesche Theorie fur¨ integrierbare Hie- rarchien von Differentialgleichungen zu entwickeln. Es gibt drei Hauptziele: erstens die Herleitung der Multi-Zeit-Euler-Lagrange-Gleichungen, zweitens die Konstruktion einer pluri-Lagrangesche Form fur¨ die Korteweg-de-Vries-Hierarchie und drittens eine kurze Besprechung der Beziehung zwischen pluri-Lagrangeschen Systemen und der bekannten Hamiltonschen Theorie der integrierbaren Systeme.

1Original: “I don’t believe there is anything in the whole earth that you can’t learn in Berlin except the German language.”

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