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On the Pluri-Lagrangian Structure of the Kdv Hierarchy Master's Thesis On the pluri-Lagrangian structure of the KdV hierarchy Mats Vermeeren Technische Universit¨atBerlin Fakult¨atII Institut f¨urMathematik 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 erkl¨are ich, dass ich die vorliegende Arbeit selbstst¨andig und ei- genh¨andig sowie ohne unerlaubte fremde Hilfe und ausschlielich unter Ver- wendung der aufgefuhrten¨ Quellen und Hilfsmittel angefertigt habe. Die selbst¨andige und eigenst¨andige 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 to^ujourspar les voies les plus simples". 1 1. Introduction n n M variation as a function δL : J X × J X ! R that satisfies 1 N M 8δu 2 C0 (R ; R ): L[u + εδu] = L[u] + εδL[u; δu] + O ("): 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 jIj @ uk (uk)I := i i ; (@t1) 1 ::: (@tN ) N α where jIj = 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 62 I (or tk 62 I) if ik = 0 and k 2 I (or tk 2 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 jIj @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 jIj @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 classi- cal 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.
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