Topic 3: Lax Pairs and Other Integrable Equations Integrable Equations As Compatibility Conditions

Topic 3: Lax Pairs and Other Integrable Equations Integrable equations as compatibility conditions.

The Lax formalism for the KdV equation. Consider the diﬀerential operators

d2 d3 d 1 L = −6 − u, B = −4 − u − u . dx2 dx3 dx 2 x L is symmetric (formally self-adjoint with respect to the L2 inner product), and B is skew-symmetric (formally skew-adjoint). Then it was noted by Peter Lax in 1968 that the operator equation

dL (1) + [L, B] = 0 dt

is equivalent to the KdV equation ut + uux + uxxx = 0 in the sense that the expression on the left-hand side of the equation is simply the operator of multiplication by the function −(ut + uux + uxxx). Here [L, B] := LB − BL denotes the operator commutator. This is a direct calculation that you should check for practice. An operator equation of this form is today called a Lax equation. As we will soon see in a concrete example of an integrable system called the Toda lattice, the Lax form of the KdV equation immediately shows (among other things) that the eigenvalues of L are independent of t. Another way to derive Lax’s form of the KdV equation is as the compatibility condition of two linear problems where now we assume that λ is a ﬁxed parameter:

Lψ = λψ, and ψt = Bψ.

Suppose that ψ = ψ(x, t) is a nonzero simultaneous solution of both of these two equations for some λ ∈ C. Then we may calculate the time derivative of the function Lψ two diﬀerent ways. First, by essentially using the operator analogue of the product rule and then eliminating ψt we have ∂ dL dL (Lψ) = ψ + Lψ = ψ + LBψ. ∂t dt t dt

Second, if we start by using Lψ = λψ and the fact that λ ∈ C is a constant we have ∂ ∂ (Lψ) = (λψ) = λψ = λBψ = B(λψ) = BLψ. ∂t ∂t t Equating these and observing that we already checked that dL/dt + [L, B] is the operator of multiplication by a function, dividing by ψ (nonzero) again gives (1). The key importance of Lax’s observation is that any equation that can be cast into such a framework for other operators L and B has automatically many of the features of the KdV equation, including an inﬁnite number of local conservation laws.

Rewriting KdV as a zero-curvature condition. Going further with the point of view of looking at integrable nonlinear problems as the compatibility conditions of two linear problems (frequently involving an arbitrary “spectral” parameter λ), we can go from linear operators to matrices at the cost of introducing some powers of λ. To do this, note that as the eigenvalue equation Lψ = λψ is second order, we may easily write it in ﬁrst-order form by introducing

w1 = ψ, and w2 = ψx, So Lψ = λψ can be rewritten as ∂ w1 0 1 w1 w1 w1 = 1 = U or, with w := , simply wx = Uw. ∂x w2 − 6 (u + λ) 0 w2 w2 w2 1 Similarly, we have

w1t = ψt 1 = −4ψ − uψ − u ψ xxx x 2 x 1 = −4(−(u + λ)ψ/6) − uψ − u ψ x x 2 x 1 2 1 = u ψ + λ − u ψ 6 x 3 3 x 1 2 1 = u w + λ − u w . 6 x 1 3 3 2

Observe how the equation Lψ = λψ was used here to express ψxxx in terms of w1 = ψ and w2 = ψx. Then, since w2t = ψxt = ψtx, we can simply diﬀerentiate the last formula with respect to x and eliminate any x derivatives of w1 and w2 with the help of wx = Uw to obtain

w2t = w1tx 1 1 2 1 1 = u w + u w + λ − u w − u w 6 x 1x 6 xx 1 3 3 2x 3 x 2 1 1 2 1 1 1 1 = u w + u w − λ − u λ + u w − u w 6 x 2 6 xx 1 3 3 6 6 1 3 x 2 1 1 1 1 1 = − λ2 − λu + u2 + u w − u w . 9 18 18 6 xx 1 6 x 2 Or, writing as a first-order system, 1 2 1 6 ux 3 λ − 3 u wt = 1 2 1 1 2 1 1 w = Vw. − 9 λ − 18 λu + 18 u + 6 uxx − 6 ux The simultaneous equations wx = Uw and wt = Vw are equivalent to the scalar differential equations Lψ = λψ and ψt = Bψ. In this new but equivalent context, the condition for compatibility is that there exists a simultaneous fundamental solution matrix W(x, t) of the equations Wx = UW and Wt = VW. We may calculate the mixed partial derivative Wxt = Wtx in two different ways: differentiating first with respect to x and then with respect to t gives ∂ ∂U ∂U W = (UW) = W + UW = W + UVW, xt ∂t ∂t t ∂t while instead differentiating first with respect to t and then with respect to x gives ∂ ∂V ∂V W = (VW) = W + VW = W + VUW. tx ∂x ∂x x ∂x Equating mixed partial derivatives and using now the fact that W is a fundamental matrix — hence invertible — shows that the compatibility condition takes the form of an equation on 2 × 2 matrices, a so-called zero- curvature condition: ∂U ∂V (2) − + [U, V] = 0. ∂t ∂x Since U is linear in λ while V is quadratic, we expect the left-hand side to be a cubic polynomial in λ. However, if we separate out the coefficients of powers of λ in this polynomial a simple calculation shows that the coefficients of λ3, λ2, and λ all vanish identically. The constant term in the zero-curvature equation (2) is the matrix equation 0 0 1 = 0. − 6 (ut + uux + uxxx) 0 The name “zero-curvature condition” comes from the following differential-geometric interpretation. The equations (∂x − U)w = 0 and (∂t − V)w = 0 define a connection on a two-dimensional vector bundle over the (x, t)-plane. The first equation describes how to “parallel-translate” a vector w in the x-direction, and the second equation describes how to “parallel-translate” a vector w in the t-direction. The matrices U and V are the connection coefficients. A connection is said to have zero curvature if parallel translation of a 2 vector w along a path from a point (x0, t0) to another point (x1, t1) gives the same result independent of path connecting the points (or in more geometric language, there exists a nontrivial globally parallel section of the vector bundle). This is the same thing as asserting the existence of a full two-dimensional basis of simultaneous solutions of the equations (∂x −U)w = 0 and (∂t −V)w = 0, which is the above zero-curvature condition that must be satisfied by the connection coefficients. Therefore, every solution of the KdV equation defines a connection with zero curvature.

Generalization. The two linear equations for w making up a zero-curvature connection are said to form a Lax pair. (Some authors use this term to refer to the pair of operators L and B, and others use it to refer to the pair of matrices U and V.) This idea is one of the most central ones in the theory of integrable systems: each integrable nonlinear problem can be represented as the compatibility condition between two linear equations of a Lax pair. It is possible to find other examples of integrable nonlinear equations by the following method: first choose a “spectral problem” involving differentiation with respect to x and generalizing Lψ = λψ or equivalently wx = Uw, and then try to find a corresponding operator B or matrix function V of λ so that the compatibility condition is independent of λ. Here are some other examples of this method, starting from the “AKNS” spectral problem for which the matrix U = U(λ) takes the form (linear in λ) −iλ q U(λ) = = λU + U . r iλ 1 0 Here q and r are some functions of x and t. Many integrable equations were first obtained from this simple- looking problem by Ablowitz, Kaup, Newell, and Segur in an excellent 1974 paper. The special case when r is plus or minus the complex conjugate of q is also known as the Zakharov-Shabat spectral problem. We will derive nonlinear equations on q and r from the AKNS spectral problem by seeking matrices V giving rise to a zero-curvature connection.

Example 1: V quadratic in λ. The cubic nonlinear Schr¨odingerequation. Suppose we seek V = V(λ) as a quadratic polynomial in λ: 2 V(λ) = λ V2 + λV1 + V0,

where Vj, j = 0, 1, 2 are matrices independent of λ to be determined. Then, the zero-curvature condition splits into several equations corresponding to powers of λ: 3 coefficient of λ :[U1, V2] = 0, ∂V coefficient of λ2: − 2 + [U , V ] + [U , V ] = 0, ∂x 1 1 0 2 ∂U ∂V coefficient of λ: 1 − 1 + [U , V ] + [U , V ] = 0, ∂t ∂x 1 0 0 1 ∂U ∂V constant term: 0 − 0 + [U , V ] = 0. ∂t ∂x 0 0 Note that U1 = −iσ3, where the Pauli matrix σ3 is defined by 1 0 σ := . 3 0 −1

−1 3 Note that σ3 = σ3. We now consider these four equations one at a time. The λ equation can be solved by choosing V2 = U1 = −iσ3 (more generally an arbitrary diagonal matrix would do). With this choice the λ2 equation is equivalent to

[iσ3, V1 − U0] = 0. OD OD D It is easy to check that for a general 2 × 2 matrix A, we have [iσ3, A] = 2iσ3A , where A and A denote respectively the oﬀ-diagonal and diagonal parts of A: A A 0 A A 0 A = 11 12 =⇒ AOD = 12 and AD = 11 . A21 A22 A21 0 0 A22 2 Therefore since σ3 is invertible, the λ equation says that V1 must diﬀer from U0 by a diagonal matrix. But let’s solve this by also choosing that V1 = U0. 3 The λ equation is then equivalent to ∂U 0 + [iσ , V ] = 0. ∂x 3 0

This equation determines the off-diagonal part of V0: i ∂U 1 OD 0 0 2 iqx V0 = σ3 = 1 . 2 ∂x − 2 irx 0 Finally, we consider the constant term equation which we split into diagonal and off-diagonal parts: the diagonal part is ∂VD − 0 + [U , VOD] = 0 ∂x 0 0 and the off-diagonal part is ∂U ∂VOD 0 − 0 + [U , VD] = 0. ∂t ∂x 0 0 OD Inserting our formula for V0 into the diagonal part gives ∂VD 1 1 0 − 2 iqrx − 2 irqx 0 = 1 1 ∂x 0 2 irqx + 2 iqrx so recalling the product rule, we may solve this by choosing 1 D − 2 iqr 0 V0 = 1 . 0 2 iqr

The matrices V2, V1, and V0 have thus been determined and the only part of the zero-curvature condition that remains is the off-diagonal part of the constant term equation. The off-diagonal part of the constant term equation reads as follows: 1 2 0 qt − 2 iqxx + iq r 1 2 = 0. rt + 2 irxx − ir q 0 2 In other words, the connection defined by the coefficient matrices λU1 + U0 and λ V2 + λV1 + V0 will have zero curvature if and only if the functions q and r satisfy the nonlinear equations 1 1 iq + q − q2r = 0, and − ir + r − r2q = 0. t 2 xx t 2 xx Generally this is a coupled system. However, it is consistent with the relation r = ±q∗, which gives 1 1 iq + q ∓ |q|2q = 0, and − iq∗ + q∗ ∓ |q|2q∗ = 0, t 2 xx t 2 xx so the second equation is just the complex conjugate of the first. Therefore we have found the Lax pair representation for the focusing (− case) and defocusing (+ case) cubic nonlinear Schrödinger equations. These are important equations in many areas of physics; they govern wave packet propagation in quite general weakly nonlinear dispersive media and have been used to accurately model physical systems ranging from water waves to light pulses in optical fibers. The focusing and defocusing nonlinear Schrödingerequations were first shown to be integrable equations like KdV is by Zakharov and Shabat in the early 1970’s, and that is why we frequently name the associated spectral problem after them in the case that r = ±q∗.

−1 Example 2: V proportional to λ . The sine-Gordon equation. Returning to the AKNS problem wx = Uw for general q and r, let’s try now an ansatz for V = V(λ) of the form: −1 V(λ) = λ V−1, where V−1 is a matrix independent of λ to be determined. The zero-curvature equation then splits into three parts: ∂U coefficient of λ: 1 = 0, ∂t ∂U constant terms: 0 + [U , V ] = 0, ∂t 1 −1 4 and ∂V coefficient of λ−1: − −1 + [U , V ] = 0. ∂x 0 −1 First observe that since U1 = −iσ3 is constant the λ equation is trivially satisfied. The first nontrivial equation is therefore the one arising from the constant terms; this equation determines the off-diagonal part of V−1: ∂U i ∂U 1 OD 0 OD 0 0 − 2 iqt 2iσ3V−1 = , so V−1 = − σ3 = 1 . ∂t 2 ∂t 2 irt 0 Moving on to the λ−1 equation, taking the diagonal part gives ∂VD − −1 + [U , VOD] = 0 ∂x 0 −1 OD because U0 is purely off-diagonal. Since V−1 has already been determined, by matrix multiplication and the use of the product rule, we see that the diagonal part takes the form ∂VD 1 −1 2 i(qr)t 0 − + 1 = 0, ∂x 0 − 2 i(qr)t D which says in particular that V−1 = ασ3 + cI, where c has to be a constant but α is a function of x and t related to q and r by 1 (3) α − i(qr) = 0. x 2 t The off-diagonal part of the λ−1 equation reads ∂VOD − −1 + [U , VD ] = 0 ∂x 0 −1 an equation in which all terms have already been determined with the exception of the constant c which cancels out. Substituting and expanding it all out gives two additional scalar equations: 1 1 (4) iq − 2αq = 0 and ir − 2αr = 0. 2 xt 2 xt So the zero curvature condition amounts to three equations ((3) and (4)) on three unknowns (α, q, and r). It is easy to check that these equations are consistent with the reduction (relating all three fields to a single field u = u(x, t)): 1 1 α = i cos(u), and q = −r = u . 4 2 x Indeed, the relation q = −r is clearly consistent with the two equations (4), and with the above substitutions the equation (3) becomes

sin(u)ux − uxuxt = 0 while both equations (4) take the same form

uxxt − cos(u)ux = 0 or equivalently uxt − sin(u) = F (t),

where F (t) is an arbitrary integration “constant”. Therefore, either ux ≡ 0 (in which case u depends on t only, in an arbitrary way since − sin(u) = F (t)) or

uxt = sin(u) (and hence also F (t) ≡ 0). This is a form of the sine-Gordon equation. Indeed, if we introduce new coordinates by ξ = x + t and τ = x − t, then ∂ ∂ ∂ ∂ u = + − u = u − u , xt ∂ξ ∂τ ∂ξ ∂τ ξξ ττ so the equation becomes

uττ − uξξ + sin(u) = 0. 5 Conservation laws from Lax pairs. Recall that one way to motivate the appearance of the linear Schrödingerequation in the theory of the KdV equation is via the linearization of the Gardner transform reinterpreted as a Riccati equation. Therefore in that example at least, the existence of an appropriate generating function for infinitely many local conservation laws implies that the KdV equation can be represented as the compatibility condition for a Lax pair. It turns out that it is possible to reverse this reasoning and therefore deduce from a Lax pair representation of a nonlinear partial differential equation a generating function for infinitely many local conservation laws. We illustrate the procedure for the case of the nonlinear Schrödingerequation. The focusing and defocusing nonlinear Schrödinger equations are special cases of the zero-curvature condition that expresses the > compatibility of the two first-order linear systems for the auxiliary unknown w = (w1, w2) : ∂w −iλ q = Uw, U := , ∂x r iλ and ∂w 2 1 1 −iλ − 2 iqr λq + 2 iqx = Vw, V := 1 2 1 . ∂t λr − 2 irx iλ + 2 iqr Setting ∂ ∂ D := log(w ),F := − log(w ), ∂x 1 ∂t 1 we automatically get the relation

Dt + Fx = 0, which has the form of a local conservation law. As the density D and flux F depend parametrically on λ we can expect that by expanding appropriately with respect to λ we may obtain an infinite hierarchy of (hopefully nontrivial) local conservation laws. Note however that the density D is the x-derivative of something, so it may appear at first glance that the conservation laws could all be trivial. It turns out that this fear is unfounded because log(w1) depends on q and r in a non-local fashion through the differential equations of the Lax pair. We will see this in detail now. First, consider the proposed flux generator, F , which can be written in the form w1t 2 1 1 w2 F = − = iλ + iqr − iqx + λq (using the first equation in the system wt = Vw). w1 2 2 w1 Similarly,

w1x w2 (5) D = = −iλ + q (using the ﬁrst equation in the system wx = Uw), w1 w1

so the ratio w2/w1 may be eliminated between these two equations to yield 1 1 D + iλ F = iλ2 + iqr − iq + λq 2 2 x q λq 1 iq = x + iqr − λ + x D. 2q 2 2q Therefore, our proposed generator for local conservation laws reads λqx 1 iqx Dt + + iqr − λ + D = 0. 2q 2 2q x This may be thought of as the nonlinear Schrödinger analogue of the Gardner equation, a conservation law involving an arbitrary parameter (here, λ). Now we need the nonlinear Schrödingeranalogue of the Gardner transform. We recall that in the KdV context the Gardner transform was a Riccati equation for the density generator D = w. We may seek a 6 Riccati equation for D in the current setting as follows: w1x Dx = w1 x w2 = −iλ + q (using the first equation in the system wx = Uw) w1 x w2 w2x w2 w1x = qx + q − q · w1 w1 w1 w1 w2 w2 w2 w1x = qx + qr + iλq − q · (using the second equation in the system wx = Uw) w1 w1 w1 w1 w2 = qr + (qx + iλq − qD) . w1

Next, we use (5) to express the quotient w2/w1 in terms of D to find D + iλ D = qr + (q + iλq − qD) x x q (6) q q = qr + iλ x − λ2 + x D − D2. q q This is the desired Riccati (first-order, quadratically nonlinear) equation for D. To obtain the local conservation laws, we need to expand D with respect to λ using the Riccati equation (6). Let us try to expand the solution D of (6) in the limit λ → ∞. A dominant balance argument suggests D ≈ −iλ in this limit, so to begin set D = −iλ(1 + v) where we expect v to be small as λ → ∞. Thus (6) becomes an equation for v: q (7) −iλv = qr + −iλ x + 2λ2 v + λ2v2. x q Now the dominant balance is to choose v to be proportional to λ−2 to leading order. Thus, set W v = − , 2λ2 and (7) becomes i q 1 (8) qr = W + − x W + W − W 2. 2λ q x 4λ2 Finally, we might change the parameter λ by setting i := − . 2λ While this latter substitution is not essential, it allows us to make an analogy with the Gardner transform for KdV; indeed, making this final substitution in (8) finally puts the original Riccati equation (6) into the form q qr = W + x W − W + 2W 2. q x We can regard this equation as the “Gardner transform for the nonlinear Schrödingerequation”. Let us try this transform out, to see how it produces nontrivial local conservation laws. Substituting the expansion 2 W ∼ W0 + W1 + W2 + ··· , → 0, and collecting powers of , we obtain q W = qr, W = − x w − w , 0 1 q 0 0,x and for k ≥ 2 we have the recursion relation k−2 qx X W = − W − W − W W . k q k−1 k−1,x j k−2−j j=0 7 The conserved quantity associated with the nontrivial density W0 = qr is sometimes called the norm because when we recall the focusing (−) and defocusing (+) special cases defined by the consistent constraints r = ±q∗, we see that Z ∞ Z ∞ 2 I0[q] := W0 dx = ± |q(x, t)| dx −∞ −∞ which is proportional to the square of the L2 norm. The second conserved density is

W1 = (qr)x − qxr = qrx. ∗ ∗ When r = ±q , we see that W1 = ±qqx is complex-valued. Therefore, we may further separate its real and imaginary parts: 1 1 Re{W } = ± (qq∗ + q∗q ) = ± (|q|2) , 1 2 x x 2 x which obviously gives a trivial conservation law, and 1 Im{W } = ± (qq∗ − q∗q ). 1 2i x x In quantum mechanics this expression has the interpretation of momentum density, and so we refer to the corresponding conserved quantity Z ∞ 1 ∗ ∗ I1[q] := ± (q(x, t)qx(x, t) − q(x, t) qx(x, t)) dx 2i −∞ as the momentum. The third conserved density is 2 2 W2 = (qrx)x − qxrx − (qr) = qrxx − (qr) . Again this is complex when r = ±q∗, so we may separate its real and imaginary parts: 1 1 Re{W } = ± (qq∗ + q∗q ) − |q|4 = ± (qq∗ + q∗q ) ∓ |q |2 − |q|4, 2 2 xx xx 2 x x x x and 1 1 Im{W } = ± (qq∗ − q∗q ) = ± (qq∗ − q∗q ) , 2 2i xx xx 2i x x x which obviously gives a trivial conservation law. Therefore at this level we learn of a new conserved quantity Z ∞ 2 4 I2[q] := − ±|qx(x, t)| + |q(x, t)| dx. −∞ This conserved quantity is the Hamiltonian. In other words, it turns out that the NLS equation can be written as an inﬁnite-dimensional Hamiltonian system with respect to this Hamiltonian functional and an appropriate Poisson bracket. This procedure can be carried out to arbitrary order, and when r = ±q∗, a nontrivial real local conservation law appears at each order.