An introduction to Lax pairs and the zero curvature representation
AAn introduction to Lax pairs and the zero curvature representation
Govind S. Krishnaswami and T. R. Vishnu
Chennai Mathematical Institute, SIPCOT IT Park, Siruseri 603103, IndiaEmail: [email protected] and [email protected]
11 April, 2020To appear in the journal Resonance published by the Indian Academy of Sciences
Abstract
Lax pairs are a useful tool in finding conserved quantities of some dynamical systems. In thisexpository article, we give a motivated introduction to the idea of a Lax pair of matrices (
L, A ) , firstfor mechanical systems such as the linear harmonic oscillator, Toda chain, Eulerian rigid body and theRajeev-Ranken model. This is then extended to Lax operators for one-dimensional field theories suchas the linear wave and KdV equations and reformulated as a zero curvature representation via a (
U, V )pair which is illustrated using the nonlinear Schr¨odinger equation. The key idea is that of realizing a(possibly) nonlinear evolution equation as a compatibility condition between a pair of linear equations.The latter could be an eigenvalue problem for the Lax operator L and a linear evolution equationgenerated by A , for the corresponding eigenfunction. Alternatively, they could be the first order linearsystem stating the covariant constancy of an arbitrary vector with respect to the 1+1 dimensional gaugepotential ( V, U ) . The compatibility conditions are then either the Lax equation ˙ L = [ L, A ] or theflatness condition U t − V x + [ U, V ] = 0 for the corresponding gauge potential. The conserved quantitiesthen follow from the isospectrality of the Lax and monodromy matrices.
Keywords:
Conserved quantities, Lax pair, Isospectral evolution, Zero curvature representation, Mon-odromy matrix, Toda chain, Euler top, Rajeev-Ranken model, KdV equation.
Contents L = − ∂ + u . . . . . . . . . . . . . . . 123.2 Korteweg-de Vries (KdV) equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 From Lax pair to zero curvature representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Conserved quantities from the zero curvature condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5 Nonlinear Schr¨odinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 A for the wave equation 18B Arriving at the rd order Lax operator A for KdV 18C Time evolution operator and the ordered exponential 19D Time evolution of the transition matrix T ( y, x ; t ) a r X i v : . [ n li n . S I] A p r Introduction
Lax pairs[1], introduced in 1968 by Peter Lax are a tool for finding conserved quantitiesof some evolutionary differential equations. A system with possibly nonlinear equations ofmotion will be said to admit a Lax pair if one can find a pair of matrices/operators ( L, A )such that the equations of motion are equivalent to the Lax equation ˙ L = [ L, A ] . If it exists,a Lax pair is not unique. For instance, we may add to A a matrix that commutes with L and add to L a time-independent matrix that commutes with A without altering the Laxequation. As we will explain, Lax pairs are based on the idea of expressing (typically) nonlinearevolution equations as ‘compatibility’ conditions for a pair of auxiliary linear equations to admitsimultaneous solutions. The idea only works for certain special systems (integrable systems),which however, play an important role in our understanding of more general dynamical systems.Unfortunately, there is no recipe for finding a Lax pair for a given system or to know inadvance whether one exists. So some knowledge of the nature of the system and its solutions(from numerical, analytical or experimental investigations) coupled with educated guessworkis involved. However, as we will see, once a Lax pair is known, it can be very helpful inunderstanding the system.In this expository article, we attempt to give an elementary introduction to the idea of a Laxpair and the associated zero curvature representation. We begin with mechanical systems withfinitely many degrees of freedom where the Lax matrices are finite dimensional. A key step isto write the equations in Lax form ˙ L = [ L, A ] , where L and A suitable square matrices whoseentries depend on the dynamical variables. This makes it easy to read off conserved quantities.Indeed, the Lax equation implies that the eigenvalues of L are independent of time (isospectralevolution). We do this for the harmonic oscillator, Toda chain, Eulerian rigid body and theRajeev-Ranken model. The latter two examples illustrate how allowing the Lax matrices todepend on an arbitrary spectral parameter gives the method additional power to find conservedquantities. As is well known, each independent conserved quantity imposes one relation amongthe phase space variables thereby confining trajectories to a hypersurface of codimension onein the phase space.We then generalize the Lax pair framework to continuum systems in one spatial dimensionsuch as the linear wave and KdV equations. L and A are now differential operators, allowingfor the possibility of infinitely many conserved quantities. We show how these are obtained forthe linear wave and KdV equations. While A may be chosen as first- and third-order operators,it turns out that both equations admit a common Lax operator L (the Schr¨odinger operator)as well as a common infinite tower of conserved quantities. What is more, these quantities areconserved for all the equations in the so-called KdV hierarchy which is obtained by choosingsuitable higher order differential operators for A .We then use the KdV example to pass to a more symmetrical formulation of the Lax pairidea where the Lax equation is viewed as a compatibility condition for a pair of linear equationsinvolving only first derivatives with respect to space and time. This compatibility conditionhas a geometric meaning: it says that a certain finite dimensional space is flat i.e., a curva-ture or finite dimensional nonabelian field strength matrix vanishes for all values of a spectralparameter. More generally, we will say that a (nonlinear) system of field equations admitsa zero curvature representation if the equations are equivalent to the condition for a certaincurvature to vanish. Remarkably, a number of interesting nonlinear field equations especiallyin one spatial dimension (such as the mKdV, nonlinear Schr¨odinger (see § Peter David Lax is an American mathematician of Hungarian origin (born 1 May, 1926). He has worked at the CourantInstitute of Mathematical Sciences (New York) on various topics including integrable systems, fluid mechanics and partialdifferential equations. He received the 2005 Abel prize “for his groundbreaking contributions to the theory and application ofpartial differential equations and to the computation of their solutions”.
In the linear harmonic oscillator the displacement x ( t ) of a particle of mass m from equilibriumevolves according to: m ¨ x = − kx, where k > . (1)We will use this example to provide an illustration of the idea of a Lax pair. Though the generalsolution x ( t ) = A cos( ωt + φ ) for constants of integration A and φ with ω = (cid:112) k/m is well-known, we will not need the explicit solution to discuss a Lax pair formulation. Introducingthe momentum p = m ˙ x , we may rewrite (1) as a pair of first order equations ˙ x = p/m and˙ p = − mω x . It is convenient to regard them as equations for the variables ωx and p/m whichhave the same dimension (of velocity): d ( ωx ) dt = ω (cid:16) pm (cid:17) and d ( p/m ) dt = − ω ( ωx ) . (2)These equations are equivalent to the Lax equation ˙ L = [ L, A ] for the pair of 2 × L = (cid:18) p/m ωxωx − p/m (cid:19) and A = (cid:18) ω/ − ω/ (cid:19) , (3)whose entries depend on the dynamical variables ωx and p/m . How did we arrive at this( L, A ) pair? We notice that (2) are linear in ω ˙ x and ˙ p/m . So, for ˙ L = [ L, A ] to reproduce(2) we choose L to be linear in ωx and p/m . The simplest possibility is to take L to be a2 × L is a traceless symmetricmatrix with entries linear in p/m and ωx as in (3). As a consequence, ˙ L is also symmetric.Since the commutator of symmetric and anti-symmetric matrices is symmetric, it is naturalto take A to be anti-symmetric . Since the RHS of (2) are linear in ωx and p/m , we take A to be independent of these variables so that [ L, A ] would also be linear in them. In termsof these variables, the RHS of (2) is independent of m and linear in ω , so the entries of theantisymmetric matrix A can depend only on ω and must be linear in it. This essentially leadsto the A appearing in (3). One then verifies that the four Lax equations following from (3)coincide with (2): ˙ L = (cid:18) ˙ p/m ω ˙ xω ˙ x − ˙ p/m (cid:19) = [ L, A ] = (cid:18) − ω x ωp/mωp/m ω x (cid:19) . (4) For [
L, A ] to be symmetric, A can differ from an anti-symmetric matrix at most by a multiple of the identity, whichwould not affect the commutator.
3e notice the following feature of the Lax matrix, tr L = 2( p /m + ω x ) is ( m/ × theconserved energy of the harmonic oscillator. It turns out that this is a general feature: onemay use the Lax matrix to obtain conserved quantities. The Lax equation ensures that the eigenvalues (spectrum) of L are independent of time. Thisproperty is known as isospectrality. To understand this, let us consider the Lax equation L t ≡ ˙ L = [ L, A ] , (5)where L and A are matrices with entries depending on the dynamical variables. We haveused subscripts to denote derivatives. Since the trace of the commutator of a pair of finitedimensional matrices vanishes, tr L is independent of time . More generally, one may showthat the eigenvalues of L are conserved. To see this, we begin with the eigenvalue problem Lψ = λψ . Differentiating in time, L t ψ + Lψ t = λ t ψ + λψ t . (7)Upon using the Lax equation (5) this becomes( LA − AL ) ψ + Lψ t = λ t ψ + λψ t . (8)Utilizing Lψ = λψ and rearranging, we get( L − λ ) Aψ + ( L − λ ) ψ t = λ t ψ or ( L − λ )( ψ t + Aψ ) = λ t ψ. (9)For the eigenvalue λ to be time-independent ( λ t = 0 ), the LHS must vanish. For this to happen, ψ t + Aψ must be an eigenvector of L with eigenvalue λ . Recall that ψ too is an eigenstate of L with the same eigenvalue. Now, for simplicity, we will assume that the λ -eigenspace of L isone-dimensional, which implies that ψ t + Aψ must be a multiple of ψ : ψ t + Aψ = βψ (10)for some (possibly time-dependent) complex number β . This equation may be viewed as anevolution equation for ψ : ψ t = ( − A + β ) ψ. (11)Here, is the identity matrix. Thus, the Lax equation L t = [ L, A ] and this evolution equationfor ψ together imply that the eigenvalue λ is a conserved quantity. We say that L evolvesisospectrally. Alternate demonstrations of isospectrality:
There are ways to show the isospectrality ofthe Lax matrix L without assuming its eigenspaces are one-dimensional. We give two of thembelow. The Lax equation ˙ L = [ L, A ] bears a resemblance to the Heisenberg equation of motion for an operator Q in theHeisenberg picture of quantum mechanics: i (cid:126) dQdt = [ Q, H ] , (6)where H is the Hamiltonian. If H and Q are finite dimensional matrices, then tr [ Q, H ] = 0 so that tr Q is conserved.While the Lax matrices L and A for mechanical systems are finite dimensional, observables of quantum systems of particlesare typically infinite dimensional and unbounded operators. The trace of the commutator of such operators may not vanish(or even be finite). In such cases, tr Q may not be a (finite) conserved quantity. We may absorb the β term into A (and write ψ t = − Aψ ) since it commutes with L and therefore does not affect theLax equation.
4. In this approach, we assume that L is hermitian so that λ is real. We take an innerproduct of ( L − λ )( ψ t + Aψ ) = λ t ψ (see Eqn. (9)) with the eigenfunction ψ and usehermiticity to get (cid:104) ( L − λ )( ψ t + Aψ ) , ψ (cid:105) = (cid:104) λ t ψ, ψ (cid:105) or (cid:104) ( ψ t + Aψ ) , ( L − λ ) ψ (cid:105) = λ t (cid:107) ψ (cid:107) . (12)The LHS vanishes as Lψ = λψ . Moreover, being an eigenfunction, || ψ || (cid:54) = 0 , so we musthave λ t = 0 .2. The isospectrality of L ( t ) may also be established by showing that L ( t ) is similar to L (0) .Indeed, suppose we define the invertible matrix S ( t ) via the equation ˙ S = − AS with theinitial condition S (0) = , then the solution of the Lax equation with initial value L (0)is L ( t ) = S ( t ) L (0) S − ( t ) . This is easily verified:˙ L ( t ) = ∂ t ( SL (0) S − ) = − ASL (0) S − − SL (0) S − ∂ t ( S ) S − = − AL ( t )+ L ( t ) A = [ L ( t ) , A ] . (13)Here we used ∂ t ( SS − ) = ∂ t = 0 , to write ∂ t ( S − ) = − S − ∂ t ( S ) S − . Finally, we observethat two matrices related by a similarity transformation have the same eigenvalues: L (0) ψ = λψ ⇒ SL (0) S − ( Sψ ) = λ ( Sψ ) or L ( t )( Sψ ) = λ ( Sψ ) . (14)Thus, the eigenvalues of L are conserved in time. Remark: In § L t = [ L, A ] is viewed as a compatibility condition among the two linear equations Lψ = λψ and ψ t = − Aψ for constant λ . We have just seen that if the equations of motion of a system can be written in Lax form L t =[ L, A ] , then the isospectrality of L gives us conserved quantities. These conserved quantities could be the eigenvalues of L or equivalently the spectral invariants det L and tr L n for n = 1 , , , . . . . For example, the familiar conserved energy of the harmonic oscillator may beexpressed in terms of the Lax matrix of Eqn. (3): E = 12 (cid:18) p m + mω x (cid:19) = − m L = m L . (15)We also notice that for any E > ± (cid:112) E/m )leading to 1D eigenspaces (one linearly independent eigenvector for each eigenvalue), as wasassumed in Eqn. (10). Furthermore, for n = 1 , , , . . . , L n = (cid:18) p m + x ω (cid:19) n = (cid:18) Em (cid:19) n and L n +1 = (cid:18) Em (cid:19) n L. (16)Thus, tr L n = 2 (2 E/m ) n while tr L n +1 = 0 so that the traces of higher powers of L do notfurnish any new conserved quantities. Indeed, a system with one degree of freedom cannot havemore than one independent conserved quantity. In fact, the conservation of energy restricts thetrajectories of the harmonic oscillator to lie on a family of ellipses in the x - p phase plane. Ifthere was an additional conserved quantity, trajectories would reduce to points which cannotdescribe nontrivial time evolution. A mechanical system with p degrees of freedom can have at most 2 p − L n is conserved for any positive integer n , not all of them may be independent.
5e now discuss some more examples of Lax representations. Our first example is the Todachain which admits a simple and elegant Lax pair. We then consider the Euler equationsfor a rigid body. They admit a simple Lax pair, which however does not allow us to obtainits conserved energy. This problem is solved by introducing a new Lax pair with a ‘spectralparameter’. We will also give a Lax pair with spectral parameter for the equations of theRajeev-Ranken model, which can be viewed as a generalization of the Euler equations to acentrally extended Euclidean algebra [3].
In 1967, Morikazu Toda introduced a model for a one-dimensional crystal in which a chain ofidentical atoms/particles of mass m interact with their nearest neighbours via nonlinear springswith exponential forces. If x i is the displacement of the i th particle from its equilibrium positionand p i its momentum, then the EOM are m ˙ x i = p i and ˙ p i = κ (cid:16) e − ( x i − x i − ) − e − ( x i +1 − x i ) (cid:17) . (17)Here, κ is a force constant and we will work in units where κ = m = 1 . We will consider an N particle Toda chain subject to periodic boundary conditions: x N + i = x i for all i . Thus,we may visualize the particles as lying on a circle and interpret x i as the angular displacement θ i from equilibrium (see Fig. 1). The exponential nonlinearity of the EOM (17) may be made
321 5 F ₆₅ F ₆₇ 𝜃 ₄
4 4 F ₆₅ = Force on atom 6 due toatom 5 𝜃 ₄ = angular displacement of atom 4 from equilibrium Figure 1:
Toda chain of N = 8 particles with periodic boundary conditions. quadratic by introducing Falschka’s variables [4] a i = 12 e − ( x i − x i − ) / and b i = − p i − , (18)which evolve according to ˙ a i = a i ( b i +1 − b i ) and ˙ b i = 2( a i − a i − ) . (19)These equations are equivalent to the Lax equation ˙ L = [ L, A ] if we define the essentiallytridiagonal matrices L and A as below L = b a · · · a N a b a a b ... . . . a N b N and A = − a · · · a N a − a · · · a · · · − a N . (20) L n for n = 1 , , . . . , N or alternatively the coefficients of the characteristicpolynomial det( L − λ ) give us N conserved quantities [5]. The first two of these may beinterpreted in terms of the total momentum and energy of the chaintr L = N (cid:88) i =1 b i = − N (cid:88) i =1 p i − = − P L = N (cid:88) i =1 (cid:0) a i + b i (cid:1) = 12 N (cid:88) i =1 (cid:18) p i + e − ( x i − x i − ) (cid:19) = E . (21) We consider a rigid body (e.g. a top) free to rotate about its center of mass (which is heldfixed) in the absence of external forces like gravity. In a frame that rotates with the body, itsEOM may be written as a system of three first order ‘Euler’ equations [6, 7] for the componentsof angular momentum (cid:126)S about its center of mass: (cid:126)S t = (cid:126)S × (cid:126) Ω or ˙ S = S Ω − S Ω , ˙ S = S Ω − S Ω and ˙ S = S Ω − S Ω . (22)Here, (cid:126) Ω = (Ω , Ω , Ω ) is the angular velocity vector which is related to (cid:126)S = ( S , S , S ) via (cid:126)S = I (cid:126)
Ω . The inertia tensor I ij = (cid:82) ( x δ ij − x i x j ) ρ ( x ) d x is a 3 × ρ ( x ) . The eigenvalues I , I and I of I are calledthe principal moments of inertia . In what follows, we will choose the axes of the co-rotatingframe to be the principal axes of inertia (eigenvectors of I ) so that the inertia tensor becomesdiagonal: I = diag( I , I , I ) .There is a straightforward way of expressing the Euler equations in Lax form if we introducethe anti-symmetric matrices S = S − S − S S S − S and Ω = − Ω − Ω Ω − Ω (23)corresponding to the vectors (cid:126)S and (cid:126) Ω . These matrices are obtained via an isomorphismfrom the R to the so (3) Lie algebra by contracting with the Levi-Civita symbol, e.g. S ij = (cid:80) k (cid:15) ijk S k while conversely S k = (cid:80) i,j (cid:15) ijk S ij / (cid:126)S × (cid:126) Ω = ( S Ω − S Ω , S Ω − S Ω , S Ω − S Ω ) , (24)then corresponds to (the negative of) the matrix commutator: [ S, Ω] = S Ω − S Ω S Ω − S Ω S Ω − S Ω S Ω − S Ω S Ω − S Ω S Ω − S Ω . (25) Thus, the Euler equations (22) take the Lax form: S t = [Ω , S ] . (26)Comparing with (5) we see that ( S, − Ω) furnish a Lax pair. What is more, the Lax equationthen implies that − (1 /
2) tr S = S + S + S (square of angular momentum) is a conservedquantity. Indeed, it is straightforward to check using (22) that S ˙ S + S ˙ S + S ˙ S = 0 . Inaddition to (cid:126)S , the Euler top is known to possess another conserved quantity, its energy: E = 12 (cid:18) S I + S I + S I (cid:19) . (27)7owever, E depends on the principal moments of inertia and cannot be obtained from the Laxmatrix S by combining the traces of its powers as S is independent of I , , .Thus, we seek a new Lax pair ( L, A ) such that both (cid:126)S and E can be obtained from traces of L . We therefore introduce a new Lax matrix which is a combination of the angular momentumand inertia matrices, weighted by a parameter λ . However, in place of I , it turns out to beconvenient to work with the diagonal matrix: I = diag( I , I , I ) with I k = (1 / I i + I j − I k )where ( i, j, k ) is any cyclic permutation of (1 , ,
3) . For example, I = (1 / I + I − I ) .Now, we postulate the new Lax pair [2] L ( λ ) = I + Sλ = I S /λ − S /λ − S /λ I S /λS /λ − S /λ I and A ( λ ) = − ( λ I + Ω) = − λ I Ω − Ω − Ω λ I Ω Ω − Ω λ I . (28) To motivate this Lax pair we first note that putting L = S/λ and A = − Ω in ˙ L = [ L, A ] givesthe desired EOM (26). For the energy to emerge as a conserved quantity from tr L , we willaugment this Lax pair by matrices involving the principal moments of inertia (or the matrix I ) while ensuring that the EOM are not affected. Since I , , are constant in time we can addany matrix function f ( I ) to L without affecting ˙ L . However, this will affect the commutator[ L, A ] . To cancel this contribution we will add another matrix function g ( I ) to A . Thus, L = S/λ + f and A = − (Ω + g ) . For the unwanted terms [ S/λ, g ] and [ f, Ω] in [
L, A ] to havea chance of cancelling, we use the relation (cid:126)S = I (cid:126)
Ω and dimensional analysis to pick f = I and g = λ I as in (28). Some algebra now shows that the Lax equation ˙ L = [ L, A ] is equivalentto (26). Indeed, ˙ L − [ L, A ] = 1 λ ( ˙ S + [ S, Ω]) + [ S, I ] + [ I , Ω] . (29)Using (cid:126)S = I (cid:126)
Ω , one finds that the sum [ S, I ] + [ I , Ω] vanishes. Thus, requiring the Laxequation to hold for any value of λ leads to the Euler equations for the angular momentumvector (cid:126)S as in (26).The trace of this new Lax matrix L is conserved, but it is not a dynamical variable as itis simply a quadratic polynomial in the material constants I , , . Pleasantly, the traces of thesecond and third powers of L involve the square of angular momentum (cid:126)S and energy E ,allowing us to deduce that both of them are conserved:tr L = tr I − λ (cid:126)S andtr L = tr (cid:2) I + λ I S (cid:3) = tr I − λ (cid:16) (tr I ) (cid:126)S − I I I E (cid:17) . (30)These conservation laws may be used to determine how (cid:126)S evolves in the corotating frame.Indeed, since both E and (cid:126)S are conserved, trajectories must lie along the intersection of theenergy ellipsoid and angular momentum sphere: E = 12 (cid:18) S I + S I + S I (cid:19) and (cid:126)S · (cid:126)S = S + S + S . (31)These two quadratic surfaces typically intersect along a closed curve which forms the periodicorbit of the tip of the angular momentum vector (cid:126)S as shown in Fig. 2. The parameter λ that appears in the Lax matrix L of Eqn. (28) is (somewhat confusingly) known as a spectral parameter.It is not to be confused with the symbol for an eigenvalue of the Lax matrix! The reason for this terminology will be clarifiedin § Traces of higher powers of L also lead to conserved quantities but they are simply functions of (cid:126)S and E . The intersection of the energy ellipsoid and angular momentum sphere is the orbit of the angular momentumvector (cid:126)S in the corotating frame of the Euler top.
Having found the evolution of the angular momentum vector, one still needs to use (cid:126)S ( t ) tosolve three first order equations for the ‘Euler angles’ ( θ, φ and ψ ) to find the instantaneousorientation of the rigid body in space. In the absence of external forces, a top displays twotypes of motion: spinning about an instantaneous axis of rotation and precession of this axisabout the fixed direction of angular momentum in the lab frame. For more on this, see thediscussion in [6]. The Rajeev-Ranken model [8, 9] describes certain nonlinear ‘continuous waves’ in a one-dimensional medium . This model is nice for our purposes since it is possible to discover aLax pair for its equations almost by inspection! Very roughly, it is a generalization of the Eulertop with two dynamical 3-vectors (cid:126)J ( t ) and (cid:126)S ( t ) and a six-dimensional phase space. Theirevolution is governed by the pair of equations˙ (cid:126)J = (cid:126)K × (cid:126)S and ˙ (cid:126)S = g (cid:16) (cid:126)S × (cid:126)J (cid:17) . (32)Here, g is a positive constant and (cid:126)K = − k ˆ z is a constant vector taken along the z -axis, where k is a constant with dimensions of wavenumber. As for the Euler top (see § J, S and K : J = (cid:126)J · (cid:126)σ i , S = (cid:126)S · (cid:126)σ i and K = − k σ i . (33)Here, (cid:126)σ is the vector whose components are the Pauli matrices σ , , . The EOM (32) now takethe form ˙ J = [ K, S ] and ˙ S = g [ S, J ] . (34)They admit a Lax representation if we postulate the Lax pair L ( λ ) = − Kλ + J λ + Sg and A ( λ ) = − Sλ . (35) The Rajeev-Ranken(RR) model is a mechanical reduction of a 1+1-dimensional scalar field theory (with field equations¨ φ = φ (cid:48)(cid:48) + λ [ ˙ φ, φ (cid:48) ] for the su (2) Lie algebra-valued field φ ( x, t ) ) dual to the SU(2) principal chiral model. It describes thedynamics of nonlinear screw-type waves of the form φ ( x, t ) = e Kx R ( t ) e − Kx + mKx . Here, K is a constant su (2) matrix, m a dimensionless parameter and λ a dimensionless coupling constant. The variables of the RR model J and S are related tothe anti-hermitian 2 × R via J = [ K, R ] + mK and S = ˙ R + K/λ . To arrive at this Lax pair we notice that ˙ L = [ L, A ] can lead to (34) if J and S appear linearly in L as coefficients ofdifferent powers of λ . L = [ L, A ] to hold for all values of the spectral parameter λ leads to Eqn. (34) along with the condition ˙ K = 0 , which is consistent with the constancyof the vector (cid:126)K . As a consequence of this Lax representation, tr L n ( λ ) must be conservedfor any λ and for any n = 1 , , . . . . Thus, each coefficient of the (2 n ) th degree polynomialtr L n ( λ ) furnishes a conserved quantity. For instance, tr L ≡ − L = λ (cid:126)K − λ (cid:126)J · (cid:126)K + 2 λ (cid:32) (cid:126)J − (cid:126)S · (cid:126)Kg (cid:33) + 2 λg (cid:126)S · (cid:126)J + (cid:126)S g (36)leads to four conserved quantities: (cid:126)K · (cid:126)J = − kJ , (cid:126)J + kg S , (cid:126)S · (cid:126)J and (cid:126)S . (37)It turns out that the traces of odd powers of L are identically zero while tr L , tr L , . . . do notlead to any new conserved quantities. We may interpret the four conserved quantities of (37)geometrically in the three-dimensional spaces of (cid:126)S and (cid:126)J vectors. For instance, the constancyof (cid:126)S implies that (cid:126)S is confined to a sphere. For (cid:126)S lying on such a sphere, the conservationof (cid:126)S · (cid:126)J implies that (cid:126)J must lie on a plane perpendicular to (cid:126)S . Similarly, constancy of J defines a horizontal plane and that of (cid:126)J / k/g ) S defines a sphere for each such vector (cid:126)S . Remarkably, it turns out that the intersection of these four surfaces is almost always atwo-dimensional torus (surface of a vada/doughnut) [3] in the six-dimensional space of ( (cid:126)S, (cid:126)J )pairs. This implies that unlike in the Euler top where (cid:126)S was periodic, here, trajectories aretypically quasi-periodic and fill up the whole torus as in Fig. 3.Figure 3: A quasi-periodic trajectory on a torus which is the intersection of level surfaces of the four conserved quantities ofthe Rajeev-Ranken model.
A Lax pair similar to the one in Eqn. (35) applies to the Neumann model [2, 9] whichdescribes the motion of a particle on a sphere subject to harmonic forces due to springs attachedto the coordinate hyperplanes.
So far, we considered systems of particles with finitely many degrees of freedom. In this section,we extend the idea of a Lax pair to certain continuum mechanical systems with infinitely manydegrees of freedom (systems of fields rather than finitely many particles). We will do this inthe context of the linear wave equation for vibrations of a stretched string and the nonlinearKorteweg de-Vries (KdV) equation for water waves. The Lax pair framework will also be Quasi-periodic refers to a superposition of two periodic motions (with incommensurate frequencies) corresponding to thetwo cycles (‘small’ and ‘large’ non-contractible loops) of the torus.
One of the simplest field equations in one dimension (1D) is the wave equation : u t + cu x = 0 for constant c. (38)Here, u ( x, t ) could represent the amplitude/height of the wave (sound/water etc.) at position x at time t . Here, subscripts on u denote partial derivatives. For c > c while maintaining their shape.Indeed, one checks that u ( x, t ) = f ( x − ct ) is a solution of (38) for any differentiable function f .We seek a Lax pair of differential operators L and A (depending on u ) such that L t = [ L, A ] isequivalent to (38). It is convenient to take L to be the Schr¨odinger operator L = − ∂ + u ( x, t ) ,where ∂ = ∂ x = ∂/∂x . L is familiar from Sturm-Liouville theory as well as from quantummechanics as the Hamiltonian of a particle moving in the potential u ( x, t ) . Since L issymmetric (hermitian), L t = u t is also symmetric, so for the Lax equation to make sense [ L, A ]must also be symmetric. As in § A to be anti-symmetric (up to the addition ofan operator that commutes with L ) guarantees this. It turns out that A = c ∂ does the job(see Appendix A). Indeed, using the commutator [ ∂, f ] = f (cid:48) for any function f , we see thatthe Lax equation is equivalent to the wave equation: L t = u t = [ L, A ] = [ − ∂ + u ( x, t ) , c∂ ] = [ u, c∂ ] = − cu x . (39)We will use this Lax pair as a stepping stone to find a Lax pair for the KdV equation whichis a nonlinear wave equation with widespread applications. As discussed in §
2, the existenceof a Lax pair is usually associated with the presence of conserved quantities. For example,integrating (38) in x , we get ddt (cid:90) ∞−∞ u dx = − c (cid:90) ∞−∞ u x dx = − c ( u ( ∞ ) − u ( −∞ )) = 0 (40)assuming u → x → ±∞ . Thus, C = (cid:82) ∞−∞ u dx is conserved. The reason this worked isthat (38) takes the form of a local conservation law: ∂ t ρ + ∂ x j = 0 with ρ = u and j = cu . In-tegrating an equation in local conservation form implies the conservation of (cid:82) ∞−∞ ρ dx , providedthe ‘flux’ of j across the ‘boundary’ vanishes: j ( ∞ ) − j ( −∞ ) = 0 . Similarly, multiplying (38)by u leads to an equation that is again in local conservation form: ∂ t ( u /
2) + c∂ x ( u /
2) = 0 .Thus, C = (cid:82) ∞−∞ u dx is also conserved. In a similar manner, we find that ∂ t u n + c∂ x u n = 0 ,so that C n = (cid:82) ∞−∞ u ( x, t ) n dx is conserved for any n = 1 , , , . . . . Thus, the wave equationadmits infinitely many constants of motion.However, unlike in § C n have not been obtained from the Lax operator L . As we willsee in § another infinite sequence of conserved quantities Q n that may be obtained from L . Unlike C n , the Q n turn out to be very special: they areconserved quantities both for the wave equation and its upcoming nonlinear generalization, theKdV equation. The 1 st order wave equation u t + cu x = 0 is related to the 2 nd order wave equation ( ∂ t − c ∂ x ) φ = 0 . Indeed, thed’Alembert wave operator ∂ t − c ∂ x may be factorized as ( ∂ t + c∂ x )( ∂ t − c∂ x ) . The first order equations u t + cu x = 0 and v t − cv x = 0 describe right/left-moving waves u = f ( x − ct ) and v = g ( x + ct ) while the 2 nd order wave equation describesbi-directional propagation: φ ( x, t ) = f ( x − ct ) + g ( x + ct ) . In one-dimensional quantum systems, bound state energy eigenvalues are nondegenerate (see p. 99 of [10]). Thus, thecorresponding eigenspaces of L = − ∂ x + u are one-dimensional. See the discussion in § .1.1 Infinitely many conserved quantities from the Lax operator L = − ∂ + u Since L = − ∂ + u ( x, t ) and A = c∂ are unbounded differential operators, we do not try tomake sense of tr L n to find conserved quantities by the method of § can be obtained from the pair of equations Lψ = λψ and ψ t = − Aψ (see § λ = k and change variables from the wavefunction ψ to a new function ρ defined via the transformation ψ ( x, t ; k ) = exp (cid:20) − ikx + (cid:90) x −∞ ρ ( y, k, t ) dy (cid:21) . (41)Then, by studying the quantum mechanical scattering problem for a plane wave with onedimensional wavevector k in the potential u (assumed to vanish at ±∞ ), it can be shown[11] that (cid:82) ∞−∞ ρ ( x, k, t ) dx (which is the reciprocal of the transmission amplitude) is conservedin time for any k . We will use this to find an infinite sequence of integrals of motion (interms of u ). Putting (41) in Lψ = k ψ , we get a Riccati-like equation relating ρ to u : ρ x + ρ − ikρ = u ( x, t ) . Since ρ is a conserved density, so are the coefficients ρ n in anasymptotic series in inverse powers of k : ρ = (cid:80) ∞ n =1 ρ n ( x, t ) / (2 ik ) n which is a bit like asemiclassical expansion. Comparing coefficients of different powers of k , one finds at O ( k ) : ρ = − u, at O (1 /k ) : ρ = ∂ρ and at O (1 /k n ) : ρ n +1 = ∂ρ n + n − (cid:88) m =1 ρ m ρ n − m . (42) Using this recursion relation we may express ρ n in terms of u and its derivatives: ρ = − u, ρ = − u x , ρ = u − u xx , ρ = (2 u − u xx ) x ,ρ = − u x + 2( u ) xx + u x + 2 uu xx − u , ρ = (cid:18) − u x + 18 uu x − u (cid:19) x etc. (43)The even coefficients integrate to zero while ρ n +1 lead to nontrivial conserved quantitiesdefined as Q n = ( − n +1 (cid:90) ∞−∞ ρ n +1 dx for n = 0 , , . . . . (44)The first few of these conserved quantities for the wave equation are: Q = (cid:90) u dx, Q = (cid:90) u dx, Q = (cid:90) (cid:18) u x u (cid:19) dx and Q = 12 (cid:90) (cid:2) u + 10 uu x + u x (cid:3) dx. (45) The KdV equation, with subscripts denoting partial derivatives, u t − uu x + u xxx = 0 , (46)describes long wavelength ( l (cid:29) h , ‘shallow-water’) surface waves of elevation u ( x, t ) (cid:28) h inwater flowing in a narrow canal of depth h (see Fig. 4). The KdV equation for the field u describes the evolution of infinitely many degrees of freedom labeled by points x lengthwisealong the canal. While the nonlinear advection term uu x can steepen the slope of a waveprofile, the dispersive u xxx term tends to spread the wave out. A balance between the two The standard form (46) of the KdV equation only admits waves of depression as its solutions. To get waves of elevationwhich we see in a canal, we need to change the sign of the advection term so that the KdV equation takes the form u t +6 uu x + u x = 0 . A linear evolutionary partial differential equation (such as the wave equation) is nondispersive if the phase velocity v p ( k ) = ω ( k ) /k of a plane wave solution e i ( kx − ω ( k ) t ) is independent of the wavevector k . This happens if the angularfrequency-wavevector dispersion relation ω = ω ( k ) is linear. For a nondispersive equation, all Fourier components (labelledby k ) travel at the same speed so that a wave packet does not spread out. Surface wave profile in a canal. effects can lead to localized solitary waves or ‘solitons’ that can propagate while maintainingtheir shape. What is more, two such solitons can collide and reemerge while retaining theirshapes. These phenomena, which were discovered via laboratory and numerical experiments,suggested that the KdV equation may possess several constants of motion.In fact, the KdV equation admits some elementary conserved quantities [12, 13]. For in-stance, integrating (46) gives ddt (cid:90) ∞−∞ u dx = (cid:90) ∞−∞ (cid:0) u − u xx (cid:1) x dx = 0 , (47)assuming u → x → ±∞ . This leads to the conservation of the mean height 2 Q = (cid:82) ∞−∞ u dx . Furthermore, one may check by differentiating in time and using (46) that2 Q = P = (cid:90) ∞−∞ u dx and Q = E = (cid:90) ∞−∞ (cid:18) u + u x (cid:19) dx (48)are also conserved. P and E can be interpreted as the momentum and energy of the waveand are related to symmetries of the KdV equation under space-time translations via Noether’stheorem (see Chapt. 1 of [14] for more on symmetries of the KdV equation). While theseconservation laws could perhaps be guessed, in what came as a major surprise, in 1967-68,Whitham and then Kruskal and Zabusky discovered a fourth ( Q from (45)) and fifth conservedquantity. Miura discovered yet more and the list grew to eleven conserved quantities. In fact,it was shown by Gardner, Kruskal and Miura [15] that the KdV equation admits an infinitesequence of independent conserved quantities . They turn out to be the same as the Q n of § L, A ) pair for the KdVequation. As for the wave equation in § L is the Schr¨odinger operator, but A is a thirdorder operator (see Appendix B for an indication of how one arrives at A ): L = − ∂ + u ( x, t ) and A = 4 ∂ − u∂ − u x . (49)As before, L t = u t . The commutator [ L, A ] receives two contributions. With u (cid:48) denoting u x ,the 3 rd order term in A gives[ − ∂ + u, ∂ ] = − u (cid:48)(cid:48)(cid:48) + 3 u (cid:48)(cid:48) ∂ + 3 u (cid:48) ∂ ) . (50)As for the first order part of A , the calculation is essentially the same as in (69) of AppendixA, with α = − u :[ − ∂ + u, − u∂ + u x )] = − u (cid:48)(cid:48)(cid:48) − u (cid:48)(cid:48) ∂ − u (cid:48) ∂ − uu (cid:48) ) . (51) The most famous solution of the KdV equation is the soliton u = − c sech (cid:104) √ c ( x − ct )2 (cid:105) . It describes a localized solitarywave of depression that travels at velocity c while retaining its shape. Observation of such a wave was reported in 1834 byScott Russell while riding along the Edinburgh-Glasgow canal. Interestingly, their work was motivated by Kruskal and Zabusky’s 1965 observation [16] of ‘recurrent behavior’ and ‘solitonscattering’ in numerical solutions of the KdV equation. L t = [ L, A ] cancel, leaving us with the KdVequation (46): u t = [ L, ∂ − u∂ − u x ] = − u (cid:48)(cid:48)(cid:48) + 6 uu (cid:48) . (52)The Lax representation helps us understand roughly why KdV admits infinitely many conservedquantities. Indeed, L = − ∂ + u may be viewed as an infinite dimensional matrix, all of whoseeigenvalues are conserved. In fact, the method of § L also applies to the KdV equation. What is more,since the two equations share the same Lax operator L , it turns out that they also possess thesame set of conserved quantities Q n . Moreover, treating Q n as a sequence of Hamiltonians,one obtains the ‘KdV’ hierarchy of field equations. The linear wave and KdV equations arethe first two in this hierarchy, while u t = u x − uu x − u x u x + 30 u u x is the third. TheSchr¨odinger operator L = − ∂ + u serves as a common Lax operator for all of them though theoperator A (which enters through ψ t = − Aψ ) differs for the various members of this hierarchy.Remarkably, it turns out that the Q n of § The zero curvature representation generalizes the idea of a Lax pair to a wider class of nonlinearevolution equations for systems especially in one spatial dimension. To understand how thisworks, we change our viewpoint and regard the nonlinear Lax equation L t = [ L, A ] as a com-patibility condition for the following pair of linear equations to admit simultaneous solutions: Lψ = λψ and ψ t = − Aψ with λ a constant . (53)Indeed, by differentiating Lψ = λψ in time and using the second equation, it is verified thatfor the eigenvalue λ of L to be time-independent, L and A must satisfy the Lax equation L t = [ L, A ] . Unlike in § λ to be a nondegenerate eigenvalue of L .In the case of the KdV equation (46), L = − ∂ x + u involves 2 nd order space derivatives, sothat the two equations in (53) are somewhat asymmetrical. There is a way of replacing (53)with a more symmetric pair of linear equations involving only 1 st order derivatives: ∂ x F = U F and ∂ t F = V F. (54)The price to be paid is that U and V are now square matrices and F a column vector (ofsize equal to the order of the differential operator L ) whose components depend on locationthrough the dynamical variables (such as u for KdV). The matrix elements of U and V alsodepend on the eigenvalue λ which is now called the spectral parameter. However, unlike L and A which are differential operators, U and V are finite dimensional matrices, a feature wewill exploit in obtaining conserved quantities.Eqn. (54) is called the auxiliary linear system of equations of Zakharov and Shabat [17,18]. ∂ x − U and ∂ t − V may be viewed as the space and time components of a ‘covariantderivative’. Thus, the auxiliary linear equations (54) require that every vector field F ( x, t )is ‘covariantly’ constant. It is overdetermined in the sense that U and V must satisfy acompatibility (consistency) condition for solutions F to exist. Indeed, equating mixed partials ∂ x ∂ t F = ∂ t ∂ x F , we get the consistency condition ∂ t U − ∂ x V + [ U, V ] = 0 . (55)The original nonlinear evolution equations are said to have a zero curvature representation ifthey are equivalent to (55) for some pair of matrices U and V . Before explaining how this14cheme may be used to find conserved quantities, let us use the KdV equation to provide anexample.To find U for KdV, we write the eigenvalue problem for the Lax operator ( − ∂ x + u ) ψ = λψ as a pair of first order equations by introducing the column vector F = ( f , f ) T = ( ψ, ψ x ) T : ∂ x (cid:18) f f (cid:19) = (cid:18) u − λ (cid:19) (cid:18) f f (cid:19) ⇒ U = (cid:18) u − λ (cid:19) , (56)upon comparing with (54). Next, we use ψ t = − Aψ with Aψ = 4 ψ xxx − uψ x − u x ψ to find V such that ∂ t F = V F . We may express ψ t = − Aψ as a system of two first order ODEs.First, we differentiate Lψ = λψ in x to express ψ xxx as u x ψ + uψ x − λψ x . Thus, Aψ canbe written in terms of ψ and ψ x : Aψ = − u + 2 λ ) ψ x + u x ψ. (57)Next, using F = ( ψ, ψ x ) T , ψ t = − Aψ takes the form ∂ t (cid:18) f f (cid:19) = V (cid:18) f f (cid:19) with V = (cid:18) − u x u + 2 λ )2 u − u xx + 2 uλ − λ u x (cid:19) . (58)Here, the second row of the matrix V is obtained by taking the x derivative of the first equationin (58) and using Lψ = λψ . The parameter λ that appears in U and V originally arose as theeigenvalue of the Lax operator L . This explains the name spectral parameter . More generally,a zero curvature representation need not arise from a Lax pair and the corresponding spectralparameter λ may not admit an interpretation as an eigenvalue. Why the name ‘zero curvature’ ?
In general relativity the gravitational field is associated tospace-time curvature. It turns out that an electromagnetic field is also associated to curvature,though not of space-time but of an internal space (U(1) principal bundle over space-time).Now, the electric and magnetic fields may be packaged in the components of the field strength: F i = E i /c and F ij = (cid:80) k (cid:15) ijk B k for 1 ≤ i, j, k ≤ c denotes the speed of light.Thus, the field strength is a measure of curvature. What is more, specializing to one spatialdimension ( x = t, x = x ) and introducing the scalar and vector potentials A and A , wehave F = ∂ t A − ∂ x A . More generally, in the non-abelian gauge theories relevant to the strongand weak interactions, A and A become square matrices and the field strength acquires anextra commutator term: F = ∂ t A − ∂ x A + [ A , A ] . Now making the substitutions A → U and A → V , we see that the consistency condition (55) states that the field strength orcurvature of this nonabelian gauge field vanishes. Hence the name zero curvature condition. Here, we will learn how the zero curvature representation may be used to construct conservedquantities. Let us consider the first of the auxiliary linear equations in (54) for the columnvector F : ∂ x F = U ( x ) F ( x ) . Let us imagine solving this equation for F from an initiallocation x to a final point y . If y = x + δx for small δx , then F ( x + δx ) ≈ [ + δx U ( x )] F ( x ) . (59)More generally, linearity suggests that the solution may be written as F ( y ) = T ( y, x ) F ( x ) . Here T ( y, x ) may be viewed as transforming F ( x ) into F ( y ) and is called the transition matrix or For a Lax operator which is an n th -order spatial differential operator, we may express the Lax equation as a systemof n first order equations for the column vector ( ψ, ψ x , ψ xx , · · · , ψ ( n − x ) T comprising the first ( n −
1) derivatives of theeigenfunction ψ . U and V then become n × n matrices. For the KdV equation, n = 2 . T ( y, x ) must satisfy the equationand boundary condition ∂ y T ( y, x ; λ ) = U ( y ; λ ) T ( y, x ; λ ) and T ( x, x ; λ ) = (60)for any value of the spectral parameter λ . This is obtained by inserting F ( y ) = T ( y, x ) F ( x )in the auxiliary linear equation ∂ y F ( y ) = U ( y ) F ( y ) and requiring it to hold for any F ( x ) .In Appendix C, we learn that the transition matrix T ( y, x ) may be expressed (essentially byiterating (59)) as an ordered exponential series which we abbreviate as T ( y, x ; λ ) = P exp (cid:90) yx U ( z ; λ ) dz. (61)For simplicity, we henceforth suppose that our one-dimensional system is defined on the spatialinterval − a ≤ x ≤ a with periodic boundary conditions , so that U ( − a ) = U ( a ) and V ( − a ) = V ( a ) . Thus, we may view our spatial coordinate x as parametrizing a circle of circumference2 a . So far, we have been working at one instant of time. It turns out that the transition matrixaround the full circle ( x = − a to y = a ), which is also called the monodromy matrix, T a ( t, λ ) = P exp (cid:90) a − a U ( z ; t, λ ) dz (62)has remarkably simple time evolution. In fact, using the derivative of the transition matrix(60) and the zero curvature condition (55), one may show (see Appendix C or § ∂ t T ( y, x ; t ) = V ( y ; t ) T ( y, x ; t ) − T ( y, x ; t ) V ( x ; t ) . (63)However, this is not quite a commutator. Nevertheless, specializing to x = − a and y = a andusing periodic boundary conditions, we find that the monodromy matrix T a ( t ) = T ( a, − a ; t )evolves via a commutator: ∂ t T a ( t, λ ) = [ V ( a ; t, λ ) , T a ( t, λ )] . (64)We are now in familiar territory: this equation has the same structure as the Lax equation (5)upon making the replacements T a (cid:55)→ L and V (cid:55)→ − A . As explained in § L is independent of time. This immediately implies that the trace of themonodromy tr T a ( t, λ ) is independent of time . Moreover, this is true for any value of thespectral parameter λ . Thus, if we expand tr T a ( λ ) in a series in (positive and negative) powersof λ , then each of the coefficients is a conserved quantity. In many interesting cases such asthe Heisenberg magnetic chain and nonlinear Schr¨odinger equations, one may obtain infinitelymany conserved quantities in this way. We now briefly illustrate the idea of a zero curvature representation by considering the one-dimensional nonlinear Schr¨odinger equation (NLSE) for the complex wave amplitude ψ ( x, t ) : i ∂ψ∂t = − ∂ ψ∂x + 2 κ | ψ | ψ. (65)Here, κ is a real parameter. For κ = 0 , it reduces to the linear Schr¨odinger wave equation(in units where (cid:126) = 1 ) for a free quantum mechanical particle of mass m = 1 / This is where the finite dimensional character of U , V and consequently the monodrmy matrix T a is useful. Unlike thetrace of the Lax differential operator L , there is no difficulty in making sense of the trace of the monodromy matrix.
16 line. The NLSE is used to model a gas of bosons with short-range pairwise interactions (ofstrength κ ) in a ‘mean field’ approximation where | ψ | is interpreted as the density of bosons.It also has applications in nonlinear optics [20]. Like KdV, NLSE too admits solitary wavesolutions. They are called bright and dark solitons depending on whether κ is negative orpositive corresponding to attractive or repulsive interactions among the bosons.The NLSE admits a zero curvature representation (55) if the U and V matrices are chosenas [19] where λ is an arbitrary spectral parameter: U = U + λU and V = V + λV + λ V where U = √ κ ( ψ ∗ σ + + ψσ − ) = − V ,U = σ i = − V and V = iκ | ψ | σ − i √ κ ( ψ ∗ x σ + − ψ x σ − ) . (66)Here, σ ± = (1 / σ ± iσ ) are built from the Pauli matrices. It may be checked that thecondition for the associated field strength (55) to vanish for all values λ is equivalent to theNLSE and its complex conjugate. Thus, ( V, U ) may be viewed as defining a flat connection inan SU(2) principal bundle over the 1+1-dimensional space-time.Using the methods of § − a ≤ x ≤ a withperiodic boundary conditions, the first four integrals of motion are N = (cid:90) a − a | ψ | dx, P = (cid:90) a − a (cid:61) ψ ∗ ψ x dx, E = (cid:90) a − a ( | ψ x | + κ | ψ | ) dx and Q = (cid:90) a − a (cid:2) ψ ∗ ψ xxx − κ | ψ | ( ψψ ∗ x + 4 ψ ∗ ψ x ) (cid:3) dx. (67)The conserved quantities N, P and E represent the number of bosons, their total momentumand energy. In this article, we have explained what a Lax pair is and how it can be used to find conservedquantities for mechanical systems such as the simple harmonic oscillator, Toda chain, Euler topand Rajeev-Ranken model as well as field theories such as the linear wave and KdV equations.Though it is not always possible or easy to find a Lax pair for a given system, it is possible togenerate lots of Lax pairs and thereby discover systems with numerous conserved quantities.Some of these turn out to be interesting ‘exactly solvable’ or ‘integrable’ systems. As one mayinfer from these examples, there is no step-by-step procedure to find a Lax pair for a givensystem or even to know whether it admits a Lax pair. One first needs to determine someproperties of the system (say numerically, analytically or experimentally as happened withKdV) to develop a feeling for whether a Lax pair might exist. As a rule of thumb, equationswhose trajectories are ‘regular’ or for which (some) analytic solutions can be obtained oftendo admit a Lax pair, while those that display irregular/chaotic behavior do not. Even if onesuspects the presence of a Lax pair, finding one may not be easy and requires playing aroundwith the equations as we have done for the harmonic oscillator, Euler top, wave equation andthe KdV equation. However, if one does find a Lax pair, it opens up a whole new window to theproblem and brings to bear new tools [21] that can be applied to its understanding. Indeed, Laxpairs are the tip of an iceberg in the study of (Hamiltonian) dynamical systems. While it helpsto have conserved quantities, one can do more if they are sufficiently numerous and generate‘commuting’ flows on the state space (i.e., if their Poisson brackets vanish). In such cases, thereis (at least in principle) a way of changing variables to so-called action-angle variables in whichthe solutions to the EOM may be written down by inspection! Moreover, continuum systemsin one spatial dimension (such as the KdV, nonlinear Schr¨odinger and sine-Gordon equations)17hich have a Lax pair and an infinite tower of conserved quantities typically admit solitarywave solutions called solitons. Two such solitons can collide with each other and interact in acomplicated way but emerge after the collision retaining their original shapes and speeds, thusmimicking the elastic scattering of particles. This soliton scattering behavior can be regardedas a generalization to nonlinear systems of the superposition principle for linear equations.These nonlinear field equations also admit a remarkable generalization of the Fourier transformtechnique of solving linear PDEs such as the heat or wave equations. This technique is basedon the Gelfand-Levitan-Marchenko equation and is called the ‘inverse scattering transform’[12, 19, 20, 22]: it can be used to solve the initial value problem of determining the fields attime t given their values at t = 0 . A Finding a Lax operator A for the wave equation The choice A = c∂ to partner the Schr¨odinger operator L in the Lax pair (39) for the waveequation (38) can be arrived at by starting from the simplest of differential operators, a firstorder differential operator α ( x, t ) ∂ + β ( x, t ) and imposing some consistency conditions. Weshall see in Appendix B, that this approach generalizes to other equations. To make A anti-symmetric, we subtract its adjoint and consider A = ( α∂ + β − ∂ † α − β ) = ( α∂ + ∂α ) = [ α, ∂ ] + = ( α (cid:48) + 2 α∂ ) where α (cid:48) = ∂α∂x . (68)Here, we used (i) ∂ † = − ∂ , (ii) g † = g for any real function g and (iii) ( ∂α )( f ) = α (cid:48) f + αf (cid:48) so that ∂α = α (cid:48) + α∂ . The commutator with the Schr¨odinger operator L is then[ L, A ] = [ − ∂ + u, α (cid:48) + 2 α∂ ] = − α (cid:48)(cid:48)(cid:48) − α (cid:48)(cid:48) ∂ − α (cid:48) ∂ − αu (cid:48) . (69)Here, we used [ ∂, α ] = α (cid:48) , the Leibnitz product rule, linearity and anti-symmetry of commuta-tors to obtain[ u, α (cid:48) ] = 0 , [ u, α∂ ] = − αu (cid:48) , [ − ∂ , α (cid:48) ] = − ∂ [ ∂, α (cid:48) ] − [ ∂, α (cid:48) ] ∂ = − α (cid:48)(cid:48)(cid:48) − α (cid:48)(cid:48) ∂ and[ − ∂ , α∂ ] = − ∂ [ ∂, α∂ ] − [ ∂, α∂ ] ∂ = − ∂ (2 α (cid:48) ∂ ) − (2 α (cid:48) ∂ ) ∂ = − α (cid:48) ∂ − α (cid:48)(cid:48) ∂. (70)In the Lax equation L t = [ L, A ] , L t = u t is multiplication by u t ( x, t ) . For [ L, A ] in (69)to also be a multiplication operator, the coefficients of ∂ and ∂ must vanish which implies α (cid:48) = α (cid:48)(cid:48) ≡ x . This implies α = α ( t ) is a function of time alone. Thus, L t = [ L, A ]becomes u t = − α ( t ) u x . For this to be equivalent to the wave equation u t + cu x = 0 , we mustpick α ( t ) = c/ A reduces to A = c∂ . B Arriving at the rd order Lax operator A for KdV Here we adapt the method of Appendix A to explain the choice of the 3 rd order differentialoperator A = 4 ∂ − u∂ − u x = 4 ∂ − u, ∂ ] + (71)in the KdV Lax pair (49). From § A = c∂ and the Schr¨odinger operator L = − ∂ + u furnish a Lax pair for the linear wave equation. To find a Lax pair for the 3 rd order KdV equation, we will retain L = − ∂ + u with L t being the multiplication operator u t , while allowing for A to be of order higher than one. The simplest possibility is a 2 nd order operator, but this does not work. Indeed, anti-symmetrization reduces it to a 1 st orderoperator which is no different from (68) with α = − ( e (cid:48) + gf (cid:48) ) : A = e∂ + f ∂g∂ − ( e∂ + f ∂g∂ ) † = e∂ + f ∂g∂ − ∂ e − ∂g∂f = − ( e (cid:48) + gf (cid:48) ) (cid:48) − e (cid:48) + gf (cid:48) ) ∂. (72)18he next possibility is a 3 rd order operator. For simplicity, we try the operator b∂ where b is a constant. Upon anti-symmetrizing, A = b∂ − ( b∂ ) † = b∂ + ∂ b = 2 b∂ . (73)As in Appendix A, using the product rule and [ ∂, h ] = h (cid:48) we find that[ L, A ] = [ − ∂ + u, b∂ ] = − b ( u (cid:48)(cid:48)(cid:48) + 3 u (cid:48)(cid:48) ∂ + 3 u (cid:48) ∂ ) . (74)While this includes a u (cid:48)(cid:48)(cid:48) term, it lacks the uu (cid:48) term in the KdV equation (46) and is notpurely a multiplication operator. Here, A from Appendix A comes to the rescue. Thus, letus consider A = A + A = 2 b∂ + 2 α∂ + α (cid:48) so that[ L, A ] = [ − ∂ + u, b∂ + 2 α∂ + α (cid:48) ] = − α (cid:48)(cid:48)(cid:48) − αu (cid:48) − bu (cid:48)(cid:48)(cid:48) − (6 bu (cid:48)(cid:48) + 4 α (cid:48)(cid:48) ) ∂ − (6 bu (cid:48) + 4 α (cid:48) ) ∂ . (75)For [ L, A ] to be a multiplication operator, the coefficients of ∂ and ∂ must vanish. Thus, α (cid:48) = − (3 / bu (cid:48) which implies α = − (3 / bu + α for an integration constant α . Eliminating α , the Lax equation becomes L t = u t = [ L, A ] = − b u (cid:48)(cid:48)(cid:48) + (3 bu − α ) u (cid:48) . (76)Comparing with the KdV equation ( u t = 6 uu x − u x ) fixes b = 2 and α = 0 so that A = 4 ∂ − u∂ − u (cid:48) as claimed. Note that we may add to A an arbitrary function of time(which would commute with L ) without affecting the Lax equation. C Time evolution operator and the ordered exponential In §
2, we encountered an equation for the time evolution operator S ( t )˙ S = − A ( t ) S, with the initial condition S (0) = , the identity matrix . (77)The same equation also arises as the second of the auxiliary linear equations in (54) and as theSchr¨odinger equation in quantum mechanics for the time dependent ‘Hamiltonian’ − i (cid:126) A ( t ) .Here, we explain how this equation may be solved. When A is independent of time the solutionis the matrix exponential S = exp( − At ) . However, for time-dependent A , this formula doesnot satisfy (77) if A ( t ) at distinct times do not commute. To solve (77), we first integrate it intime form 0 to t to get an integral equation that automatically encodes the initial condition: S ( t ) − = − (cid:90) t A ( t ) S ( t ) dt . (78) S appears on both sides, so this is not an explicit solution. Iterating once, we get S ( t ) = − (cid:90) t dt A ( t ) (cid:18) − (cid:90) t dt A ( t ) S ( t ) (cid:19) . (79)Repeating this process, we get an infinite sum of multiple integrals, S ( t ) = − (cid:90) t dt A ( t )+ (cid:90) t (cid:90) t dt dt A ( t ) A ( t ) −· · · = ∞ (cid:88) n =0 ( − n (cid:90) · · · (cid:90)
D Time evolution of the transition matrix T ( y, x ; t ) Recall from § T ( y, x ; t ) ‘propagates’ vectors in the auxiliarylinear space from x to y : F ( y ; t ) = T ( y, x ; t ) F ( x ; t ) and may be expressed as a path orderedexponential as in (61). To obtain Eqn. (63) for its time evolution, we first differentiate Eqn.(60) [ ∂ y T ( y, x ; t ) = U ( y ; t ) T ( y, x ; t ) ] in time: ∂ t ∂ y T ( y, x ; t ) = ∂ t U ( y ; t ) T ( y, x ; t ) + U ( y ; t ) ∂ t T ( y, x ; t ) . (84)Then we use the zero curvature condition ∂ t U ( y ) − ∂ y V ( y ) + [ U ( y ) , V ( y )] = 0 and Eqn. (60)again to get: ∂ t ∂ y T ( y, x ; t ) = ( ∂ y V ) T + V U T − U V T + U ( ∂ t T ) = ∂ y ( V T ) + U ( ∂ t T − V ( y ) T ) . (85)Thus, we have ∂ y W ( y, x ; t ) = U ( y ) W ( y, x ; t ) where W ( y, x ; t ) = ∂ t T − V ( y ) T. (86)Thus both W ( y, x ; t ) and T ( y, x ; t ) satisfy the same differential equation (60) though theyobey different ‘boundary conditions’ W ( x, x ; t ) = − V ( x ) while T ( x, x ; t ) = I . We now usethis to check that ˜ W ( y, x ; t ) = − T ( y, x ; t ) V ( x ) also satisfies the same differential equation withthe desired boundary condition ˜ W ( x, x ; t ) = − V ( x ) . Exploiting the uniqueness of solutionsto (60) for a given boundary condition, we conclude that W ( y, x ; t ) = ˜ W = − T ( y, x ; t ) V ( x ) .Substituting this in the definition of W ( y, x ; t ) (86), we obtain the evolution equation (63) forthe transition matrix: ∂ t T ( y, x ; t ) = V ( y ; t ) T ( y, x ; t ) − T ( y, x ; t ) V ( x ; t ) . Acknowledgements:
We thank an anonymous referee for useful comments and references.This work was supported in part by the Infosys Foundation, J N Tata Trust and grants(MTR/2018/000734, CRG/2018/002040) from the Science and Engineering Research Board,Govt. of India.
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