Lax pair

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In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the Lax equation. Lax pairs were introduced by Peter Lax to discuss solitons in continuous media. The inverse scattering transform makes use of the Lax equations to solve such systems.

Definition[edit]

A Lax pair is a pair of matrices or operators L(t), P(t) dependent on time and acting on a fixed Hilbert space, and satisfying Lax's equation:

\frac{dL}{dt}=[P,L]

where [P,L]=PL-LP is the commutator. Often, as in the example below, P depends on L in a prescribed way, so this is a nonlinear equation for L as a function of t.

Isospectral property[edit]

It can then be shown that the eigenvalues and more generally the spectrum of L are independent of t. The matrices/operators L are said to be isospectral as t varies.

The core observation is that the matrices L(t) are all similar by virtue of

L(t)=U(t,s) L(s) U(t,s)^{-1}

where U(t,s) is the solution of the Cauchy problem

 \frac{d}{dt} U(t,s) = P(t) U(t,s), \qquad U(s,s) = I,

where I denotes the identity matrix. Note that if L(t) is self-adjoint and P(t) is skew-adjoint, then U(t,s) will be unitary.

In other words, to solve the eigenvalue problem Lψ = λψ at time t, it is possible to solve the same problem at time 0 where L is generally known better, and to propagate the solution with the following formulas:

\lambda(t)=\lambda(0) (no change in spectrum)
\frac{\partial \psi}{\partial t}=P \psi.

Link with the inverse scattering method[edit]

The above property is the basis for the inverse scattering method. In this method, L and P act on a functional space (thus ψ = ψ(t,x)), and depend on an unknown function u(t,x) which is to be determined. It is generally assumed that u(0,x) is known, and that P does not depend on u in the scattering region where \Vert x \Vert\to \infty. The method then takes the following form:

  1. Compute the spectrum of L(0), giving \lambda and \psi(0,x),
  2. In the scattering region where P is known, propagate \psi in time by using \frac{\partial \psi}{\partial t}(t,x)=P \psi(t,x) with initial condition \psi(0,x),
  3. Knowing \psi in the scattering region, compute L(t) and/or u(t,x).

Example[edit]

The Korteweg–de Vries equation is

u_t=6uu_x-u_{xxx}.\,

It can be reformulated as the Lax equation

L_t=[P,L]\,

with

L=-\partial_{x}^2+u\, (a Sturm–Liouville operator)
P= -4\partial_{x}^3+3(u\partial_{x}+\partial_{x} u)\,

where all derivatives act on all objects to the right. This accounts for the infinite number of first integrals of the KdV equation.

Equations with a Lax pair[edit]

Further examples of systems of equations that can be formulated as a Lax pair include:

References[edit]

  • Lax, P. (1968), Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Applied Math. 21 (5): 467–490, doi:10.1002/cpa.3160210503  archive
  • P. Lax and R.S. Phillips, Scattering Theory for Automorphic Functions, (1976) Princeton University Press.