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 A Lax pair is a pair of matrices or operators dependent on time and acting on a fixed Hilbert space, and satisfying Lax's equation: where is the commutator. Often, as in the example below, depends on in a prescribed way, so this is a nonlinear equation for as a function of . Isospectral property 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 varies. The core observation is that the matrices are all similar by virtue of where is the solution of the Cauchy problem where I denotes the identity matrix. Note that if P(t) is skew-adjoint, 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: (no change in spectrum) Link with the inverse scattering method 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 . The method then takes the following form: Compute the spectrum of , giving and , In the scattering region where is known, propagate in time by using with initial condition , Knowing in the scattering region, compute and/or . Zero-curvature representation Any PDE which admits a Lax pair representation also admits a zero-curvature representation. In fact, the zero-curvature representation is more general and for other integrable PDEs, such as the sine-Gordon equation, the Lax pair refers to matrices that satisfy the zero-curvature equation rather than the Lax equation. Furthermore, the zero-curvature representation makes the link between integrable systems and geometry manifest, culminating in Ward's programme to formulate known integrable systems as solutions to