Inmathematics, in the theory ofintegrable systems, aLax pair is a pair of time-dependent matrices oroperators that satisfy a correspondingdifferential equation, called theLax equation. Lax pairs were introduced byPeter Lax to discusssolitons incontinuous media. Theinverse scattering transform makes use of the Lax equations to solve such systems.

A Lax pair is a pair of matrices or operators${\displaystyle L(t),P(t)}$ dependent on time, acting on a fixedHilbert space, and satisfyingLax's equation:

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

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

It can then be shown that theeigenvalues and more generally thespectrum ofL are independent oft. The matrices/operatorsL are said to beisospectral as${\displaystyle t}$ varies.

The core observation is that the matrices${\displaystyle L(t)}$ are all similar by virtue of

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

where${\displaystyle U(t,s)}$ is the solution of theCauchy problem

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

whereI denotes the identity matrix. Note that ifP(t) isskew-adjoint,U(t, s) will beunitary.

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

${\displaystyle \lambda (t)=\lambda (0)}$ (no change in spectrum),

${\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }t}=P\psi .}$

The result can also be shown using the invariants${\displaystyle \mathrm{tr}(L^n)}$ for any${\displaystyle n}$. These satisfy$${\displaystyle \frac{d}{dt}\mathrm{tr}(L^n)=0}$$due to the Lax equation, and since thecharacteristic polynomial can be written in terms of these traces, the spectrum is preserved by the flow.^[1]

The above property is the basis for the inverse scattering method. In this method,L andP act on afunctional space (thusψ =ψ(t, x)) and depend on an unknown functionu(t, x) which is to be determined. It is generally assumed thatu(0, x) is known, and thatP does not depend onu in the scattering region where${\displaystyle \Vert x\Vert \to \mathrm{\infty }.}$The method then takes the following form:

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

If the Lax matrix additionally depends on a complex parameter${\displaystyle z}$ (as is the case for, say,sine-Gordon), the equation$${\displaystyle det(wI-L(z))=0}$$defines analgebraic curve in${\displaystyle {\mathbb{C}}^2}$ with coordinates${\displaystyle w,z.}$ By the isospectral property, this curve is preserved under time translation. This is thespectral curve. Such curves appear in the theory ofHitchin systems.^[2]

Any PDE which admits a Lax-pair representation also admits a zero-curvature representation.^[3] In fact, the zero-curvature representation is more general and for other integrable PDEs, such as thesine-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 inWard's programme to formulate known integrable systems as solutions to theanti-self-dual Yang–Mills (ASDYM) equations.

The zero-curvature equations are described by a pair of matrix-valued functions${\displaystyle A_x(x,t),A_t(x,t),}$ where the subscripts denote coordinate indices rather than derivatives. Often the${\displaystyle (x,t)}$ dependence is through a single scalar function${\displaystyle \phi (x,t)}$ and its derivatives. The zero-curvature equation is then$${\displaystyle {\mathrm{\partial }}_t{A}_x-{\mathrm{\partial }}_x{A}_t+[A_x,A_t]=0.}$$It is so called as it corresponds to the vanishing of thecurvature tensor, which in this case is${\displaystyle F_{\mu \nu }=[{\mathrm{\partial }}_{\mu }-A_{\mu },{\mathrm{\partial }}_{\nu }-A_{\nu }]=-{\mathrm{\partial }}_{\mu }{A}_{\nu }+{\mathrm{\partial }}_{\nu }{A}_{\mu }+[A_{\mu },A_{\nu }]}$. This differs from the conventional expression by some minus signs, which are ultimately unimportant.

For an eigensolution to the Lax operator${\displaystyle L}$, one has$${\displaystyle L\psi =\lambda \psi ,{\psi }_t+A\psi =0.}$$If we instead enforce these, together with time independence of${\displaystyle \lambda }$, instead the Lax equation arises as a consistency equation for an overdetermined system.

The Lax pair${\displaystyle (L,P)}$ can be used to define the connection components${\displaystyle (A_x,A_t)}$. When a PDE admits a zero-curvature representation but not a Lax equation representation, the connection components${\displaystyle (A_x,A_t)}$ are referred to as the Lax pair, and the connection as a Lax connection.

TheKorteweg–de Vries equation

${\displaystyle u_t=6u{u}_x-u_{xxx}}$

can be reformulated as the Lax equation

${\displaystyle L_t=[P,L]}$

with

${\displaystyle L=-{\mathrm{\partial }}_x^2+u}$ (aSturm–Liouville operator),

${\displaystyle P=-4{\mathrm{\partial }}_x^3+6u{\mathrm{\partial }}_x+3u_x,}$

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

The previous example used an infinite-dimensional Hilbert space. Examples are also possible with finite-dimensional Hilbert spaces. These includeKovalevskaya top and the generalization to include an electric field${\displaystyle \overrightarrow{h}}$.^[4]

In theHeisenberg picture ofquantum mechanics, anobservableA without explicit timet dependence satisfies

${\displaystyle \frac{d}{dt}A(t)=\frac{i}{\hslash }[H,A(t)],}$

withH theHamiltonian andħ the reducedPlanck constant. Aside from a factor, observables (without explicit time dependence) in this picture can thus be seen to form Lax pairs together with the Hamiltonian. TheSchrödinger picture is then interpreted as the alternative expression in terms of isospectral evolution of these observables.

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

• Benjamin–Ono equation
• One-dimensional cubicnon-linear Schrödinger equation
• Davey–Stewartson system
• Integrable systems with contact Lax pairs^[5]
• Kadomtsev–Petviashvili equation
• Korteweg–de Vries equation
• KdV hierarchy
• Marchenko equation
• Modified Korteweg–de Vries equation
• Sine-Gordon equation
• Toda lattice
• Lagrange, Euler, and Kovalevskaya tops
• Belinski–Zakharov transform, in general relativity.

The last is remarkable, as it implies that both theSchwarzschild metric and theKerr metric can be understood as solitons.

• Lax, P. (1968),"Integrals of nonlinear equations of evolution and solitary waves",Communications on Pure and Applied Mathematics,21 (5):467–490,doi:10.1002/cpa.3160210503archive
• P. Lax and R.S. Phillips,Scattering Theory for Automorphic Functions[1], (1976) Princeton University Press.

• Differential equations
• Automorphic forms
• Spectral theory
• Exactly solvable models