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In [[Mathematics|mathematics]], in the theory of [[Physics:Integrable systems|integrable systems]], a '''Lax pair''' is a pair of time-dependent matrices or [[Operator (mathematics)|operator]]s that satisfy a corresponding [[Differential equation|differential equation]], called the ''Lax equation''. Lax pairs were introduced by [[Biography:Peter Lax|Peter Lax]] to discuss [[Physics:Soliton|soliton]]s in continuous media. The [[Inverse scattering transform|inverse scattering transform]] makes use of the Lax equations to solve such systems.
==Definition==
A Lax pair is a pair of matrices or operators <math>L(t), P(t)</math> dependent on time and acting on a fixed [[Hilbert space]], and satisfying '''Lax's equation''':

:<math>\frac{dL}{dt}=[P,L]</math>

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

==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|isospectral]]'' as <math>t</math> varies.

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

:<math>L(t)=U(t,s) L(s) U(t,s)^{-1}</math>

where <math>U(t,s)</math> is the solution of the [[Cauchy problem]]

:<math> \frac{d}{dt} U(t,s) = P(t) U(t,s), \qquad U(s,s) = I,</math>

where ''I'' denotes the identity matrix. Note that if ''P(t)'' is skew-adjoint, ''U(t,s)'' will be [[Unitary operator|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:
:<math>\lambda(t)=\lambda(0)</math> (no change in spectrum)
:<math>\frac{\partial \psi}{\partial t}=P \psi.</math>

=== Through principal invariants ===
The result can also be shown using the invariants <math>\mathrm{tr}(L^n)</math> for any <math>n</math>. These satisfy 
<math display=block>\frac{d}{dt}\mathrm{tr}(L^n) = 0</math>
due to the Lax equation, and since the [[Characteristic polynomial|characteristic polynomial]] can be written in terms of these traces, the spectrum is preserved by the flow.<ref>{{cite book |last1=Hitchin |first1=N. J. |title=Integrable systems : twistors, loop groups, and Riemann surfaces |date=1999 |publisher=Clarendon Press |location=Oxford |isbn=0198504217}}</ref>

=== 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 <math>\Vert x \Vert\to \infty</math>.
The method then takes the following form:
# Compute the spectrum of <math>L(0)</math>, giving <math>\lambda</math> and <math>\psi(0,x)</math>,
# In the scattering region where <math>P</math> is known, propagate <math>\psi</math> in time by using <math>\frac{\partial \psi}{\partial t}(t,x)=P \psi(t,x)</math> with initial condition <math>\psi(0,x)</math>,
# Knowing <math>\psi</math> in the scattering region, compute <math>L(t)</math> and/or <math>u(t,x)</math>.

=== Spectral curve ===
If the Lax matrix additionally depends on a complex parameter <math>z</math> (as is the case for say sine-Gordon), the equation
<math display = block> \det(wI - L(z)) = 0</math>
defines an [[Algebraic curve|algebraic curve]] in <math>\mathbb{C}^2</math> with coordinates <math>w, z</math>. By the isospectral property, this curve is preserved under time translation. This is the '''spectral curve'''. Such curves appear in the theory of [[Hitchin system]]s.<ref>{{cite book |last1=Hitchin |first1=N. J. |title=Integrable systems : twistors, loop groups, and Riemann surfaces |date=1999 |publisher=Clarendon Press |location=Oxford |isbn=9780198504214}}</ref>

== Zero-curvature representation ==
Any PDE which admits a Lax pair representation also admits a zero-curvature representation.<ref name="SIT">{{cite book |last1=Dunajski |first1=Maciej |title=Solitons, instantons, and twistors |date=2010 |publisher=Oxford University Press |location=Oxford |isbn=978-0-19-857063-9 |pages=54-56}}</ref> In fact, the zero-curvature representation is more general and for other integrable PDEs, such as the [[Physics:Sine-Gordon equation|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 the [[Yang–Mills equations|anti-self dual Yang–Mills]] (ASDYM) equations.

=== Zero-curvature equation ===
The zero-curvature equations are described by a pair of matrix-valued functions <math>A_x(x,t), A_t(x,t)</math>, where the subscripts denote coordinate indices rather than derivatives. Often the <math>(x,t)</math> dependence is through a single scalar function <math>\varphi(x,t)</math> and its derivatives. The zero-curvature equation is then
<math display=block>\partial_t A_x - \partial_x A_t + [A_x, A_t] = 0.</math>
It is so called as it corresponds to the vanishing of the [[Curvature form|curvature tensor]], which in this case is <math>F_{\mu\nu} = [\partial_\mu - A_\mu, \partial_\nu - A_\nu] = -\partial_\mu A_\nu + \partial_\nu A_\mu + [A_\mu, A_\nu]</math>. This differs from the conventional expression by some minus signs, which are ultimately unimportant.

=== Lax pair to zero-curvature ===

For an eigensolution to the Lax operator <math>L</math>, one has
<math display=block>L\psi = \lambda \psi, \psi_t + A\psi = 0.</math>
If we instead enforce these, together with time independence of <math>\lambda</math>, instead the Lax equation arises as a consistency equation for an overdetermined system.

The Lax pair <math>(L, P)</math> can be used to define the connection components <math>(A_x, A_t)</math>. When a PDE admits a zero-curvature representation but not a Lax equation representation, the connection components <math>(A_x, A_t)</math> are referred to as the Lax pair, and the connection as a Lax connection.

== Examples ==
=== Korteweg–de Vries equation ===
The [[Korteweg–de Vries equation]] 
:<math>u_t=6uu_x-u_{xxx}.\,</math>
can be reformulated as the Lax equation
:<math>L_t=[P,L]\,</math>
with
:<math>L=-\partial_{x}^2+u\,</math> (a Sturm–Liouville operator)
:<math>P= -4\partial_{x}^3+6u\partial_{x}+3u_x\,</math>
where all derivatives act on all objects to the right. This accounts for the infinite number of first integrals of the KdV equation.

=== Kovalevskaya top ===
The previous example used an infinite dimensional Hilbert space. Examples are also possible with finite dimensional Hilbert spaces. These include Kovalevskaya top and the generalization to include an electric Field <math>\vec{h}</math>.<ref>{{Cite journal|last1=Bobenko|first1=A. I.|last2=Reyman|first2=A. G.|last3=Semenov-Tian-Shansky|first3=M. A.|date=1989|title=The Kowalewski top 99 years later: a Lax pair, generalizations and explicit solutions|url=https://projecteuclid.org/euclid.cmp/1104178400|journal=Communications in Mathematical Physics|volume=122|issue=2|pages=321–354|issn=0010-3616|bibcode=1989CMaPh.122..321B|doi=10.1007/BF01257419|s2cid=121752578 }}</ref>

<math>\begin{align}
L &= \begin{pmatrix}
g_1 + h_2 & g_2 + h_1 & g_3 & h_3\\
g_2 + h_1 & -g_1 + h_2 & h_3 & -g_3\\
g_3 & h_3 & -g_1 - h_2 & g_2 - h_1\\
h_3 & -g_3 & g_2 - h_1 & g_1 + h_2\\
\end{pmatrix} \lambda^{-1}\\
&+ \begin{pmatrix}
0 & 0 & -l_2 & -l_1\\
0 & 0 & l_1 & -l_2\\
l_2 & -l_1 & -2 \lambda & -2 l_3 \\
l_1 & l_2 & 2 l_3 & 2 \lambda\\
\end{pmatrix}
\\
P &= \frac{-1}{2} \begin{pmatrix}
0 & -2 l_3 & l_2 & l_1\\
2 l_3 & 0 & -l_1 & l_2\\
-l_2 & l_1 & 2 \lambda & 2 l_3 + \gamma\\
-l_1 & -l_2 & -2 l_3 & -2\lambda\\
\end{pmatrix}
\end{align}</math>

=== Heisenberg picture ===

In the [[Physics:Heisenberg picture|Heisenberg picture]] of [[Physics:Quantum mechanics|quantum mechanics]], an [[Physics:Observable|observable]] {{mvar|A}} without explicit time {{mvar|t}} dependence satisfies

<math>\frac{d}{dt}A(t)=\frac{i}{\hbar}[H,A(t)],</math>

with {{mvar|H}} the [[Physics:Hamiltonian (quantum mechanics)|Hamiltonian]] and {{math|''ħ''}} the reduced [[Physics:Planck constant|Planck 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. The [[Physics:Schrödinger picture|Schrödinger picture]] is then interpreted as the alternative expression in terms of isospectral evolution of these observables.

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

* [[Benjamin–Ono equation]]
* One-dimensional cubic non-linear Schrödinger equation
* [[Davey–Stewartson equation|Davey–Stewartson system]]
* Integrable systems with contact Lax pairs<ref>A. Sergyeyev, New integrable (3+1)-dimensional systems and contact geometry, Lett. Math. Phys. 108 (2018), no. 2, 359-376, {{arXiv|1401.2122}} {{doi|10.1007/s11005-017-1013-4}}</ref>
* [[Kadomtsev–Petviashvili equation]]
* [[Korteweg–de Vries equation]]
* [[KdV hierarchy]]
* [[Physics:Marchenko equation|Marchenko equation]]
* Modified Korteweg–de Vries equation
* [[Physics:Sine-Gordon equation|Sine-Gordon equation]]
* [[Toda lattice]]
* [[Physics:Lagrange, Euler, and Kovalevskaya tops|Lagrange, Euler, and Kovalevskaya tops]]
* [[Physics:Belinski–Zakharov transform|Belinski–Zakharov transform]], in general relativity.

The last is remarkable, as it implies that both the [[Astronomy:Schwarzschild metric|Schwarzschild metric]] and the [[Astronomy:Kerr metric|Kerr metric]] can be understood as solitons.

== References ==
{{Reflist}}
* {{citation|first=P.|last= Lax|title=Integrals of nonlinear equations of evolution and solitary waves|journal=Communications on Pure and Applied Mathematics|volume=21|year=1968|pages= 467–490|doi=10.1002/cpa.3160210503|issue=5 }} [https://archive.org/details/integralsofnonli00laxp archive]
* P. Lax and R.S. Phillips, ''Scattering Theory for Automorphic Functions''[https://projecteuclid.org/euclid.bams/1183546232], (1976) Princeton University Press.

{{Functional analysis}}

[[Category:Differential equations]]
[[Category:Automorphic forms]]
[[Category:Spectral theory]]
[[Category:Exactly solvable models]]

{{Sourceattribution|Lax pair}}