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.
A Lax pair is a pair of matrices or operators
L(t),P(t)
| dL | |
| dt |
=[P,L],
where
[P,L]=PL-LP
P
L
L
t
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
The core observation is that the matrices
L(t)
L(t)=U(t,s)L(s)U(t,s)-1,
where
U(t,s)
| d | |
| dt |
U(t,s)=P(t)U(t,s), U(s,s)=I,
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:
λ(t)=λ(0)
| \partial\psi | |
| \partialt |
=P\psi.
See also: Invariants of tensors. The result can also be shown using the invariants
\operatorname{tr}(Ln)
n
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
\|x\|\toinfty.
L(0)
λ
\psi(0,x).
P
\psi
| \partial\psi | |
| \partialt |
(t,x)=P\psi(t,x)
\psi(0,x).
\psi
L(t)
u(t,x).
If the Lax matrix additionally depends on a complex parameter
z
C2
w,z.
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 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 the anti-self-dual Yang–Mills (ASDYM) equations.
The zero-curvature equations are described by a pair of matrix-valued functions
Ax(x,t),At(x,t),
(x,t)
\varphi(x,t)
F\mu\nu=[\partial\mu-A\mu,\partial\nu-A\nu]=-\partial\muA\nu+\partial\nuA\mu+[A\mu,A\nu]
For an eigensolution to the Lax operator
L
λ
The Lax pair
(L,P)
(Ax,At)
(Ax,At)
ut=6uux-uxxx
Lt=[P,L]
L=
| 2 | |
| -\partial | |
| x |
+u
P=
| 3 | |
| -4\partial | |
| x |
+6u\partialx+3ux,
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
\vec{h}
\begin{align} L&=\begin{pmatrix} g1+h2&g2+h1&g3&h3\\ g2+h1&-g1+h2&h3&-g3\\ g3&h3&-g1-h2&g2-h1\\ h3&-g3&g2-h1&g1+h2\\ \end{pmatrix}λ-1\\ &+\begin{pmatrix} 0&0&-l2&-l1\\ 0&0&l1&-l2\\ l2&-l1&-2λ&-2l3\\ l1&l2&2l3&2λ\\ \end{pmatrix}, \\ P&=
| -1 | |
| 2 |
\begin{pmatrix} 0&-2l3&l2&l1\\ 2l3&0&-l1&l2\\ -l2&l1&2λ&2l3+\gamma\\ -l1&-l2&-2l3&-2λ\\ \end{pmatrix}. \end{align}
In the Heisenberg picture of quantum mechanics, an observable without explicit time dependence satisfies
| d | |
| dt |
A(t)=
| i | |
| \hbar |
[H,A(t)],
with the Hamiltonian and the reduced 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 Schrödinger 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:
The last is remarkable, as it implies that both the Schwarzschild metric and the Kerr metric can be understood as solitons.