nLab
Lax equation

Lax equation is used in integrable systems; namely some systems are equivalent to the Lax equation.

Lax equation is a linear ordinary differential equation of the form

dLdt=[M,L] \frac{d L}{d t} = [M, L]

for n×nn\times n-matrix-valued function L=L(t)L = L(t), where MM is also a n×nn\times n matrix. The pair (L,M)(L,M) is also called a Lax pair.

The Lax equation is the compatibility condition for the system

λψ=Lψ \lambda \psi = L \psi
dψdt=Mψ \frac{d \psi}{d t} = M\psi

where ψ=ψ(t)\psi = \psi(t) is a vector function which is an eigenvector for LL with eigenvalue λ\lambda. To see this make a derivative of LψL\psi and use the Leibniz rule.

MLψ=Mλψ=λMψ=λdψdt=d(λψ)dt=d(Lψ)dt=dLdtψ+Ldψdt=dLdtψ+LMψ M L \psi = M \lambda \psi = \lambda M \psi = \lambda \frac{d \psi}{d t} = \frac{d (\lambda\psi)}{d t} = \frac{d (L\psi)}{d t} = \frac{d L}{d t} \psi + L \frac{d \psi}{d t} = \frac{d L}{d t} \psi + L M\psi
  • Peter Lax, Integrals of nonlinear equation of evolution and solitary waves, Commun. on Pure and Applied Mathematics 21:5, 467–490, 1968 doi
Revised on October 15, 2012 21:47:23 by Zoran Škoda (193.198.162.13)