Lax Pair

# Lax pair

to get instant updates about 'Lax Pair' on your MyPage. Meet other similar minded people. Its Free!

X

All Updates

Description:
In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that describe the corresponding differential equations. They 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 [itex]L(t), P(t)[/itex] dependent on time and acting on a fixed Hilbert space, and verifying the Lax's equation:

[itex]frac=[/itex]

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

## 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 [itex]t[/itex] varies.

The core observation is that the matrices [itex]L(t)[/itex] are all similar by virtue of

[itex]L(t)=U(t,s) L(s) U(t,s)^[/itex]

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

[itex] frac U(t,s) = P(t) U(t,s), qquad U(s,s) = I,[/itex]

where I denotes the identity matrix. Note that if L(t) is self-adjoint and P(t) is skew-adjoint, then U(t,s) will be unitary.

In other words, to solve...
Read More

No feeds found

All
Posting your question. Please wait!...

No updates available.
No messages found
Suggested Pages
Tell your friends >
about this page
Create a new Page
for companies, colleges, celebrities or anything you like.Get updates on MyPage.
Create a new Page
Find your friends
Find friends on MyPage from