Questions tagged [lax-pairs]

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Spectrum of a Lax Pair and conservation laws of a PDE

I would like to ask a question that I had asked a few days ago on the site math.stackexchange and I still have not received an answer. If we have a Lax operator, we know that the spectrum of this ...
Niser's user avatar
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5 votes
1 answer
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Recovering the nonlinear Schrödinger equation from its Lax pair

My question is less concerned with the physical aspects of the nonlinear Schrödinger equation and more with the mathematical mechanics of using a Lax pair. I am considering how to recover the ...
Talmsmen's user avatar
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1 answer
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Lax pair of an integrable non-linear PDE

The following is a fourth-order non-linear PDE that passes the Painleve integrability test $$\left(1+x^{2}\right)^{2}u_{xxxx} + 8x\left(1+x^{2}\right)u_{xxx} + 4\left(1+3x^{2}\right)u_{xx}+ t\left(...
Spoilt Milk's user avatar
6 votes
0 answers
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Why does the Lax pair formalism look so similar to the Hamiltonian equations, and what is the significance of this?

If we have a Lax pair for a system, which we'll call operators $L$ and $B$, then the system \begin{align*}L\psi&=\lambda\psi\\ \psi_t&=B\psi\end{align*} has as its integrability condition ...
user41208's user avatar
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1 vote
2 answers
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Integrability - conditions of lax pairs

I'm trying to understand what the conditions are for the Lax pairs for the zero-curvature representation: $$ \partial_t U - \partial_x V + [U,V]=0 $$ where $U=U(x,t,\lambda)$ and $V=V(x,t,\lambda)$ ...
Hunter's user avatar
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3 votes
1 answer
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How to find a Lax Pair for the modified KdV equation

Question I am having trouble trying to find a matrix $T$ so that with $X$, they form a Lax pair for the modified KdV equation $u_t - 6 u^2 u_x + u_{xxx} = 0$. Where $X$ is defined as: $ X = \begin{...
User's user avatar
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7 votes
1 answer
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Lax pairs for linear PDEs

I'm trying to understand the discussion around equation (2.1) of the paper http://www.jstor.org/stable/53053. It says that the linear PDE $M(\partial_x,\partial_y)q=0$ with constant coefficients has ...
Matthew Dodelson's user avatar