conservation laws for nonlinear schrodinger equations

gylpm · **Joined:** Mon, 12 Jan 2009 14:48:04 UTC **Posts:** 13

nonlinear schrodinger equations
$i\psi_t+\psi_{xx}+\gamma |\psi|^2\psi=0$
what's the invariants? how to prove?
the first three :
$\int_{-\infty}^\infty |\psi|^2 dx$
$\int_{-\infty}^\infty( \psi \overline{\psi}_x-\overline{\psi}\psi_x)dx$
$\int_{-\infty}^\infty (|\psi_x|^2-\frac{\gamma}{2}|\psi|^4)dx$

outermeasure · **Posted:** Thu, 18 Jun 2009 09:56:04 UTC

gylpm wrote:

nonlinear schrodinger equations
$i\psi_t+\psi_{xx}+\gamma |\psi|^2\psi=0$
what's the invariants? how to prove?
the first three :
$\int_{-\infty}^\infty |\psi|^2 dx$
$\int_{-\infty}^\infty( \psi \overline{\psi}_x-\overline{\psi}\psi_x)dx$
$\int_{-\infty}^\infty (|\psi_x|^2-|\psi|^4)dx$

You are looking at Hamiltonian density
$\displaystyle H=\lvert\psi_x\rvert^2+\frac{\gamma}{2}\lvert\psi\rvert^4$
So it is just a matter of finding self-adjoint operators commuting with Hamiltonian, then take integrals.

Of course, if you are asking how to prove the three you gave are invariants, then it is simpler to just differentiate them with respect to t.

gylpm · **Joined:** Mon, 12 Jan 2009 14:48:04 UTC **Posts:** 13

Not only these three invariants, it is an infinit sequence,
Can you tell me where i can get the knowledge about this topic, i am an autodidact

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conservation laws for nonlinear schrodinger equations

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