A new Hamiltonian amplitude equation governing modulated wave instabilities.(English)Zbl 1112.35354

Summary: A new Hamiltonian amplitude equation, $$i\psi_x+\psi_{tt}+2\sigma|\psi|^2\psi-\epsilon\psi_{xt}=0$$, $$\sigma=\pm 1$$, $$\epsilon\ll 1$$, is introduced. The equation governs certain instabilities of modulated wave-trains, and the addition of the term $$-\epsilon\psi_{xt}$$ overcomes the ill-posedness of the unstable nonlinear Schrödinger equation. This new equation is apparently not integrable, but it is a Hamiltonian analogue of the Kuramoto-Sivashinsky equation, which arises in dissipative systems.

MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations)
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