In this paper we establish the nonlinear orbital instability of ground state standing waves for a Benney-Roskes/Zakharov-Rubenchik system that models the interaction of low amplitude high frequency waves, acustic type waves in N = 2 and N = 3 spatial directions. For N = 2, we follow M. Weinstein's approach used in the case of the Schrödinger equation, by establishing a virial identity that relates the second variation of a momentum type functional with the energy (Hamiltonian) on a class of solutions for the Benney-Roskes/Zakharov-Rubenchik system. From this identity, it is possible to show that solutions for the Benney-Roskes/Zakharov-Rubenchik system blow up in finite time, in the case that the energy (Hamiltonian) of the initial data is negative, indicating a possible blow-up result for non radial solutions to the Zakharov equations. For N = 3, we establish the instability by using a scaling argument and the existence of invariant regions under the flow due to a concavity argument.
In this paper we show that solutions of the cubic nonlinear Schrödinger equation are asymptotic limit of solutions to the Benney system. Due to the special characteristic of the one-dimensional transport equation same result is obtained for solutions of the onedimensional Zakharov and 1d-Zakharov-Rubenchik systems. Convergence is reached in the topology L 2 (R) × L 2 (R) and with an approximation in the energy space H 1 (R) × L 2 (R). In the case of the Zakharov system this is achieved without the condition ∂tn(x, 0) ∈ Ḣ−1 (R) for the wave component, improving previous results.
We obtain local well-posedness for the one-dimensional Schrödinger-Debye interactions in nonlinear optics in the spaces L 2 × L p , 1 ≤ p < ∞. When p = 1 we show that the local solutions extend globally. In the focusing regime, we consider a family of solutionsx whenever uτ 0 converges to u 0 in H 1 as long as τ tends to 0, where u is the solution of the one-dimensional cubic non-linear Schrödinger equation with initial data u 0 . The convergence of vτ for −|u| 2 in the space L ∞ [0,T ] L 2 x is shown under compatibility conditions of the initial data. For non compatible data we prove convergence except for a corrector term which looks like an initial layer phenomenon.
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