On the Stability of the Periodic Waves for the Benney System

Abstract

We analyze the Benney model for interaction of short and long waves in resonant water wave interactions. Our particular interest is in the periodic traveling waves, which we construct and study in detail. The main results are that, for all natural values of the parameters, the periodic dnoidal waves are spectrally stable with respect to perturbations of the same period. For another natural set of parameters, we construct the snoidal waves, which exhibit instabilities, in the same setup. Our results are the first instability results in this context. On the other hand, the spectral stability established herein improves significantly on the work of Angulo, Corcho, and Hakkaev [Adv. Difference Equ., 16 (2011), pp. 523--550], which established stability of the dnoidal waves, on a subset of parameter space, by relying on the Grillakis--Shatah theory. Our approach, which turns out to give definite answer for the entire domain of parameters, relies on the instability index theory, as developed by Kapitula, Kevrekidis, and Sandstede [Phys. D, 3--4 (2004), pp. 263--282]; Kapitula, Kevrekidis, and Sandstede [Phys. D, 195 (2004), no. 3--4, 263--282], Phys. D, 201 (2005), pp. 199--201]; Lin and Zeng [Instability, Index Theorem, and Exponential Trichotomy for Linear Hamiltonian PDEs, 2021]; and Pelinovsky [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), pp. 783--812]. Interestingly, end even though the linearized operators are explicit, our spectral analysis requires subtle and detailed analysis of matrix Schrödinger operators in the periodic context, which support some interesting features.