Sevdzhan Hakkaev

Our particular interest is in the periodic traveling waves, which we construct and study in detail. The question of the spectral stability of spatially periodic waves for the Boussinesq and symmetric regularized long-wave equations with fifth-order nonlinearity are studied. The simultaneous consideration of the two equations is justified, because they have many similarities. However, they have certain differences as well. The main result of this paper is a theorem which establishes spectral stability (instability) of the found periodic waves for the studied equations.

In the seminal work of Benjamin (1974 Nonlinear Wave Motion (American Mathematical Society)), in the late 70s, he has derived the ubiquitous Benjamin model, which is a reduced model in the theory of water waves. Notably, it contains two parameters in its dispersion part and under some special circumstances, it turns into the celebrated KdV or the Benjamin–Ono equation, During the 90s, there was renewed interest in it. Benjamin (1992 J. Fluid Mech. 245 401–11; 1996 Phil. Trans. R. Soc. A 354 1775–806) studied the problem for existence of solitary waves, followed by works of Bona–Chen (1998 Adv. Differ. Equ. 3 51–84), Albert–Bona–Restrepo (1999 SIAM J. Appl. Math. 59 2139–61), Pava (1999 J. Differ. Equ. 152 136–59), who have showed the existence of travelling waves, mostly by variational, but also bifurcation methods. Some results about the stability became available, but unfortunately, those were restricted to either small waves or Benjamin model, close to a distinguished (i.e. KdV or BO) limit. Quite recently, in 2024 (arXiv:2404.04711 [math.AP]), Abdallah et al, proved existence, orbital stability and uniqueness results for these waves, but only for large values of c γ 2 ≫ 1 . In this article, we present an alternative constrained maximization procedure for the construction of these waves, for the full range of the parameters, which allows us to ascertain their spectral stability. Moreover, we extend this construction to all L ² subcritical cases (i.e. power nonlinearities ( | u | p − 2 u ) x , 2 < p ⩽ 6 ). Finally, we propose a different procedure, based on a specific form of the Sobolev embedding inequality, which works for all powers 2 < p < ∞ , but produces some unstable waves, for large p. Some open questions and a conjecture regarding this last result are proposed for further investigation.

We study the periodic Schrödinger-Korteweg de Vries system. We describe the two-parametetric family of 2 T 2T periodic traveling waves of dnoidal type. The main objective of the paper is to establish their spectral stability with respect to co-periodic perturbations. In the limit T → ∞ T\to \infty , we recover the results of Chen, [J. Partial Differential Equations 12 (1999), pp. 11–25] for the stability of the soliton solutions of this system on the real line.

This paper is concerned with the stability of periodic traveling waves of dnoidal type, of the Zakharov system. This problem was considered in a study of Angulo and Brango [Nonlinearity 24, 2913 (2011)]. In particular, it was shown that under a technical condition on the perturbation, such waves are orbitally stable, with respect to perturbations of the same period. Our main result fills up the gap created by the aforementioned technical condition. More precisely, we show that for all natural values of the parameters, the periodic dnoidal waves are spectrally stable.