Orbital Stability of Solitary Wave Solutions for an Interaction Equation of Short and Long Dispersive Waves

Existence and linearized stability of solitary waves for a quasilinear Benney system

“…Now, in the case where p ≥ 0, the first equation in (6) shows that φ ′′ ∈ L 2 (R), that is, φ ∈ H 2 (R). And again, by differentiating the second equation,…”

“…The mathematical study of system (1), namely the well-posedness of the associated Cauchy Problem or the existence and stability of solitary waves, has been extensively conducted over the years by many authors (see for instance [6], [12], [26], [32], [37], [38] and references therein).…”

A Newton-Type Approach to Approximate Travelling Wave Solutions of a Schrödinger-Benjamin-Ono System

“…We point out that the constant in the SBO system is positive for physical meaning [1]. However, as we mentioned above, in previous works [7,[10][11][12]], the SBO system has been studied from a mathematical point of view, establishing well-posedness and that travelling wave solutions for the SBO system are possible when the parameter takes both signs. This is an interesting mathematical fact, and thus we will conduct some numerical experiments which illustrate the family of travelling wave solutions for both > 0 and < 0.…”

“…For instance, Angulo and Montenegro [7] established the existence of even solitary wave solutions by employing the concentration compactness method (Lions [8,9]). Existence and stability of a new family of solitary waves was established in [10] for system (1) and a coupled Schrödinger-KdV model. On the other hand, when | | ̸ = 1, the nonperiodic initial value problem corresponding to the SBO system has been considered by Bekiranov et al [11], who proved wellposedness in the Sobolev space C (R) × −1/2 R (R), with ≥ 0.…”

Long time behavior for the weakly damped driven long-wave–short-wave resonance equations

“…These results are improved by Bekiranov et al [3,4] which investigate the well-posedness of solutions with low regularity via Bourgain's method. For the solitary waves and their stability, we refer to Angulo and Montenegro [1], Guo and Chen [13], Guo and Pan [15], Laurençot [19], Pava and Montenegro [23], etc.…”