Existence and Orbital Stability of Cnoidal Waves for a 1D Boussinesq Equation

Abstract

We will study the existence and stability of periodic travelling-wave solutions of the nonlinear one-dimensional Boussinesq-type equationPeriodic travelling-wave solutions with an arbitrary fundamental period T 0 will be built by using Jacobian elliptic functions. Stability (orbital) of these solutions by periodic disturbances with period T 0 will be a consequence of the general stability criteria given by M. Grillakis, J. Shatah, and W. Strauss. A complete study of the periodic eigenvalue problem associated to the Lame equation is set up.

Linear Stability of Solitary Waves for the One‐Dimensional Benney–Luke and Klein–Gordon Equations

“…In the last two decades, there has been considerable research on model water wave equations and the stability of their solitary waves. Among them, the Boussinesq-type models such as [1], [2], [3], and [4] describe small amplitude long waves in water of finite length. In this paper, we will study the one-dimensional Benney-Luke equation and the one-dimensional Klein-Gordon equation.…”

“…where u is a complex-valued function. We are interested in the stability of traveling standing wave solutions of (1), which are in the form e iωt e iq(x−ct) ϕ ω,c (x − ct), (2) for some real parameters ω, c, q and a real-valued function ϕ, which depends on these parameters. For studies of other special solutions for the Klein-Gordon equation in the solitary or periodic case see for example [8][9][10][11].…”

“…For studies of other special solutions for the Klein-Gordon equation in the solitary or periodic case see for example [8][9][10][11]. Plugging in the ansatz (2) in (1) yields the ODE (c 2 − 1)ϕ + 2i(c(ω + qc) − q)ϕ + (q 2 − (ω + qc) 2 + 1)ϕ −ϕ p = 0, x ∈ R 1 .…”

“…Let ω 2 + c 2 < 1. For p ≤ 5, the traveling standing waves for the Klein-Gordon equation (1) are stable if and only if…”

Solitons, Shock Waves, Conservation Laws and Bifurcation Analysis of Boussinesq Equation with Power Law Nonlinearity and Dual Dispersion

“…The dynamics of shallow water waves along ocean and sea shores are modeled by several forms of nonlinear evolution equations (NLEEs) [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. A few of the well-known models are Korteweg-de Vries (KdV) equation, Peregrine equation, Benjamin-Bona-Mahoney equation.…”