In this work, we consider the initial-value problem associated with a coupled system of generalized Korteweg–de Vries equations. We present relationship between best constant for Gagliardo–Nirenberg type inequality and criterion existence global solutions in energy space. prove that such is directly related to solitary wave minimal mass, so-called ground state solutions. A characterization stat...

We study the dynamics of solitons as solutions to the perturbed KdV (pKdV) equation ∂tu = −∂x(∂ xu + 3u − bu), where b(x, t) = b0(hx, ht), h 1 is a slowly varying, but not small, potential. We obtain an explicit description of the trajectory of the soliton parameters of scale and position on the dynamically relevant time scale δh−1 log h−1, together with an estimate on the error of size h. In a...

In this paper, we study the orbital stability for a four-parameter family of periodic stationary traveling wave solutions to the generalized Korteweg-de Vries (gKdV) equation ut = uxxx + f(u)x. In particular, we derive sufficient conditions for such a solution to be orbitally stable in terms of the Hessian of the classical action of the corresponding traveling wave ordinary differential equatio...

We provide an accurate description of the long time dynamics solutions generalized Korteweg–De Vries (gKdV) and Benjamin–Ono (gBO) equations on one dimension torus, without external parameters, that are issued from almost any (in probability in density) small smooth initial data. In particular, we prove a long-time stability result Sobolev norm: given large constant r sufficiently parameter $$\...

The linear stability of solitary-wave or front solutions of Hamiltonian evolutionary equations, which are equivariant with respect to a Lie group, is studied. The organizing centre for the analysis is a multisymplectic formulation of Hamiltonian partial differential equations (PDEs) where a distinct symplectic operator is assigned for time and space. This separation of symplectic structures lea...

with general C nonlinearity f . Under an explicit condition on f and c > 0, there exists a solution in the energy space H of (0.1) of the type u(t, x) = Qc(x − x0 − ct), called soliton. In this paper, under general assumptions on f and Qc, we prove that the family of soliton solutions around Qc is asymptotically stable in some local sense in H , i.e. if u(t) is close to Qc (for all t ≥ 0), then...

We consider a higher-dimensional version of the Benjamin-Ono (HBO) equation in 2D setting: , which is -critical, and investigate properties solutions both analytically numerically. For generalized (fractional gKdV) after deriving Pohozaev identities, we obtain nonexistence conditions for solitary wave solutions, then prove uniform bounds energy space or conditional global existence, radiation r...

The family of the KdV equations, the most famous equations embodying both nonlinearity and dispersion, has attracted enormous attention over the years and has served as the model equation for the development of soliton theory. In this paper we present a comparative study between two different methods for solving the general KdV equation, namely the numerical Crank Nicolson method, and the semi-...