-
Article
Numerical Study of the Second-Order Correct Hamiltonian Model for Unidirectional Water Waves
Second-order correct versions of the usual KdV–BBM models for unidirectional propagation of long-crested, surface water waves are considered here. The class of models studied here has a Hamiltonian structure a...
-
Article
Finite Element Methods for a System of Dispersive Equations
The present study is concerned with the numerical approximation of periodic solutions of systems of Korteweg–de Vries type, coupled through their nonlinear terms. We construct, analyze and numerically validate...
-
Article
Stability of Solitary-Wave Solutions of Systems of Dispersive Equations
The present study is concerned with systems $$\begin{aligned} \left\{ \begin{array}{ll} &{} \frac{\partial u}{\partial t} +\frac{\part...
-
Article
Preface
-
Article
Dispersive blow-up II. Schrödinger-type equations, optical and oceanic rogue waves
Addressed here is the occurrence of point singularities which owe to the focusing of short or long waves, a phenomenon labeled dispersive blow-up. The context of this investigation is linear and nonlinear, str...
-
Article
Long Wave Approximations for Water Waves
In this paper, we obtain new nonlinear systems describing the interaction of long water waves in both two and three dimensions. These systems are symmetric and conservative. Rigorous convergence results are pr...
-
Chapter and Conference Paper
Continuous evolution of functions and measures toward fixed points of contraction map**s
Let T be a contraction map** on an appropriate Banach space B(X). Then the evolution equation y t =T y −
-
Article
A model system for strong interaction between internal solitary waves
A mathematical theory is mounted for a complex system of equations derived by Gear and Grimshaw that models the strong interaction of two-dimensional, long, internal gravity waves propagating on neighboring py...
-
Article
Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation
Certain generalizations of one of the classical Boussinesq-type equations, $$u_{tt} = u_{xx} - (u^2 + u_{xx} )_{xx} $$ ...
-
Article
Convergence of periodic wavetrains in the limit of large wavelength
The Korteweg-de Vries equation was originally derived as a model for unidirectional propagation of water waves. This equation possesses a special class of traveling-wave solutions corresponding to surface soli...