We describe a conservative numerical scheme with the property of uniform exponential decay for the critical case of the Generalized Korteweg-de Vrie equation (p = 4), with damping.

Extended shallow water wave equations are derived, using the method of asymptotic expansions, from Euler (or wave) equations. These extended models valid one order beyond usual weakly nonlinear, long approximation, incorporating all appropriate dispersive and nonlinear terms. Specifically, first we derive Korteweg–de Vries (KdV) equation, then proceed with Benjamin–Bona–Mahony Camassa–Holm in (...

An extended Korteweg-de Vries (KdV) equation is derived that describes the evolution and propagation of long interfacial gravity waves in the presence of a strong, space-time varying background. Provision is made in the derivation for a spatially varying lower depth so that some topographic eeects can also be included. The extended KdV model is applied to some simple scenarios in basins of cons...

This class of equations was introduced by Kichenassamy and Olver [16]. It contains in particular the Kawahara equation [14] introduced to model magneto-acoustic waves, the various models derived by Olver [18] for the unidirectional propagation of waves in shallow water when the third order term appearing in the Korteweg-de Vries equation is small, and many other models. See [16] for a discussio...

Robbert Dijkgraaf, Sergei Gukov, Vladimir A. Kazakov, and Cumrun Vafa Institute for Theoretical Physics & Korteweg–de Vries Institute for Mathematics, University of Amsterdam, 1018 XE Amsterdam, The Netherlands Jefferson Physical Laboratory, Harvard University, Cambridge, Massachusetts 02138, USA Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris CEDEX,...

In this paper, we consider the integrable extended complex modified Korteweg–de Vries equation. Based on Darboux transformation, obtain soliton molecules, positon solutions, rational solutions and rogue waves for Further, under standard decomposition, divide into three patterns: fundamental pattern, triangular pattern ring pattern. On basis of define length width discuss effect different parame...

The interaction of a strongly nonlinear solitary wave with an external force is studied using the extended Korteweg-de Vries equation as a model. This equation has several different families of nonlinear wave solutions: solitons, the so-called ”thick” solitons, algebraic solitons, and breathers, depending upon the sign of the cubic nonlinear term. A simple nonlinear dynamical system of the seco...

The initial-value problem for box-like initial disturbances is studied within the framework of an extended Korteweg-de Vries equation with both quadratic and cubic nonlinear terms, also known as the Gardner equation, for the case when the cubic nonlinear coefficient has the same sign as the linear dispersion coefficient. The discrete spectrum of the associated scattering problem is found, which...

Embedded solitons are solitary waves residing inside the continuous spectrum of a wave system. They have been discovered in a wide array of physical situations recently. In this article, we present the first comprehensive theory on the dynamics of embedded solitons and nonlocal solitary waves in the framework of the perturbed fifth-order Korteweg–de Vries (KdV) hierarchy equation. Our method is...