We solve the Cauchy problem for the Korteweg–de Vries equation with initial conditions which are steplike Schwartz-type perturbations of finitegap potentials under the assumption that the mutual spectral bands either coincide or are disjoint.

We solve the Cauchy problem for the modified Korteweg–de Vries equation with steplike quasi-periodic, finite-gap initial conditions under the assumption that the perturbations have a given number of derivatives and moments finite.

The goal of this note is to construct a class of traveling solitary wave solutions for the compound Burgers-Korteweg-de Vries equation by means of a hyperbolic ansatz. A computational error in a previous work has been clarified.

In this paper, we prove that there exist no blow-up solutions of the critical generalized Korteweg–de Vries (gKdV) equation with minimal L2-mass, assuming an L2-decay on the right on the initial data.

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,...

solitons are ubiquitous and exist in almost every area from sky to bottom. for solitons to appear, the relevant equation of motion must be nonlinear. in the present study, we deal with the korteweg-devries (kdv), modied korteweg-de vries (mkdv) and regularised longwave (rlw) equations using homotopy perturbation method (hpm). the algorithm makes use of the hpm to determine the initial expansio...

The connection of curves and surfaces in R3 to some nonlinear partial differential equations is very well known in differential geometry [1], [2]. Motion of curves on two dimensional surfaces in differential geometry lead to some integrable nonlinear differential equations such as nonlinear Schrödinger (NLS) equation [3], Korteweg de Vries (KdV) and modified Korteweg de Vries (mKdV) equations [...

The Ostrovsky equation is a modification of the Korteweg-de Vries equation which takes account of the effects of background rotation. It is well known that then the usual Korteweg-de Vries solitary wave decays and is replaced by radiating inertia-gravity waves. Here we show through numerical simulations that after a long-time a localized wave packet emerges as a persistent and dominant feature....

Purely dispersive equations as the Korteweg-de Vries and the nonlinear Schrödinger equation in the limit of small dispersion have solutions to Cauchy problems with smooth initial data which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blowup. Fourth order time-stepping in combination with spectral methods is benefic...

We consider the (KdV)/(KP-I) asymptotic regime for the Nonlinear Schrödinger Equation with a general nonlinearity. In a previous work, we have proved the convergence to the Korteweg-de Vries equation (in dimension 1) and to the Kadomtsev-Petviashvili equation (in higher dimensions) by a compactness argument. We propose a weakly transverse Boussinesq type system formally equivalent to the (KdV)/...