Near-linear evolution in the Korteweg-de-Vries (KdV) equation with periodic boundary conditions is established under the assumption of high frequency initial data. This result is obtained by the method of normal form reduction. Mathematics Subject Classification: 35Q53

Differential equations are very popular mathematical models of real world problems. Not every differential equation has a well behaved solution. For those equations whose well behaved solutions exist, we are interested in how they can be computed. Thus, the computability of the solution operators for different types of nonlinear differential equations becomes one of the most exciting topics in ...

Due to the horizontal variability of oceanic hydrology (density and current stratification) and the variable depth over the continental shelf, internal solitary waves transform as they propagate shorewards into the coastal zone. If the background variability is smooth enough, a solitary wave possesses a soliton-like form with varying amplitude and phase. This stage is studied in detail in the f...

The conventional Lie group approach is extended successfully to give out the group explanation to the new conditional similarity reductions obtained by modifying the Clarkson and Kruskal's (CK's) direct method for the (2+1)-dimensional Korteweg–de Vries (KdV) equation.

In this paper, we study a compound Korteweg-de Vries-Burgers equation with a higher-order nonlinearity. A class of solitary wave solutions is obtained by means of a series expansion.

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

In this letter, applying a series of coordinate transformations, we obtain a new class of solutions of the Korteweg–de Vries–Burgers equation, which arises in the theory of ferroelectricity. © 2005 Elsevier Ltd. All rights reserved.

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 respective spectral bands either coincide or are disjoint.

A proper bilinear form is proposed for the N = 1 supersymmetric modified Korteweg-de Vries equation. The bilinear Bäcklund transformation for this system is constructed. As applications, some solutions are presented for it.

linear control system, 51-67Cauchy problem, 52-54controllability, 55-62Korteweg-de Vries equation, 65—67transport equation, 62-65adjoint operator, 373a d ^ y (definition), 130admissibility condition, 52, see also regularity propertyapproximate controllability, 55, 57, 61Ascoli's theorem, 152, 170attractor, 280averaging, 332-333 backstepping, 334-337, ...