We prove that the Korteweg-de Vries initial-value problem is globally well-posed in H−3/4(R) and the modified Korteweg-de Vries initial-value problem is globally well-posed in H1/4(R). The new ingredient is that we use directly the contraction principle to prove local well-posedness for KdV equation at s = −3/4 by constructing some special resolution spaces in order to avoid some ’logarithmic d...

A new set of modified Legendre rational functions which are mutually orthogonal in L2(0,+∞) is introduced. Various projection and interpolation results using the modified Legendre rational functions are established. These results form the mathematical foundation of related spectral and pseudospectral methods for solving partial differential equations on the half line. A spectral scheme using th...

Time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system is studied. Concerning linearized problem, with diffusion wave property for an initial data derived. As application, time nonlinear problem given. In contrast Navier-Stokes system, linear regularities are lower and independent order derivative owing smoothing effect from Korteweg tensor. Furthermore, obtained hav...

In this study, we aim to construct a traveling wave solution for nonlinear partial differential equations. In this regards, a cosine-function method is used to find and generate the exact solutions for three different types of nonlinear partial differential equations such as general regularized long wave equation (GRLW), general Korteweg-de Vries equation (GKDV) and general equal width wave equ...

We describe new classes of nonlinear Galilean–invariant equations of Burgers and Korteweg–de Vries type and study symmetry properties of these equations.

We study local and global well posedness of the k-generalized Korteweg-de Vries equation in weighted Sobolev spaces Hs(R) ∩ L2(|x|2rdx).

We show that the smooth traveling waves of the Camassa-Holm equation naturally correspond to traveling waves of the Korteweg-de Vries equation.

We show that the well-known order reduction phenomenon affecting implicit Runge-Kutta methods does not occur when approximating periodic solutions of the Korteweg-de Vries equation.

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.