Evolution of perturbed embedded solitons in the general Hamiltonian fifth-order Korteweg–de Vries (KdV) equation is studied. When an embedded soliton is perturbed, it sheds a one-directional continuous-wave radiation. It is shown that the radiation amplitude is not minimal in general. A dynamical equation for velocity of the perturbed embedded soliton is derived. This equation shows that a neut...

In this paper, we explore the travelling wave solutions for some nonlinear partial differential equations by using recently established rational (G' /G)-expansion method. We apply method to combined KdV-mKdV equation, reaction-diffusion equation and coupled Hirota-Satsuma KdV equations. The are expressed hyperbolic functions, trigonometric functions functions. When parameters taken as special v...

The KdV and modified KdV integrable hierarchies are shown to be different descriptions of the same 2D gravitational system – open-closed string theory. Nonperturbative solutions of the multi-critical unitary matrix models map to nonsingular solutions of the ‘renormalisation group’ equation for the string susceptibility, [P̃ , Q] =Q . We also demonstrate that the large N solutions of unitary matr...

Statistical properties of the noisy Burgers and KdV-Burgers equations are numerically studied. It is found that shock-like structures appear in the timeaveraged patterns for the case of stepwise fixed boundary conditions. Our results show that the shock structure for the noisy KdV-Burgers equation has an oscillating tail, even for the time averaged pattern. Also, we find that the width of the s...

Geometric discretizations that preserve certain Hamiltonian structures at the discrete level has been proven to enhance the accuracy of numerical schemes. In particular, numerous symplectic and multi-symplectic schemes have been proposed to solve numerically the celebrated Korteweg-de Vries (KdV) equation. In this work, we show that geometrical schemes are as much robust and accurate as Fourier...

It is shown here that the possibility of the existence of new (2 + 1) dimensional integrable equations of the modified KdV equation using the Painlevé test.

Nonlinear science is a fundamental frontier that includes research in the common properties of nonlinear phenomena. This article devoted for study new extended hyperbolic function method (EHFM) to attain exact soliton solutions perturbed Boussinesq equation (PBE) and KdV–Caudery–Dodd–Gibbon (KdV-CDG) equation. We can claim these are not previously presented literature. In addition, 2d 3d graphi...

We prove in this paper the stability and asymptotic stability in H of a decoupled sum of N solitons for the subcritical generalized KdV equations ut + (uxx + u )x = 0 (1 < p < 5). The proof of the stability result is based on energy arguments and monotonicity of local L norm. Note that the result is new even for p = 2 (the KdV equation). The asymptotic stability result then follows directly fro...

This article presents a numerical investigation of overtaking collisions between two solitary waves in the context Schamel equation. Our study reveals different regimes characterized by behavior wave interactions. In certain regimes, maintain well-separated crests consistently over time, while other number local maxima undergoes variations following patterns 2→1→2→1→2 or 2→1→2. These findings d...

Some techniques of bilinearization of the non-autonomous 1+1 dimensional discrete soliton equations are discussed by taking the discrete KdV equation, the discrete Toda lattice equation, and the discrete Lotka-Volterra equation as examples. Casorati determinant solutions to those equations are also constructed explicitly. §