Related papers: Multi-cnoidal Solutions of Korteweg-de Vries Evolu…

We study the nonlocal modified Korteweg-de Vries (mKdV) equations obtained from AKNS scheme by Ablowitz-Musslimani type nonlocal reductions. We first find soliton solutions of the coupled mKdV system by using the Hirota direct method. Then…

We consider a randomly perturbed Korteweg-de Vries equation. The perturbation is a random potential depending both on space and time, with a white noise behavior in time, and a regular, but stationary behavior in space. We investigate the…

We generalize the non-linear one-dimensional equation of a fluid layer for any depth and length as an infinite order differential equation for the steady waves. This equation can be written as a q-differential one, with its general solution…

We apply a multiple-time version of the reductive perturbation method to study long waves as governed by the shallow water wave model equation. As a consequence of the requirement of a secularity-free perturbation theory, we show that the…

We present a fundamental solution to an initial value problem for the KdV-Whitham system in an explicit integral form. Monotonically decreasing initial data with finite number of breaking points are considered. Generating function for the…

By applying the Craig-Wayne-Bourgain (CWB) method, we establish the persistence of periodic solutions to multi-dimensional nonlinear wave equations (NLW) with unbounded perturbation.

We study existence of helical solitons in the vector modified Korteweg-de Vries (mKdV) equations, one of which is integrable, whereas another one is non-integrable. The latter one describes nonlinear waves in various physical systems,…

In this paper, nonlocal symmetries and exact solutions of variable coefficient Korteweg-de Vries (KdV) equation are studied for the first time. Using pseudo-potential, high order nonlocal symmetries of time-dependent coefficient KdV…

The two-dimensional evolution of perturbed long weakly-nonlinear surface plane, ring, and hybrid waves, consisting, to leading order, of a part of a ring and two tangent plane waves, is modelled numerically within the scope of the 2D…

We study one particular asymptotic behaviour of a solution of the fractional modified Korteweg-de Vries equation (also known as the dispersion generalised modified Benjamin-Ono equation): \begin{align}\tag{fmKdV} \partial_t u + \partial_x…

Considering lateral influence from adjacent lane, an improved car-following model is developed in this paper. Then linear and non-linear stability analyses are carried out. The modified Korteweg-de Vries (MKdV) equation is derived with the…

We study a deformation of the defocusing nonlinear Schr\"odinger (NLS) equation, the defocusing Camassa- Holm NLS, hereafter referred to as CH-NLS equation. We use asymptotic multiscale expansion methods to reduce this model to a…

A one-dimensional Bose-Einstein condensate may experience nonlinear periodic modulations known as "cnoidal waves". We argue that such structures represent promising candidates for the study of supersolidity-related phenomena in a…

We study the variable bottom generalized Korteweg-de Vries (bKdV) equation dt u=-dx(dx^2 u+f(u)-b(t,x)u), where f is a nonlinearity and b is a small, bounded and slowly varying function related to the varying depth of a channel of water.…

We study the parameter space of cnoidal waves -- the periodic solitons of the Korteweg-de Vries equation -- from the perspective of Virasoro coadjoint orbits. The monodromy method familiar from inverse scattering implies that many, but not…

We report on the experimental observation of solitons propagating along a torus of fluid. We show that such a periodic system leads to significant differences compared to the classical plane geometry. In particular, we highlight the…

We prove that Mathieu's N=2 supersymmetric Korteweg-de Vries equations with a=1 or a=4 admit Hirota's n-supersoliton solutions, whose nonlinear interaction does not produce any phase shifts. For initial profiles that can not be…

We study the long-time asymptotic behavior of the Cauchy problem for the modified Korteweg - de Vries equation with an initial function of the step type. This function rapidly tends to zero as $x\to+\infty$ and to some positive constant c…

The Korteweg-de Vries (KdV) equation is a fundamental partial differential equation that models wave propagation in shallow water and other dispersive media. Accurately solving the KdV equation is essential for understanding wave dynamics…

Under investigation in this paper is the nonisospectral and variable coefficients modified Kortweg-de Vries (vc-mKdV) equation, which manifests in diverse areas of physics such as fluid dynamics, ion acoustic solitons and plasma mechanics.…