Related papers: On connection between the splitting parameters of …
We discuss a new numerical schema for solving the initial value problem for the Korteweg-de Vries equation for large times. Our approach is based upon the Inverse Scattering Transform that reduces the problem to calculating the reflection…
The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\epsilon^2$, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order $\epsilon$. These oscillations are…
For slowly-varying initial data, solutions to the Ablowitz-Ladik system have been proven to converge to solutions of the cubic Schr\"odinger equation. In this paper we show that in the continuum limit, solutions to the Ablowitz-Ladik system…
This paper considers properties of nonlinear waves and solitons of Korteweg-de Vries equation in the presence of external perturbation. For time-periodic hamiltonian perturbation the width of the stochastic layer is calculated. The…
Periodic waves in the fractional Korteweg-de Vries equation have been previously characterized as constrained minimizers of energy subject to fixed momentum and mass. Here we characterize these periodic waves as constrained minimizers of…
We present a novel approximation method that can predict the number of solitons asymptotically appearing under arbitrary rapidly decreasing initial wave packets. The number of solitons can be estimated without integration of the original…
We solve the direct scattering problem for the ultradiscrete Korteweg de Vries (udKdV) equation, over $\mathbb R$ for any potential with compact (finite) support, by explicitly constructing bound state and non-bound state eigenfunctions. We…
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…
Harmonically modulated complex solitary waves which are a generalized type of envelope soliton (herein coined oscillatory solitons) are studied for the two U(1)-invariant integrable generalizations of the modified Korteweg-de Vries…
It is shown that equations of the Korteweg-de Vries hierarchy and their conservation laws can be expressed via the whole powers of an integro-differential operator and functions provided by them.
In the scalar 1D case, conservation laws and Hamilton-Jacobi equations are deeply related. For both, we characterize those profiles that can be attained as solutions at a given positive time corresponding to at least one initial datum.…
The Kadomtsev-Petviashvili (KP) equation describes weakly dispersive and small amplitude waves propagating in a quasi-two dimensional situation. Recently a large variety of exact soliton solutions of the KP equation has been found and…
In the context of the full line Schrodinger equation, we revisit the binary Darboux transformation (double commutation method) which inserts or removes any number of positive eigenvalues embedded into the absolutely continuous spectrum…
We study the generalized Korteweg-DeVries equations derivable from the Lagrangian: $ L(l,p) = \int \left( \frac{1}{2} \varphi_{x} \varphi_{t} - { {(\varphi_{x})^{l}} \over {l(l-1)}} + \alpha(\varphi_{x})^{p} (\varphi_{xx})^{2} \right) dx, $…
Generalizations of the Korteweg-de Vries equation are considered, and some explicit solutions are presented. There are situations where solutions engender the interesting property of being chiral, that is, of having velocity determined in…
Partial differential equations endowed with a Hamiltonian structure, like the Korteweg--de Vries equation and many other more or less classical models, are known to admit rich families of periodic travelling waves. The stability theory for…
A variable-coefficient forced Korteweg-de Vries equation with spacial inhomogeneity is investigated in this paper. Under constraints, this equation is transformed into its bilinear form, and multi-soliton solutions are derived. Effects of…
We derive the equations governing the motion of Kerr solitons in pair waveforms. Recent experiments in microresonators have studied a variety of interaction effects in multisoliton waveforms, including collisions and formation of soliton…
The Korteweg-de Vries (KdV) equation is a non-linear wave equation that has played a fundamental role in diverse branches of mathematical and theoretical physics. In the present paper, we consider its significance to cosmology. It is found…
Soliton gas or soliton turbulence is a subject of intense studies due to its great importance to optics, hydrodynamics, electricity, chemistry, biology and plasma physics. Usually, this term is used for integrable models where solitons…