Solitons in a system of coupled Korteweg-de Vries equations (original) (raw)
Related papers
“Leapfrogging” solitons in a system of coupled KdV equations
Wave Motion, 1987
A system of two KdV equations coupled by small linear dispersive terms is considered. This system describes, for example, resonant interaction of two transverse gravity internal wave modes in a shallow stratified liquid. In the framework of an approach based on Hamilton's equations of motion, evolution equations for parameters of two solitons belonging to different wave modes are obtained in the adiabatic approximation. It is demonstrated that when the solitons' velocities are sufficiently close, the solitons may form a breather-like oscillatory bound state, which provides a natural explanation for recent numerical experiments demonstrating "leapfrogging" motion of the two solitons. The frequency and the maximum amplitude of the "breather" 's internal oscillations are obtained. For the case when the relative velocity of the solitons is not small, perturbationinduced phase shifts of the two colliding free solitons are calculated. Then emission of radiation (small-amplitude quasilinear waves) by an oscillating "breather," also detected in the numerical experiments, is investigated in the framework of the perturbation theory based on the inverse scattering transform. The intensity of the emission is calculated. Radiative effects accompanying collision of the free solitons are also investigated.
Dispersion management for solitons in a Korteweg–de Vries system
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2002
The existence of "dispersion-managed solitons," i.e., stable pulsating solitary-wave solutions to the nonlinear Schrodinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics. Our purpose here is to investigate whether similar structures exist for other well-known nonlinear wave models. Hence, here we consider as a basic model the variable-coefficient Korteweg-de Vries equation; this has the form of a Korteweg-de Vries equation with a periodically varying third-order dispersion coefficient, that can take both positive and negative values. More generally, this model may be extended to include fifth-order dispersion. Such models may describe, for instance, periodically modulated waveguides for long gravity-capillary waves. We develop an analytical approximation for solitary waves in the weakly nonlinear case, from which it is possible to obtain a reduction to a relatively simple integral equation, which is readily solved numerically. Then, we describe some systematic direct simulations of the full equation, which use the soliton shape produced by the integral equation as an initial condition. These simulations reveal regions of stable and unstable pulsating solitary waves in the corresponding parametric space. Finally, we consider the effects of fifth-order dispersion. (c) 2002 American Institute of Physics.
On the generation of solitons and breathers in the modified Korteweg–de Vries equation
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2000
We consider the evolution of an initial disturbance described by the modified Korteweg-de Vries equation with a positive coefficient of the cubic nonlinear term, so that it can support solitons. Our primary aim is to determine the circumstances which can lead to the formation of solitons and/or breathers. We use the associated scattering problem and determine the discrete spectrum, where real eigenvalues describe solitons and complex eigenvalues describe breathers. For analytical convenience we consider various piecewise-constant initial conditions. We show how complex eigenvalues may be generated by bifurcation from either the real axis, or the imaginary axis; in the former case the bifurcation occurs as the unfolding of a double real eigenvalue. A bifurcation from the real axis describes the transition of a soliton pair with opposite polarities into a breather, while the bifurcation from the imaginary axis describes the generation of a breather from the continuous spectrum. Within the class of initial conditions we consider, a disturbance of one polarity, either positive or negative, will only generate solitons, and the number of solitons depends on the total mass. On the other hand, an initial disturbance with both polarities and very small mass will favor the generation of breathers, and the number of breathers then depends on the total energy. Direct numerical simulations of the modified Korteweg-de Vries equation confirms the analytical results, and show in detail the formation of solitons, breathers, and quasistationary coupled soliton pairs. Being based on spectral theory, our analytical results apply to the entire hierarchy of evolution equations connected with the same eigenvalue problem.
Theoretical and Mathematical Physics, 1999
We investigate the interaction process for two solitons with close amplitudes under a smaU perturbation. The leading term of the formal asymptotic solution is found as the sum of two solitons with slowly varying parameters. The equations of slow variations are derived for the soliton phase shifts. The effects related to the interaction between the perturbed solitons can compensate the velocity difference in some conditions, which can result in the formation of the so-called quasi-stationary soliton pair.
Soliton interaction with external forcing within the Korteweg–de Vries equation
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2019
We revise the solutions of the forced Korteweg–de Vries equation describing a resonant interaction of a solitary wave with external pulse-type perturbations. In contrast to previous work where only the limiting cases of a very narrow forcing in comparison with the initial soliton or a very narrow soliton in comparison with the width of external perturbation were studied, we consider here an arbitrary relationship between the widths of soliton and external perturbation of a relatively small amplitude. In many particular cases, exact solutions of the forced Korteweg–de Vries equation can be obtained for the specific forcings of arbitrary amplitude. We use the earlier developed asymptotic method to derive an approximate set of equations up to the second-order on a small parameter characterising the amplitude of external force. The analysis of exact solutions of the derived equations is presented and illustrated graphically. It is shown that the theoretical outcomes obtained by the asym...
Gap-soliton hunt in a coupled Korteweg-de Vries system
1995
We report results of systematic numerical simulations of a system of two linearly coupled Kotteweg-de Vries equations with opposite signs of the dispersion coefficients, in which existence of a new type of gap soliton with decaying oscillatory tails has been recently predicted by means of asymptotic analysis. We demonstrate that stable solitary waves of this type indeed exist in this system in a form close to that predicted analytically, while the usual Korteweg-de Vries solitary waves quickly decay into radiation. Obtaining the new solitary waves requires preparation of an initial state well-fitted to the analytically predicted wave form. We also demonstrate that the solitary waves do not emerge in a relatively narrow vicinity of a special parametric point, where the asymptotic analysis predicts a singularity.
Coupled solitons of intense high-frequency and low-frequency waves in Zakharov-type systems
Chaos (Woodbury, N.Y.), 2016
One-parameter families of exact two-component solitary-wave solutions for interacting high-frequency (HF) and low-frequency (LF) waves are found in the framework of Zakharov-type models, which couple the nonlinear Schrödinger equation for intense HF waves to the Boussinesq (Bq) or Korteweg-de Vries (KdV) equation for the LF component through quadratic terms. The systems apply, in particular, to the interaction of surface (HF) and internal (LF) waves in stratified fluids. These solutions are two-component generalizations of the single-component Bq and KdV solitons. Perturbed dynamics and stability of the solitary waves are studied in detail by means of analytical and numerical methods. Essentially, they are stable against separation of the HF and LF components if the latter one is shaped as a potential well acting on the HF field, and unstable, against splitting of the two components, with a barrier-shaped LF one. Collisions between the solitary waves are studied by means of direct s...
Analytical and Numerical Computations of Multi-Solitons in the Korteweg-de Vries (KdV) Equation
Applied Mathematics
In this paper, an analytical and numerical computation of multi-solitons in Korteweg-de Vries (KdV) equation is presented. The KdV equation, which is classic of all model equations of nonlinear waves in the soliton phenomena, is described. In the analytical computation, the multi-solitons in KdV equation are computed symbolically using computer symbolic manipulator-Wolfram Mathematica via Hirota method because of the lengthy algebraic computation in the method. For the numerical computation, Crank-Nicolson implicit scheme is used to obtain numerical algorithm for the KdV equation. The simulations of solitons in MATLAB as well as results concerning collision or interactions between solitons are presented. Comparing the analytical and numerical solutions, it is observed that the results are identically equal with little ripples in solitons after a collision in the numerical simulations; however there is no significant effect to cause a change in their properties. This supports the existence of solitons solutions and the theoretical assertion that solitons indeed collide with one another and come out without change of properties or identities.
Soliton-like solutions of higher order wave equations of the Korteweg–de Vries type
Journal of Mathematical Physics, 2002
In this work we study second and third order approximations of water wave equations of the Korteweg-de Vries ͑KdV͒ type. First we derive analytical expressions for solitary wave solutions for some special sets of parameters of the equations. Remarkably enough, in all these approximations, the form of the solitary wave and its amplitude-velocity dependence are identical to the sech 2 formula of the onesoliton solution of the KdV. Next we carry out a detailed numerical study of these solutions using a Fourier pseudospectral method combined with a finite-difference scheme, in parameter regions where soliton-like behavior is observed. In these regions, we find solitary waves which are stable and behave like solitons in the sense that they remain virtually unchanged under time evolution and mutual interaction. In general, these solutions sustain small oscillations in the form of radiation waves ͑trailing the solitary wave͒ and may still be regarded as stable, provided these radiation waves do not exceed a numerical stability threshold. Instability occurs at high enough wave speeds, when these oscillations exceed the stability threshold already at the outset, and manifests itself as a sudden increase of these oscillations followed by a blowup of the wave after relatively short time intervals.
Soliton solution of korteweg-de vries equation
2019
The Korteweg-de Vries (K-dV) equation plays an important role in studying of the propagation of low amplitude water waves in shallow water bodies and the remarkable discovery of soliton solution K-dV equation that leads to solitary waves. The importance of soliton solution one can predict how energy is transported from one part of medium to another and soliton carries energy away from its sources. Soliton solution has become a breakthrough in mechanics, nonlinear analysis and many developments in algebra, analysis, geometry and physics. We present the analytic solution of K-dV equation and then using finite element analysis to predict the soliton behavior in shallow water bodies. The propagation of long water wave equation is close to the soliton solution of our said equation has been investigated in this study. The valid analytical solution for k-dv equation is restricted to time and hence close to the initial position and time as well. Finite element analysis that leads to the sol...