Primitive solutions of the Korteweg–de Vries equation

Theoretical and Mathematical Physics - We survey recent results connected with constructing a new family of solutions of the Korteweg-de Vries equation, which we call primitive solutions. These...     

Blow-up of smooth solutions of the Korteweg–de Vries equation

The paper is devoted both to some initial–boundary value problems and to the Cauchy problem for the KdV equation.     

A simple and robust boundary treatment for the forced Korteweg–de Vries equation

In this paper, we propose a simple and robust numerical method for the forced Korteweg–de Vries (fKdV) equation which models free surface waves of an incompressible and inviscid fluid flow over a bump. The fKdV equation is defined in an infinite domain. However, to solve the equation numerically we must truncate the infinite domain to a bounded domain by introducing an artificial boundary and imposing boundary conditions there. Due to unsuitable artificial boundary conditions, most wave propagation problems have numerical difficulties (e.g., the truncated computational domain must be large enough or the numerical simulation must be terminated before the wave approaches the artificial boundary for the quality of the numerical solution). To solve this boundary problem, we develop an absorbing non-reflecting boundary treatment which uses outward wave velocity. The basic idea of the proposing algorithm is that we first calculate an outward wave velocity from the solutions at the previous and present time steps and then we obtain a solution at the next time step on the artificial boundary by moving the solution at the present time step with the velocity. And then we update solutions at the next time step inside the domain using the calculated solution on the artificial boundary. Numerical experiments with various initial conditions for the KdV and fKdV equations are presented to illustrate the accuracy and efficiency of our method.     

Complexiton solutions of the Korteweg–de Vries equation with self-consistent sources

Complexiton solutions to the Korteweg–de Vires equation with self-consistent sources are presented. The basic technique adopted is the Darboux transformation. The resulting solutions provide evidence that soliton equations with self-consistent sources can have complexiton solutions, in addition to soliton, positon and negaton solutions. This also implies that soliton equations with self-consistent sources possess some kind of analytical characteristics that linear differential equations possess and brings ideas toward classification of exact explicit solutions of nonlinear integrable differential equations.     

On the second-order approximate symmetry classification and optimal systems of subalgebras for a forced Korteweg–de Vries equation

We investigate a forced Korteweg–de Vries (fKdV) equation, $u_{\text{,}t}+{\mathit{cu}}_{\text{,}x}+\alpha {\mathit{uu}}_{\text{,}x}+\beta u_{\text{,}\mathit{xxx}}=\beta F(t)$, which arises in the modelling of tsunami generation by submarine landslides. Approximate symmetries are found for the fKdV equation using the method as proposed by Fushchich and Shtelen [6]. Symmetries are found to second order in the perturbation parameter using the MAPLE symmetry package ASP [11], an add-on to the symmetry package DESOLVII [18]. ASP also allows particular forms of the arbitrary function $F(t)$ to be found which extend the symmetry algebra and hence a full approximate symmetry classification of the fKdV equation is obtained. Optimal systems of one-dimensional subalgebras are also determined. Corresponding approximate invariant solutions to the fKdV equation are then constructed for particular $F(t)$ using DESOLVII routines.     

Random modulation of solitons for the stochastic Korteweg–de Vries equation

We study the asymptotic behavior of the solution of a Korteweg–de Vries equation with an additive noise whose amplitude ε   tends to zero. The noise is white in time and correlated in space and the initial state of the solution is a soliton solution of the unperturbed Korteweg–de Vries equation. We prove that up to times of the order of 1/ε²$1/{\epsilon }^2$, the solution decomposes into the sum of a randomly modulated soliton, and a small remainder, and we derive the equations for the modulation parameters. We prove in addition that the first order part of the remainder converges, as ε tends to zero, to a Gaussian process, which satisfies an additively perturbed linear equation.     

On completely integrable coupled Burgers and coupled Korteweg–de Vries systems

In this work, we study two completely integrable equations, namely, coupled Burgers and Korteweg–de Vries systems. The modified form of Hirota’s bilinear method, established by Hereman, is employed to formally derive multiple-soliton solutions and multiple-singular-soliton solutions for each system. Hirota’s bilinear method is reliable and effective and can also be applied to solve other types of higher-dimensional integrable and non-integrable systems.     

Asymptotic behavior of the generalized Korteweg–de Vries–Burgers equation

Cauchy’s problem for a generalization of the KdV–Burgers equation is considered in Sobolev spaces H¹(\mathbbR){H^1(\mathbb{R})} and H²(\mathbbR){H^2(\mathbb{R})}. We study its local and global solvability and the asymptotic behavior of solutions (in terms of the global attractors). The parabolic regularization technique is used in this paper which allows us to extend the strong regularity properties and estimates of solutions of the fourth order parabolic approximations onto their third order limit—the generalized Korteweg–de Vries–Burgers (KdVB) equation. For initial data in H²(\mathbbR){H^2(\mathbb{R})} we study the notion of viscosity solutions to KdVB, while for the larger H¹(\mathbbR){H^1(\mathbb{R})} phase space we introduce weak solutions to that problem. Finally, thanks to our general assumptions on the nonlinear term f guaranteeing that the global attractor is usually nontrivial (i.e., not reduced to a single stationary solution), we study an upper semicontinuity property of the family of global attractors corresponding to parabolic regularizations when the regularization parameter e{\epsilon} tends to 0⁺ (which corresponds the passage to the KdVB equation).     

Nonautonomous soliton solutions of the modified Korteweg–de Vries–sine-Gordon equation

Multisoliton solutions of the modified Korteweg–de Vries–sine-Gordon (mKdV–SG) equation with time-dependent coefficients are considered. Cases describing changes in the shape of soliton solutions (kinks and breathers) observed in gradual transitions between the mKdV, SG, and mKdV–SG equations are numerically studied.     

Painlevé analysis and exact solutions of the Korteweg–de Vries equation with a source

We consider the Korteweg–de Vries equation with a source. The source depends on the solution as polynomials with constant coefficients. Using the Painlevé test we show that the generalized Korteweg–de Vries equation is not integrable by the inverse scattering transform. However there are some exact solutions of the generalized Korteweg–de Vries equation for two forms of the source. We present these exact solutions.     

The superposition of algebraic solitons for the modified Korteweg–de Vries equation

Many real nonlinear evolution equations exhibiting soliton properties display a special superposition principle, where an infinite array of equally spaced, identical solitons constitutes an exact periodic solution. This arrangement is studied for the modified Korteweg–de Vries equation with positive cubic nonlinearity, which possesses algebraic solitons with nonvanishing far field conditions. An infinite sum of equally spaced, identical algebraic pulses is evaluated in closed form, and leads to a complex valued solution of the nonlinear evolution equation.     

Solitary wave solution of higher-order Korteweg–de Vries equation

An attempt has been made to obtain exact analytical traveling wave solution or simple wave solution of higher-order Korteweg–de Vries (KdV) equation by using tanh-method or hyperbolic method. The higher-order equation can be derived for magnetized plasmas by using the reductive perturbation technique. It is found that the exact solitary wave solution of higher-order KdV equation is obtained by tanh-method. Using this method, different kinds of nonlinear wave equations can be evaluated. The higher-order nonlinearity and higher-order dispersive effect can be observed from the solutions of the equations. The method is applicable for other nonlinear wave equations.     

A continuous analog of the binary Darboux transformation for the Korteweg–de Vries equation

In the Korteweg–de Vries equation (KdV) context, we put forward a continuous version of the binary Darboux transformation (aka the double commutation method). Our approach is based on the Riemann–Hilbert problem and yields a new explicit formula for perturbation of the negative spectrum of a wide class of step-type potentials without changing the rest of the scattering data. This extends the previously known formulas for inserting/removing finitely many bound states to arbitrary sets of negative spectrum of arbitrary nature. In the KdV context, our method offers same benefits as the classical binary Darboux transformation does.     

Integration of the nonlinear Korteweg–de Vries equation with an additional term

Theoretical and Mathematical Physics - We use the method of the inverse spectral problem to integrate the nonlinear Korteweg–de Vries equation with an additional term in the class of periodic...