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.     

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.     

Asymptotic solutions of the one-dimensional linearized Korteweg–de Vries equation with localized initial data

The Cauchy problem with localized initial data for the linearized Korteweg–de Vries equation is considered. In the case of constant coefficients, exact solutions for the initial function in the form of the Gaussian exponential are constructed. For a fairly arbitrary localized initial function, an asymptotic (with respect to the small localization parameter) solution is constructed as the combination of the Airy function and its derivative. In the limit as the parameter tends to zero, this solution becomes the exactGreen function for the Cauchy problem. Such an asymptotics is also applicable to the case of a discontinuous initial function. For an equation with variable coefficients, the asymptotic solution in a neighborhood of focal points is expressed using special functions. The leading front of the wave and its asymptotics are constructed.     

Rapidly oscillating solutions of a generalized Korteweg–de Vries equation

For a generalized Korteweg–de Vries equation, the existence of families of rapidly oscillating periodic solutions is proved and their asymptotic representation is found. The asymptotics of tori of different dimensions are examined. Formulas for solutions depending on all parameters of the problem are derived.     

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.     

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...     

Numerical computation of the finite-genus solutions of the Korteweg–de Vries equation via Riemann–Hilbert problems

In this letter we describe how to compute the finite-genus solutions of the Korteweg–de Vries equation using a Riemann–Hilbert problem that is satisfied by the Baker–Akhiezer function corresponding to a Schrödinger operator with finite-gap spectrum. The recovery of the corresponding finite-genus solution is performed using the asymptotics of the Baker–Akhiezer function. This method has the benefit that the space and time dependence of the Baker–Akhiezer function appear in an explicit, linear and computable way. We make use of recent advances in the numerical solution of Riemann–Hilbert problems to produce an efficient and uniformly accurate numerical method for computing all finite-genus solutions of the KdV equation.     

A note on solutions of the Korteweg–de Vries hierarchy

olutions of the Korteweg–de Vries hierarchy are discussed. It is shown that results by Wazwaz [Wazwaz AM. Multiple-soliton solutions of the perturbed KdV equation. Commun Nonlinear Sci Simul 2010;15911:3270–73] are the well-known consequences of the full integrability for the Korteweg–de Vries hierarchy.     

Algebro-geometric solutions of the coupled modified Korteweg–de Vries hierarchy

Based on the stationary zero-curvature equation and the Lenard recursion equations, we derive the coupled modified Korteweg–de Vries (cmKdV) hierarchy associated with a 3×3$3\times 3$ matrix spectral problem. Resorting to the Baker–Akhiezer function and the characteristic polynomial of Lax matrix for the cmKdV hierarchy, we introduce a trigonal curve with three infinite points and two algebraic functions carrying the data of the divisor. The asymptotic properties of the Baker–Akhiezer function and the two algebraic functions are studied near three infinite points on the trigonal curve. Algebro-geometric solutions of the cmKdV hierarchy are obtained in terms of the Riemann theta function.     

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).     

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 the non-isospectral Kadomtsev–Petviashvili equation with self-consistent sources

A new procedure called ‘source generation’ is applied to construct non-isospectral soliton equations with self-consistent sources. As results, the non-isospectral Kadomtsev–Petviashvili equation with self-consistent sources (KPESCS) and its Gramm-type determinant solutions are obtained. Furthermore, the non-isospectral Pfaffianized-KP equation with self-consistent sources is constructed. This coupled system can not only be reduced to the non-isospectral Pfaffianized-KP equation, but also reduced to the non-isospectral KPESCS.     

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.     

Nonautonomous soliton solutions for a nonintegrable Korteweg–de Vries equation with variable coefficients by the variational approach

A nonintegrable Korteweg–de Vries equation with variable coefficients is investigated in this paper. Due to the existence of variable coefficients, the equation becomes nonintegrable, which leads to the invalidity of the traditional analytical methods to obtain soliton solutions. In order to overcome this difficulty, the variational approach is employed in this paper. The variational principle corresponding to this nonintegrable equation is established. Based on that, the first- and second-order nonautonomous soliton solutions are derived. We note that the obtained solutions can be degenerated to the integrable cases. Properties of the nonautonomous solitons and influence of the variable coefficients are discussed.     

Soliton,rational and special solutions of the Korteweg–de Vries hierarchy

Soliton, rational and special solutions for any member of the Korteweg–de Vries hierarchy are presented. It is shown that the soliton solutions are found by means of the Hirota formulae with change of the dispersion relation, special solutions for the Korteweg–de Vries hierarchy are expressed via the transcendents of the first and the second Painlevé hierarchies and rational solutions of members for the Korteveg–de Vries hierarchy can be presented using the Yablonski–Vorobév polynomials.