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 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.     

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.     

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...     

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.     

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...     

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.     

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.     

Bäcklund transformation,superposition formulae and N-soliton solutions for the perturbed Korteweg–de Vries equation

With symbolic computation, under investigation in this paper is the perturbed Korteweg–de Vries equation for the nonlocal solitary waves and arrays of wave crests. Via the Hirota method, the bilinear form, Bäcklund transformation and superposition formulae are obtained. N-soliton solutions in terms of the Wronskian are constructed. Asymptotic analysis is used to analyze the collision dynamics, and figures are plotted to illustrate the influence of the perturbation. We find that the perturbation affects the propagation velocities of the solitons, but does not affect the amplitudes and widths of the solitons. Besides, the solitonic collisions turn out to be elastic.     

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.     

Polynomial dynamical systems and the Korteweg—de Vries equation

We explicitly construct polynomial vector fields L_k, k = 0, 1, 2, 3, 4, 6, on the complex linear space C⁶ with coordinates X = (x₂, x₃, x₄) and Z = (z₄, z₅, z₆). The fields L_k are linearly independent outside their discriminant variety Δ ? C⁶ and are tangent to this variety. We describe a polynomial Lie algebra of the fields L_k and the structure of the polynomial ring C[X,Z] as a graded module with two generators x₂ and z₄ over this algebra. The fields L₁ and L₃ commute. Any polynomial P(X,Z) ∈ C[X,Z] determines a hyperelliptic function P(X,Z)(u₁, u₃) of genus 2, where u₁ and u₃ are the coordinates of trajectories of the fields L₁ and L₃. The function 2x₂(u₁, u₃) is a two-zone solution of the Korteweg–de Vries hierarchy, and ?z₄(u₁, u₃)/?u₁ = ?x₂(u₁, u₃)/?u₃.     

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.