Periodic solutions of Wick-type stochastic Korteweg–de Vries equations

Nonlinear stochastic partial differential equations have a wide range of applications in science and engineering. Finding exact solutions of the Wick-type stochastic equation will be helpful in the theories and numerical studies of such equations. In this paper, Kudrayshov method together with Hermite transform is implemented to obtain exact solutions of Wick-type stochastic Korteweg–de Vries equation. Further, graphical illustrations in two- and three-dimensional plots of the obtained solutions depending on time and space are also given with white noise functionals.     

Positon and hybrid solutions for the(2+1)-dimensional complex modified Korteweg–de Vries equations

Solving nonlinear partial differential equations have attracted intensive attention in the past few decades. In this paper, the Darboux transformation method is used to derive several positon and hybrid solutions for the(2+1)-dimensional complex modified Korteweg–de Vries equations. Based on the zero seed solution, the positon solution and the hybrid solutions of positon and soliton are constructed. The composition of positons is studied, showing that multi-positons of(2+1)-dimensional equations...     

Formation of freak waves in a soliton gas described by the modified Korteweg–de Vries equation

The nonlinear dynamics of multisoliton, differently polar fields is investigated within the framework of the modified Korteweg–de Vries equation. It is shown that the occurrence of abnormally large waves (freak waves) is possible in similar fields, which is associated with the modulation instability of cnoidal waves. The statistical moments of wave fields are investigated. It is shown that an increase in the coefficient of excess due to the interaction of solitons correlates with an increase in the probability of occurrence of freak waves. It is shown that the nonlinear interaction of differently polar solitons results in variation of the distribution functions of peak characteristics: the fraction of low-amplitude waves decreases, while that of the waves with large amplitudes increases. The dependence of the intensity of the density of the characteristics of the soliton gas is shown.     

Nonlinear structures for extended Korteweg–de Vries equation in multicomponent plasma

Using the fluid hydrodynamic equations of positive and negative ions, as well as q-nonextensive electron density distribution, an extended Korteweg–de Vries (EKdV) equation describing a small but finite amplitude dust ion-acoustic waves (DIAWs) is derived. Extended homogeneous balance method is used to obtain a new class of solutions of the EKdV equation. The effects of different physical parameters on the propagating nonlinear structures and their relevance to particle acceleration in space plasma are reported.     

Dynamics of new higher-order rational soliton solutions of the modified Korteweg–de Vries equation

By virtue of the bilinear method and the KP hierarchy reduction technique, exact explicit rational solutions of the multicomponent Mel’nikov equation and the multicomponent Schrödinger–Boussinesq equation are constructed, which contain multicomponent short waves and single-component long wave. For the multicomponent Mel’nikov equation, the fundamental rational solutions possess two different behaviours: lump and rogue wave. It is shown that the fundamental (simplest) rogue waves are line localised waves which arise from the constant background with a line profile and then disappear into the constant background again. The fundamental line rogue waves can be classified into three: bright, intermediate and dark line rogue waves. Two subclasses of non-fundamental rogue waves, i.e., multirogue waves and higher-order rogue waves are discussed. The multirogue waves describe interaction of several fundamental line rogue waves, in which interesting wave patterns appear in the intermediate time. Higher-order rogue waves exhibit dynamic behaviours that the wave structures start from lump and then retreat back to it. Moreover, by taking the parameter constraints further, general higher-order rogue wave solutions for the multicomponent Schrödinger–Boussinesq system are generated.     

Variational principles for two kinds of extended Korteweg—de Vries equations

Variational principles are constructed using the semi-inverse method for two kinds of extended Korteweg—de Vries (KdV) equations, which can be regarded as simple models of the nonlinear oceanic internal waves and atmospheric long waves, respectively. The obtained variational principles have also been proved to be correct.     

Periodic and chaotic behaviour in a reduced form of the perturbed generalized Korteweg–de Vries and Kadomtsev–Petviashvili equations

The dynamical behaviour of a reduced form of the perturbed generalized Korteweg–de Vries and Kadomtsev–Petviashvili equations (extension of the Korteweg–de Vries equation to two space variables) are studied in this paper. Harmonic solutions of non-resonance and primary resonance are obtained using the perturbation method. Chaotic motion under harmonic excitations is studied using the Melnikov method.A wide range of solutions for the reduced perturbed generalized Korteweg–de Vries equations, in which non-linear phenomena appearing within transition from regular harmonic response (periodic solutions) to chaotic motion, are obtained using the time integration Runge–Kutta method. When chaos is found, it is detected by examining the phase plane, the Poincaré map, the sensitivity solution of the solution to initial conditions, and by calculating the largest Lyapunov exponent.     

Discrete integrable couplings associated with modified Korteweg——de Vries lattice and two hierarchies of discrete soliton equations

A direct way to construct integrable couplings for discrete systems is presented by use of two semi-direct sum Lie algebras. As their applications, the discrete integrable couplings associated with modified Korteweg--de Vries (m-KdV) lattice and two hierarchies of discrete soliton equations are developed. It is also indicated that the study of integrable couplings using semi-direct sums of Lie algebras is an important step towards the complete classification of integrable couplings.     

On the modulation equations and stability of periodic generalized Korteweg–de Vries waves via Bloch decompositions

In this paper, we consider the relation between Evans-function-based approaches to the stability of periodic travelling waves and other theories based on long-wavelength asymptotics together with Bloch wave expansions. In previous work it was shown by rigorous Evans function calculations that the formal slow modulation approximation resulting in the linearized Whitham averaged system accurately describes the spectral stability to long-wavelength perturbations. To clarify the connection between Bloch-wave-based expansions and Evans-function-based approaches, we reproduce this result without reference to the Evans function by using direct Bloch expansion methods and spectral perturbation analysis. One of the novelties of this approach is that we are able to calculate the relevant Bloch waves explicitly for arbitrary finite-amplitude solutions. Furthermore, this approach has the advantage of being applicable in the more general multi-periodic setting where no conveniently computable Evans function has yet been devised.     

From Vlasov–Poisson to Korteweg–de Vries and Zakharov–Kuznetsov

We introduce a long wave scaling for the Vlasov–Poisson equation and derive, in the cold ions limit, the Korteweg–de Vries equation (in 1D) and the Zakharov–Kuznetsov equation (in higher dimensions, in the presence of an external magnetic field). The proofs are based on the relative entropy method.     

Riemann-Hilbert problem for the fifth-order modified Korteweg–de Vries equation with the prescribed initial and boundary values

In this paper, we investigate the fifth-order modified Korteweg–de Vries(mKdV) equation on the half-line via the Fokas unified transformation approach. We show that the solution u(x, t) of the fifth-order m Kd V equation can be represented by the solution of the matrix Riemann-Hilbert problem constructed on the plane of complex spectral parameter θ. The jump matrix L(x, t, θ) has an explicit representation dependent on x, t and it can be represented exactly by the two pairs of spectral functions...     

Long-Time Dynamics of Korteweg–de Vries Solitons Driven by Random Perturbations

This paper deals with the transmission of a soliton in a random medium described by a randomly perturbed Korteweg–de Vries equation. Different kinds of perturbations are addressed, depending on their specific time or position dependences, with or without damping. We derive effective evolution equations for the soliton parameter by applying a perturbation theory of the inverse scattering transform and limit theorems of stochastic calculus. Original results are derived that are very different compared to a randomly perturbed Nonlinear Schrödinger equation. First the emission of a soliton gas is proved to be a very general feature. Second some perturbations are shown to involve a speeding-up of the soliton, instead of the decay that is usually observed in random media.     

Three dimensional Lifshitz black hole and the Korteweg–de Vries equation

We consider a solution of three dimensional New Massive Gravity with a negative cosmological constant and use the AdS/CTF correspondence to inquire about the equivalent two dimensional model at the boundary. We conclude that there should be a close relation of the theory with the Korteweg–de Vries equation.     

On solitary wave solutions of the compound Burgers–Korteweg–de Vries equation

The goal of this note is to construct a class of traveling solitary wave solutions for the compound Burgers–Korteweg–de Vries equation by means of a hyperbolic ansatz. A computational error in a previous work has been clarified.     

Formulation of the Korteweg–de Vries and the Burgers equations expressing mass transports from the generalized Kawasaki–Ohta equation

By generalization of the Kawasaki–Ohta equation representing the interface dynamics, we report formulation of equations, which express mass transports, deterministic and stochastic, for nonlinear lattices. The equations are written characteristically by flow variable representations defined in the Letter. We found that the KdV equation and the Burgers equation, formulated by the flow variables, express mass transports in hydrodynamics and in stochastic processes, respectively. The representations lead to the conclusion that in nonequilibria we should observe a change not in a concentration but in concentration flows.     

Exact solutions of stochastic fractional Korteweg de–Vries equation with conformable derivatives

We deal with the Wick-type stochastic fractional Korteweg de–Vries(KdV) equation with conformable derivatives.With the aid of the Exp-function method, white noise theory, and Hermite transform, we produce a novel set of exact soliton and periodic wave solutions to the fractional KdV equation with conformable derivatives. With the help of inverse Hermite transform, we get stochastic soliton and periodic wave solutions of the Wick-type stochastic fractional KdV equation with conformable derivatives. Eventually, by an application example, we show how the stochastic solutions can be given as Brownian motion functional solutions.     

The description of reflection coefficients of the scattering problems for finding solutions of the Korteweg–de Vries equations

The results of inverse scattering problem associated with the initial-boundary value problem (IBVP) for the Korteweg–de Vries (KdV) equation with dominant surface tension are formulated. The necessary and sufficient conditions for given functions to be the left- and right-reflection coefficients of the scattering problem are established. The time-dependence t, t > 0 of each element of the scattering matrix s(k,t) is found in respective sector of the k-spectral plane by expansion formulas which are constructed from the known initial and boundary conditions of the IBVP. Knowing the right-reflection coefficient calculated from the elements of s(k,t), we solve the Gelfand–Levitan–Marchenko (GLM) equation in the inverse problem. Then the solution of the IBVP is expressible through the solution of the GLM equation. The asymptotic behavior at infinity of time of the solution of the IBVP is shown