KdV-type Equations Research Articles

The Traveling Wave Solutions of Space-Time Fractional Differential Equation Using Fractional Riccati Expansion Method

In this paper, we firstly give a counterexample to indicate that the chain rule is lack of accuracy. After that, we put forward the fractional Riccati expansion method. No need to use the chain rule, we apply this method to fractional KdV-type and fractional Telegraph equations and obtain the tangent and cotangent functions solutions of these fractional equations for the first time.

A numerical study of the 3-periodic wave solutions to KdV-type equations

In this paper, by using the direct method of calculating periodic wave solutions proposed by Akira Nakamura, we present a numerical process to calculate the 3-periodic wave solutions to several KdV-type equations: the Korteweg–de Vries equation, the Sawada–Koterra equation, the Boussinesq equation, the Ito equation, the Hietarinta equation and the (2+1)-dimensional Kadomtsev–Petviashvili equation. Some detailed numerical examples are given to show the existence of the three-periodic wave solutions numerically.

Physical uniqueness of higher-order Korteweg-de Vries theory for continuously stratified fluids without background shear

The 2nd-order Korteweg-de Vries (KdV) equation and the Gardner (or extended KdV) equation are often used to investigate internal solitary waves, commonly observed in oceans and lakes. However, application of these KdV-type equations for continuously stratified fluids to geophysical problems is hindered by nonuniqueness of the higher-order coefficients and the associated correction functions to the wave fields. This study proposes to reduce arbitrariness of the higher-order KdV theory by considering its uniqueness in the following three physical senses: (i) consistency of the nonlinear higher-order coefficients and correction functions with the corresponding phase speeds, (ii) wavenumber-independence of the vertically integrated available potential energy, and (iii) its positive definiteness. The spectral (or generalized Fourier) approach based on vertical modes in the isopycnal coordinate is shown to enable an alternative derivation of the 2nd-order KdV equation, without encountering nonuniqueness. Comparison with previous theories shows that Parseval’s theorem naturally yields a unique set of special conditions for (ii) and (iii). Hydrostatic fully nonlinear solutions, derived by combining the spectral approach and simple-wave analysis, reveal that both proposed and previous 2nd-order theories satisfy (i), provided that consistent definitions are used for the wave amplitude and the nonlinear correction. This condition reduces the arbitrariness when higher-order KdV-type theories are compared with observations or numerical simulations. The coefficients and correction functions that satisfy (i)-(iii) are given by explicit formulae to 2nd order and by algebraic recurrence relationships to arbitrary order for hydrostatic fully nonlinear and linear fully nonhydrostatic effects.

On Lie symmetry analysis and invariant subspace methods of coupled time fractional partial differential equations

Lie symmetry analysis and invariant subspace methods of differential equations play an important role separately in the study of fractional partial differential equations. The former method helps to derive point symmetries, symmetry algebra and admissible exact solution, while the later one determines admissible invariant subspace as well as to derive exact solution of fractional partial differential equations. In this article, a comparison between Lie symmetry analysis and invariant subspace methods is presented towards deriving exact solution of the following coupled time fractional partial differential equations: (i) system of fractional diffusion equation, (ii) system of fractional KdV type equation, (iii) system of fractional Whitham-Broer-Kaup’s type equation, (iv) system of fractional Boussinesq-Burgers equation and (v) system of fractional generalized Hirota-Satsuma KdV equation.

Analysis of the generalized (2+1)-dimensional Nizhnik–Novikov–Veselov equations with variable coefficients in an inhomogeneous medium

Korteweg-de Vries (KdV)-type equations are seen to describe the shallow-water waves, lattice structures and ion-acoustic waves in plasmas. Hereby, we consider an extension of the KdV-type equations called the generalized (2[Formula: see text]+[Formula: see text]1)-dimensional Nizhnik–Novikov–Veselov equations with variable coefficients in an inhomogeneous medium. Via the Hirota bilinear method and symbolic computation, we derive the bilinear forms, N-soliton solutions and Bäcklund transformation. Effects of the first- and higher-order dispersion terms are investigated. Soliton evolution and interaction are graphically presented and analyzed: Both the propagation velocity and direction of the soliton change when the dispersion terms are time-dependent; The interactions between/among the solitons are elastic, independent of the forms of the coefficients in the equations.

Some Remarks on the Stability Condition of Numerical Scheme of the KdV-type Equation

This paper has employed a comparative study between the numerical scheme and stability condition. Numerical calculations are carried out based on three different numerical schemes, namely the central finite difference, fourier leap-frog, and fourier spectral RK4 schemes. Stability criteria for different numerical schemes are developed for the KdV equation, and numerical examples are put to test to illustrate the accuracy and stability between the solution profile and numerical scheme.

Ion-acoustic dressed solitary structures in two component plasma with Tsallis-nonthermal velocity distribution of electrons

The present investigation deals with the study of small amplitude nonlinear ion-acoustic dressed solitary structures in two component plasma model consisting of cold positive ions and electrons. The electrons are assumed as featuring hybrid q-nonextensive nonthermal distribution also known as Cairns–Tsallis velocity distribution. In the lowest order of potential, the basic set of fluid equations is reduced to the well known Korteweg-de Vries (KdV) equation. By including the contribution of higher order potential, an inhomogeneous KdV-type nonliner equation results that contains fifth order dispersion. The numerical values of the parameters like nonthermal ( $$\alpha$$ ) and nonextensive (q) are according to the range suggested by Williams et al. (Phys Rev E 88:023103, 2013). The finite value of nonthermal parameter results into the formation of rarefactive type solitary structures. A comparison of amplitudes of higher order correction, dressed and KdV solitons has been presented pictorially. The combined effect of nonextensive parameter q and nontermal parameter on the soliton dynamics has also been studied and the results are shown in the form of two and three dimensional profiles.

A GENERALIZED KDV TYPE EQUATION FOR UNEVEN DEPTHS

A Korteweg & de Vries type equation with improved dispersion characteristics for uneven water depths is presented. The new KdV type equation contains mixed dispersion and shoaling terms, which extend its applicable range of relative depths to virtually deep waters. The wave equation also satisfies an important consistency condition that there is an exact agreement between the shoaling rate of the equation itself and the rate obtained from the constancy of energy flux. A finite-difference scheme is devised for simulating several linear and nonlinear cases over varying bathymetry. The performance the new KdV type equation is observed to be quite satisfactory.

Universal dynamical phase diagram of lattice spin models and strongly correlated ultracold atoms in optical lattices

We study semiclassical dynamics of anisotropic Heisenberg models in two and three dimensions. Such models describe lattice spin systems and hard core bosons in optical lattices. We solve numerically Landau–Lifshitz type equations on a lattice and show that in the phase diagram of magnetization and interaction anisotropy, one can identify several distinct regimes of dynamics. These regions can be distinguished based on the character of one dimensional solitonic excitations, and stability of such solitons to transverse modulation. Small amplitude and long wavelength perturbations can be analyzed analytically using mapping of non-linear hydrodynamic equations to KdV type equations. Numerically we find that properties of solitons and dynamics in general remain similar to our analytical results even for large amplitude and short distance inhomogeneities, which allows us to obtain a universal dynamical phase diagram. As a concrete example we study dynamical evolution of the system starting from a state with magnetization step and show that formation of oscillatory regions and their stability to transverse modulation can be understood from the properties of solitons. In regimes unstable to transverse modulation we observe formation of lump type solutions with modulation in all directions. We discuss implications of our results for experiments with ultracold atoms.

Reverse Space-Time Nonlocal Sasa–Satsuma Equation and Its Solutions

The Sasa–Satsuma equation is an integrable high-order nonlinear Schrodinger (NLS) equation, and also is a complex modified KdV-type equation. It can describe the propagation of femtosecond pulses in optical fibers. Very recently, Ablowitz and Mussliman introduced a class of reverse space-time and reverse time nonlinear integrable equations, including the reverse space nonlocal NLS equation, the real and complex reverse space-time nonlocal mKdV, sine-Gordon, Davey–Stewartson equations, etc. So, what is nonlocal version of high-order NLS? In this paper, we introduce a reverse space-time nonlocal Sasa–Satsuma equation, i.e., a reverse space-time nonlocal high-order NLS equation, and derive its solutions with the binary Darboux transformation method. Periodic solutions, and some localized solutions, such as dark soliton, W-shaped soliton, M-shaped soliton and breather soliton of the reverse space-time nonlocal Sasa–Satsuma equation are constructed.

Well-Posedness Results and Dissipative Limit of High Dimensional KdV-Type Equations

Considered in this work is an n-dimensional dissipative version of the Korteweg–de Vries (KdV) equation. Our goal here is to investigate the well-posedness issue for the associated initial value problem in the anisotropic Sobolev spaces. We also study well-posedness behavior of this equation when the dissipative effects are reduced.

Effect of a damping force on dust acoustic waves simulated by particle-in-cell method

Damping dust acoustic waves described by the Korteweg-de Vries-type (KdV-type) equation and the nonlinear Schrödinger equation-type (quasi-NLSE) have been studied by the particle-in-cell (PIC) simulation method. The KdV-type equation and the quasi-NLSE with dust-neutral collision are analytically obtained by the reductive perturbation method. The PIC simulation methods for dust acoustic waves with damping force are shown. The PIC simulation results are compared with the analytical one. The relationship of the damping coefficient with the collision frequency is obtained. It is found that amplitudes of KdV-type solitary waves and quasienvelope solitary waves with damping force decrease exponentially.

Abundant general solitary wave solutions to the family of KdV type equations

This work explores the construction of more general exact traveling wave solutions of some nonlinear evolution equations (NLEEs) through the application of the (G′/G, 1/G)-expansion method. This method is allied to the widely used (G′/G)-method initiated by Wang et al. and can be considered as an extension of the (G′/G)-expansion method. For effectiveness, the method is applied to the family of KdV type equations. Abundant general form solitary wave solutions as well as periodic solutions are successfully obtained through this method. Moreover, in the obtained wider set of solutions, if we set special values of the parameters, some previously known solutions are revived. The approach of this method is simple and elegantly standard. Having been computerized it is also powerful, reliable and effective.

Wave Collision for the gKdV-4 equation. Asymptotic Approach

We consider an approach which allows to describe uniformly in time the process of collision of solitary waves. Next we apply it to the KdV-type equation with nonlinearity u4 for three interacting waves assuming that all wave trajectories intersect at the same point. The constructed asymptotic solution satisfies the equation in a weak sense and it can be treated as a classical asymptotics in the sense that it satisfies some conservation and balance laws associated with the gKdV-4 equation. Results of numerical simulation confirm the theoretical conclusion about the elastic type of the wave interaction.

Lie symmetry analysis, conservation laws and analytical solutions of a time-fractional generalized KdV-type equation

Under investigation in this work is the time-fractional generalized KdV-type equation, which occurs in different contexts in mathematical physics. Lie group analysis method is presented to explicitly study its vector fields and symmetry reductions. Furthermore, two straightforward methods are employed to consider its travelling wave solutions and power series solutions, respectively. Finally, based on the new conservation theorem, conservation laws of the equation are well constructed with a detailed derivation.

Numerical generation of periodic traveling wave solutions of some nonlinear dispersive wave systems

In this paper a numerical procedure to generate approximations to periodic traveling wave profiles of some nonlinear dispersive wave systems is introduced. The method is based on a suitable modification of a fixed point algorithm of Petviashvili type and solves several drawbacks of some previous algorithms presented in the literature. By way of illustration, the method is applied to generate numerically periodic traveling waves of two problems of interest: the fractional KdV type equations and the extended Boussinesq system.

The simplified Hirota’s method for studying three extended higher-order KdV-type equations

In this work we study three extended higher-order KdV-type equations. The Lax-type equation, the Sawada–Kotera-type equation and the CDG-type equation are derived from the extended KdV equation. We use the simplified Hirota’s direct method to derive multiple soliton solutions for each equation. We show that each model gives multiple soliton solutions, where the structures of the obtained solutions differ from the solutions of the canonical form of these equations.

Three kinds of periodic wave solutions and their limit forms for a modified KdV-type equation

A modified KdV-type equation is studied by using the bifurcation theory of dynamical system. By investigating the dynamical behavior with phase space analysis, all possible explicit exact traveling wave solutions including peakon solutions, kink and anti-kink wave solutions, blow-up wave solutions, smooth periodic wave solutions, periodic cusp wave solutions, and periodic blow-up wave solutions are obtained. When the first integral varies, we also show the convergence of the periodic wave solutions, such as the smooth periodic wave solutions converge to the kink and anti-kink wave solutions, the periodic cusp wave solutions converge to the peakon solution, the periodic blow-up wave solutions converge to the blow-up wave solution, the blow-up wave solutions converge to the blow-up wave solution, and the periodic blow-up wave solutions converge to the periodic blow-up wave solution.

Sharp local well-posedness of KdV type equations with dissipative perturbations

In this work, we study the initial value problems associated to some linear perturbations of KdV equations. Our focus is in the well-posedness issues for initial data given in the $L^2$-based Sobolev spaces. We derive bilinear estimate in a space with weight in the time variable and obtain sharp local well-posedness results.

Preventing Blow up by Convective Terms in Dissipative PDE’s

We study the impact of the convective terms on the global solvability or finite time blow up of solutions of dissipative PDEs. We consider the model examples of 1D Burger's type equations, convective Cahn-Hilliard equation, generalized Kuramoto-Sivashinsky equation and KdV type equations, we establish the following common scenario: adding sufficiently strong (in comparison with the destabilizing nonlinearity) convective terms to equation prevents the solutions from blowing up in finite time and makes the considered system globally well-posed and dissipative and for weak enough convective terms the finite time blow up may occur similarly to the case when the equation does not involve convective term. This kind of result has been previously known for the case of Burger's type equations and has been strongly based on maximum principle. In contrast to this, our results are based on the weighted energy estimates which do not require the maximum principle for the considered problem.