Dimensional KdV Research Articles

Painlevé analysis, integrability property and multiwave interaction solutions for a new (4+1)-dimensional KdV–Calogero–Bogoyavlenkskii–Schiff equation

In this letter a novel (4+1)-dimensional KdV–Calogero–Bogoyavlenkskii–Schiff (KdV-CBS) equation is investigated. The Painlevé analysis shows that this higher dimensional nonlinear equation passes the integrability test. Its integrable properties, including bilinear Bäcklund transformation, Lax pair as well as N-soliton solution, are systematically derived using the binary Bell polynomial method. Furthermore, incorporating the inheritance solving technique into direct algebraic method, we construct the multiwave solutions of 1-lump and (N−2)-soliton (N>2). Multiple choices of parameters in the obtained solutions lead to rich interactions among lump waves, kink waves and breather waves. Several interesting interactions of the multiwave solutions are shown by graphs.

A NEW (3+1)-DIMENSIONAL KDV EQUATION AND MKDV EQUATION WITH THEIR CORRESPONDING FRACTIONAL FORMS

In this work, new [Formula: see text]-dimensional Korteweg–de Vries (KdV) equation and modified KdV (mKdV) equation as well as the corresponding fractional forms are presented. These two equations are derived for the first time relying on the extended [Formula: see text]-dimensional zero curvature equation. In addition, symmetries and conservation laws are displayed. Meanwhile, one-soliton solution for the time fractional of [Formula: see text]-dimensional KdV equation and mKdV equation is obtained.

Lie symmetry analysis, optimal system and exact solutions of a new (2 + 1)-dimensional KdV equation

This work attempts to apply the Lie symmetry approach to an updated (2+1)-dimensional KdV equation, recently updated in A.-M. Wazwaz, Nucl. Phys. B 954 (2020) 115009. The equation can be considered as one of the famous examples of the soliton equation. The infinitesimal generators for the governing equation have been found using the invariance property of Lie groups. The commutator table, adjoint table, invariant functions and one-dimensional optimal system of subalgebras are then derived using Lie point symmetries. Some group invariant solutions are derived based on various subalgebras, symmetry reductions and an optimal system. To demonstrate the physical acceptability of the results, the obtained solutions are evaluated using numerical simulation.

Extended exp (-φ (ξ))-expansion method for some exact solutions of (2+1) and (3+1)-dimensional constant coefficients KdV equations

We gained some exact solutions of the (2+1)- and (3+1)-dimensional constant coefficients KdV equations by using extended exp(−φ(ξ))-expansion method. These solutions we gained are hyperbolic, trigonometric, exponential and rational solutions. In addition, we provided both two-dimensional and three-dimensional graphics of the solutions we found in −10≤x≤10 and −1≤x≤1 intervals. Then, we observed, via Mathematica 11.2, that the solutions prove the equations. This method has been used for obtaining exact travelling wave solutions of nonlinear partial differential equations (NLPDEs).

Painlevé integrability, auto-Bäcklund transformations, new abundant analytic solutions including multi-soliton solutions for time-dependent extended KdV8 equation in nonlinear physics

In this paper, a new eighth-order (1+1)-dimensional time-dependent extended KdV equation has been developed. This considered equation is being found completely integrable by using the Painlevé analysis method. Moreover, three auto-Bäcklund transformations have been generated by truncating the Painlevé series at a constant level. These auto-Bäcklund transformations have been used to derive various analytic solution families for the newly developed equation. These solutions include the kink-antikink soliton, kink-soliton, antikink-soliton, periodic-soliton, complex kink-soliton and complex periodic-soliton solutions. Multi-soliton solutions including N-soliton solution, have been obtained by using the simplified Hirota’s method for the considered equation. All the results are being expressed graphically to signify the physical importance of the considered equation.

Soliton-like solutions of general variable coefficient (2+1)-dimensional KdV equation with linear damping term

A general variable coefficient (2+1)-dimensional KdV equation with linear damping term (VC(2+1)d damping KdV equation for short) has been investigated by using the simplified homogeneous balance method. It has been proven firstly that if the coefficients of the equation satisfy a certain constraint condition, then the VC(2+1)d damping KdV equation has a nonlinear transformation that converts the VC(2+1)d damping KdV equation itself into a homogeneous quadratic equation which admits a series of exponential function solutions. By means of the nonlinear transformation derived here, one and two soliton-like solutions of the VC(2+1)d damping KdV equation are obtained successfully. As an illustrative example, a concrete VC(2+1)d damping KdV equation is studied, the nonlinear transformation, and one and two soliton-like solutions of the equation are all given.

A (2+1)-dimensional combined KdV–mKdV equation: integrability, stability analysis and soliton solutions

A (2+1)-dimensional combined KdV–mKdV equation: integrability, stability analysis and soliton solutions

The Only Valid (2+1)-Dimensional Kdv, Kadomtsev-Petviashvili, Fifth-Order Kdv, and Gardner Equations Derived from the Ideal Fluid Model

The Only Valid (2+1)-Dimensional Kdv, Kadomtsev-Petviashvili, Fifth-Order Kdv, and Gardner Equations Derived from the Ideal Fluid Model

Using Riccati Equation to Construct New Solitary Solutions of Nonlinear Difference Differential Equations

In this paper, we use Riccati equation to construct new solitary wave solutions of the nonlinear evolution equations (NLEEs). Through the new function transformation, the Riccati equation is solved, and many new solitary wave solutions are obtained. Then it is substituted into the (2 + 1)-dimensional BLMP equation and (2 + 1)-dimensional KDV equation as an auxiliary equation. Many types of solitary wave solutions are obtained by choosing different coefficient p1 and q1 in the Riccati equation, and some of them have not been found in other documents. These solutions that we obtained in this paper will be helpful to understand the physics of the NLEEs.

M-lump solutions and interactions phenomena for the (2+1)-dimensional KdV equation with constant and time-dependent coefficients

In this paper, Hirota’s simple method and long-wave method are used to study the (2+1)-dimensional KdV equation including constant and time-dependent coefficients. One-, two-, and three-M-lump solutions are constructed for both cases. Also, the interaction phenomena of M-lump solution with one-soliton, and two-soliton waves are derived for these equations. Moreover, to investigate the physical characteristics of the obtained results, the 3-dimensional figures and contour plots are graphed. All gained solutions verify the proposed equations.

Lie symmetry analysis, optimal system and conservation laws of a new (2+1)-dimensional KdV system

In this paper, Lie point symmetries of a new (2+1)-dimensional KdV system are constructed by using the symbolic computation software Maple. Then, the one-dimensional optimal system, associated with corresponding Lie algebra, is obtained. Moreover, the reduction equations and some explicit solutions based on the optimal system are presented. Finally, the nonlinear self-adjointness is provided and conservation laws of this KdV system are constructed.

M-breather, M-lump, breather molecules and their interaction solutions for a (2+1)-dimensional KdV equation

Date-Jimbo-Kashiwara-Miwa (DJKM) equation is an integrable (2+1)- dimensional extension of the KdV equation and describes two-dimensional nonlinear dispersive waves. In this paper, we first derive the M-breather solution in terms of determinants for the DJKM equation applying the nonlinear superposition formula and analyze the dynamical properties of the breathers. The M-lump waves for the DJKM equation are obtained through the full degeneration of the breathers and hybrid solutions composed of line solitons, breathers and lumps are constructed. By using the velocity resonance mechanism, we also show that the DJKM equation possesses some resonant structures for breathers and solitons such as breather molecules and breather-soliton molecules. Furthermore, based on the N-soliton solution, the interactions between breather/breather-soliton molecules and breathers/lumps are investigated.

Space-Curved Resonant Line Solitons in a Generalized (2 + 1)-Dimensional Fifth-Order KdV SystemSupported by the National Natural Science Foundation of China (Grant Nos. 11775121 and 11435005), and the K. C. Wong Magna Fund at Ningbo University.

On the basis of N-soliton solutions, space-curved resonant line solitons are derived via a new constraint proposed here, for a generalized (2+1)-dimensional fifth-order KdV system. The dynamic properties of these new resonant line solitons are studied in detail. We then discuss the interaction between a resonance line soliton and a lump wave in greater detail. Our results highlight the distinctions between the generalized (2+1)-dimensional fifth-order KdV system and the classical type.

Formula omitted]-soliton solution and the Hirota condition of a (2+1)-dimensional combined equation

Within the Hirota bilinear formulation, we construct N-soliton solutions and analyze the Hirota N-soliton conditions in (2+1)-dimensions. A generalized algorithm to prove the Hirota conditions is presented by comparing degrees of the multivariate polynomials derived from the Hirota function in N wave vectors, and two weight numbers are introduced for transforming the Hirota function to achieve homogeneity of the related polynomials. An application is developed for a general combined nonlinear equation, which provides a proof of existence of its N-soliton solutions. The considered model equation includes three integrable equations in (2+1)-dimensions: the (2+1)-dimensional KdV equation, the Kadomtsev–Petviashvili equation, and the (2+1)-dimensional Hirota–Satsuma–Ito equation, as specific examples.

Two new Painlevé integrable KdV–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation and new negative-order KdV-CBS equation

In this work, we develop two new (3+1)-dimensional KdV–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation and (3+1)-dimensional negative-order KdV-CBS (nKdV-nCBS) equation. The newly developed equations pass the Painlevé integrability test via examining the compatibility conditions for each developed model. We examine the dispersion relation and derive multiple soliton solutions for each new equation.

Mixed Rational Lump-Solitary Wave Solutions to an Extended (2+1)-Dimensional KdV Equation

Based on the bilinear method, rational lump and mixed lump-solitary wave solutions to an extended (2+1)-dimensional KdV equation are constructed through the different assumptions of the auxiliary function in the trilinear form. It is found that the rational lump decays algebraically in all directions in the space plane and its amplitude possesses one maximum and two minima. One kind of the mixed solution describes the interaction between one lump and one line solitary wave, which exhibits fission and fusion phenomena under the different parameters. The other kind of the mixed solution shows one lump interacting with two paralleled line solitary waves, in which the evolution of the lump gives rise to a two-dimensional rogue wave. This shows that these three interesting phenomena exist in the corresponding physical model.

Lie symmetry, nonlocal symmetry analysis, and interaction of solutions of a (2+1)-dimensional KdV–mKdV equation

Lie symmetry, nonlocal symmetry analysis, and interaction of solutions of a (2+1)-dimensional KdV–mKdV equation

On the exact and numerical solutions to a new (2 + 1)-dimensional Korteweg-de Vries equation with conformable derivative

Abstract The aim of this paper is to introduce a novel study of obtaining exact solutions to the (2+1) - dimensional conformable KdV equation modeling the amplitude of the shallow-water waves in fluids or electrostatic wave potential in plasmas. The reduction of the governing equation to a simpler ordinary differential equation by wave transformation is the first step of the procedure. By using the improved tan(φ/2)-expansion method (ITEM) and Jacobi elliptic function expansion method, exact solutions including the hyperbolic function solution, rational function solution, soliton solution, traveling wave solution, and periodic wave solution of the considered equation have been obtained. We achieve also a numerical solution corresponding to the initial value problem by conformable variational iteration method (C-VIM) and give comparative results in tables. Moreover, by using Maple, some graphical simulations are done to see the behavior of these solutions with choosing the suitable parameters.

Fusion and fission phenomena for [formula omitted]-dimensional fifth-order KdV system

Based on N-soliton solutions, fusion and fission waves are obtained through a new constraint among parameters. With this new constraint, fusion and fission phenomena can be reduced from N-soliton solutions in most (2+1)-dimensional integrable systems. This paper takes the (2+1)-dimensional fifth-order KdV equation as an example to introduce how to use this constraint to generate these fusion and fission waves in detail. At the same time, by continuing to introduce constraints like velocity resonance, module resonance, and long-wave limit, we can generate all kinds of hybrid solutions for granted, such as a hybrid of fissionable waves and breather waves, and a nonlinear superposition between fissionable waves and higher-order lump waves.

N-soliton solutions and the Hirota conditions in (2+1)-dimensions

We compute N-soliton solutions and analyze the Hirota N-soliton conditions, in (2+1)-dimensions, based on the Hirota bilinear formulation. An algorithm to check the Hirota conditions is proposed by comparing degrees of the polynomials generated from the Hirota function in N wave vectors. A weight number is introduced while transforming the Hirota function to achieve homogeneity of the resulting polynomial. Applications to three integrable equations: the (2+1)-dimensional KdV equation, the Kadomtsev–Petviashvili equation, the (2+1)-dimensional Hirota–Satsuma–Ito equation, are made, thereby providing proofs of the existence of N-soliton solutions in the three model equations.