Dimensional Korteweg-de Vries Equation Research Articles

New coherent structures and interaction behavior for the new [formula omitted]-dimensional Korteweg-de Vries equation

A new type of variable separation solutions for the (2+1)-dimensional Korteweg–de Vries equation is derived by means of an improved mapping approach. The lump solutions are constructed and the interaction phenomena between a lump wave and muliti-kink waves are discussed, and some of these solutions exhibit a rich dynamics, with a wide variety of qualitative behavior and structures that are exponentially localized, showing some novel features and interesting behaviors.

Wronskian rational solutions to the generalized (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa equation in fluid dynamics

The main topic of the paper is to investigate the generalized (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) and Korteweg–de Vries (KdV) equations, which are widely used in many physical areas, especially in fluids. A new Wronskian formulation is presented for these two equations associated with the bilinear Bäcklund transformation. Based on Wronskian identities of the bilinear Kadomtsev–Petviashvili (KP) hierarchy, the Wronskian determinant solution is verified by a direct and concise calculation. The newly introduced Wronskian formulation provides a comprehensive way for building rational solutions. A few rational Wronskian solutions of lower order are computed for the generalized (2 + 1)-dimensional DJKM equation. Our work can show that the extended (2 + 1)-dimensional KdV equation possesses the similar rational Wronskian solutions through the corresponding logarithmic transformation.

Line solitons, lumps, and lump chains in the (2+1)-dimensional generalization of the Korteweg–de Vries equation

This article focuses on the exploration of novel soliton molecules for the (2+1)-dimensional Korteweg–de Vries equation. Specifically, Hirota bilinear form is derived through Bell polynomial method, and the hybrid solution comprising one lump soliton and an arbitrary number of line solitons or/and lump chains is derived through the introduction of a long wave limit and new constraint conditions between the parameters of the N-soliton solutions and velocity resonance. The paper presents both analytical and graphical demonstrations of a range of interactions, showcasing the propagation of nonlinear localized waves. The findings of this study could contribute to a better understanding of physical phenomena associated with the propagation of nonlinear localized waves.

Line-rogue waves, transformed nonlinear waves and their interactions for a (3+1)-dimensional Korteweg-de Vries equation in a fluid

Nonlinear phenomena are studied in such disciplines as meteorology, engineering, oceanography and astrophysics. In this Letter, we study a (3+1)-dimensional Korteweg-de Vries equation in a fluid. By virtue of the long-wave-limit method, the higher-order rational solutions are given. Based on the first-order rational solutions, we convert a lump into a line-rogue wave under certain condition. We investigate the effects of the coefficients in the equation for the velocities and amplitudes on the lump and line-rogue wave. Interactions between two line-rogue waves/one lump and one line-rogue wave are worked out with the second-order rational solutions. Through the characteristic-line analysis, we convert the breathers into five different types of the transformed nonlinear waves. Based on certain semi-rational solutions, interactions between the line-rogue wave and soliton/breather/transformed nonlinear wave are graphically illustrated.

(2+1)-dimensional KdV, fifth-order KdV, and Gardner equations derived from the ideal fluid model. Soliton, cnoidal and superposition solutions

We study the problem of gravity surface waves for an ideal fluid model in the (2+1)-dimensional case. We apply a systematic procedure to derive the Boussinesq equations for a given relation between the orders of four expansion parameters, the amplitude parameter α, the long-wavelength parameter β, the transverse wavelength parameter γ, and the bottom variation parameter δ. Such an approach, known in (1+1)-dimensional theory, is extended for the first time for (2+1)-dimensional shallow water waves. Using this procedure, we derived the only possible (2+1)-dimensional extensions of the Korteweg–de Vries equation, the fifth-order KdV equation and the Gardner equation in three cases of the relationship between these parameters. All these equations are non-local. When the bottom is flat, the (2+1)-dimensional KdV equation can be transformed to the Kadomtsev–Petviashvili equation in a fixed reference frame and next to the classical KP equation in a moving frame. Thus, the Kadomtsev–Petviashvili equation gained a derivation from the fundamental laws of hydrodynamics. We have found soliton, cnoidal, and superposition solutions (essentially one-dimensional) to the (2+1)-dimensional Korteweg–de Vries equation and the Kadomtsev–Petviashvili equation.

Non classical interaction aspects to a nonlinear physical model

This study deals the dynamics of waves to the conformable fractional (2+1)-dimensional Nizhnik–Novikov–Veselov (NNV) equations. The (2+1)-dimensional NNV equations are the isotropic Lax integrable extension of the (1+1)-dimensional Korteweg–de Vries equations. Fractional differential models (FDMs) from the corresponding integer order model can describes more complex behavior and even cover all properties of integer order model. By the usage of different test approaches, the Hirota bilinear method (HBM) has been successfully applied. The Hirota method is a well-known and reliable mathematical tool for finding the soliton solutions of fractional nonlinear partial differential equations in many fields. However, it demands bilinearization of nonlinear fractional PDE. A class of results in the shapes of lump-periodic, breather-type and two wave solutions have been extracted. Numerical visualizations of the results are also used to demonstrate the implications of the fractionality and parameters. It is acceptable to assume that many experts in engineering models will learn from this research. The results prove the used algorithm is effective, quick, easy, and flexible in its application to various systems.

New (3+1)-dimensional conformable KdV equation and its analytical and numerical solutions

In this research, we used the Riccati equation approach to establish new exact solution sets for a new [Formula: see text]-dimensional Korteweg–de Vries (KdV) equation, which is an enlarged version of the time-fractional Kadomtsev–Petviashvili equation. Additionally, using the Mathematica symbolic computation program, approximate solutions to the equation, applying the residual power series method (RPSM), have been obtained. A comparison table and various graphical representations have been presented to demonstrate the validity of the solutions. The numerical findings indicate the effectiveness of both approaches in finding exact and approximate solutions to nonlinear fractional differential equations.

Deformation conjecture: deforming lower dimensional integrable systems to higher dimensional ones by using conservation laws

Utilizing some conservation laws of (1+1)-dimensional integrable local evolution systems, it is conjectured that higher dimensional integrable equations may be regularly constructed by a deformation algorithm. The algorithm can be applied to Lax pairs and higher order flows. In other words, if the original lower dimensional model is Lax integrable (possesses Lax pairs) and symmetry integrable (possesses infinitely many higher order symmetries and/or infinitely many conservation laws), then the deformed higher order systems are also Lax integrable and symmetry integrable. For concreteness, the deformation algorithm is applied to the usual (1 + 1)-dimensional Korteweg-de Vries (KdV) equation and the (1 + 1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) system (including nonlinear Schrödinger (NLS) equation as a special example). It is interesting that the deformed (3+1)-dimensional KdV equation is also an extension of the (1 + 1)-dimensional Harry-Dym (HD) type equations which are reciprocal links of the (1+1)-dimensional KdV equation. The Lax pairs of the (3 + 1)-dimensional KdV-HD system and the (2 + 1)-dimensional AKNS system are explicitly given. The higher order symmetries, i.e., the whole (3 + 1)-dimensional KdV-HD hierarchy, are also explicitly obtained via the deformation algorithm. The single soliton solution of the (3 + 1)-dimensional KdV-HD equation is implicitly given. Because of the effects of the deformation, the symmetric soliton shape of the usual KdV equation is no longer conserved and deformed to be asymmetric and/or multi-valued. The deformation conjecture holds for all the known (1 +1)-dimensional integrable local evolution systems that have been checked, and we have not yet found any counter-example so far. The introduction of a large number of (D + 1)-dimensional integrable systems of this paper explores a serious challenge to all mathematicians and theoretical physicists because the traditional methods are no longer directly valid to solve these integrable equations.

Formula omitted]-dressing method for the (2+1)-dimensional Korteweg–de Vries equation

In this paper, we investigate the (2+1)-dimensional Korteweg–de Vries equation by applying the ∂̄-dressing method. A new ∂̄ problem is obtained through the characteristic functions and Green’s functions by Lax representation. A solution of ∂̄ problem is constructed by applying Cauchy–Green formula and selecting appropriate spectral transformation. Finally, we can give the solution of the (2+1) dimensional Korteweg–de Vries equation after determining the time evolution of the spectral data.

New approach for soliton solutions for the (2 + 1)-dimensional KdV equation describing shallow water wave

PurposeThe purpose of this study is to produce families of exact soliton solutions (2+1)-dimensional Korteweg-de Vries (KdV) equation, that describes shallow water waves, using an ansätze approach.Design/methodology/approachThis article aims to introduce a recently developed ansätze for creating soliton and travelling wave solutions to nonlinear nonintegrable partial differential equations, especially those with physical significance.FindingsA recently developed ansätze solution was used to successfully construct soliton solutions to the (2 + 1)-dimensional KdV equation. This straightforward method is an alternative to the Painleve test analysis, yielding similar results. The strategy demonstrated the existence of a single soliton solution, also known as a localized wave or bright soliton, as well as singular solutions or kink solitons.Originality/valueThe ansätze solution used to construct soliton solutions to the (2 + 1)-dimensional KdV equation is novel. New soliton solutions were also obtained.

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.

New lump interaction complexitons to the (2+1)-dimensional Korteweg-de Vries equation with electrostatic wave potential in plasmas

A soliton is a packet of self-reinforcing waves that maintains its structure when moving at a constant speed. Solitons are caused by the cancellation of the medium’s nonlinear and dispersive effects. In plasmas, the bilinear form of Hirota will be utilized to investigate the (2+1)-dimensional Korteweg-de Vries equation with electrostatic wave potential. Solutions for complexiton lump interaction have been developed. To throw further light on the physical qualities of the recorded data, certain 3-dimensional and contour plots are presented to illustrate the interaction elements of these solutions.

Different Wave Structures for the (2+1)-Dimensional Korteweg-de Vries Equation

In this article, a (2+1)-dimensional Korteweg-de Vries equation is investigated. Abundant periodic wave solutions are obtained based on the Hirota’s bilinear form and a direct test function. Meanwhile, the interaction solutions between lump and periodic waves are presented. What is more, we derive the interaction solutions among lump, periodic, and solitary waves. Based on the extended homoclinic test technique, some new double periodic-soliton solutions are presented. Finally, some 3D and density plots are simulated and displayed to respond the dynamic behavior of these obtained solutions.

Bäcklund transformation, residual symmetry and exact interaction solutions of an extended (2+1)-dimensional Korteweg–de Vries equation

Starting from the truncated Painlevé expansion, auto- and non-auto Bäcklund transformations are constructed for an extended (2+1)-dimensional Korteweg–de Vries equation. Then nonlocal residual symmetry is derived and localized to obtain the finite group transformation. By virtue of a simple connection between the truncated Painlevé expansion and the consistent tanh expansion (CTE) method, exact single soliton and resonant two-soliton solutions as well as interaction solutions between soliton and cnoidal wave are presented explicitly.

A Hausdorff fractal Nizhnik-Novikov-Veselov model arising in the incompressible fluid

PurposeNizhnik–Novikov–Veselov system (NNVS) is a well-known isotropic extension of the Lax (1 + 1) dimensional Korteweg-deVries equation that is also used as a paradigm for an incompressible fluid. The purpose of this paper is to present a fractal model of the NNVS based on the Hausdorff fractal derivative fundamental concept.Design/methodology/approachA two-scale transformation is used to convert the proposed fractal model into regular NNVS. The variational strategy of well-known Chinese scientist Prof. Ji Huan He is used to generate bright and exponential soliton solutions for the proposed fractal system.FindingsThe NNV fractal model and its variational principle are introduced in this paper. Solitons are created with a variety of restriction interactions that must all be applied equally. Finally, the three-dimensional diagrams are displayed using an appropriate range of physical parameters. The results of the solitary solutions demonstrated that the suggested method is very accurate and effective. The proposed methodology is extremely useful and nearly preferable for use in such problems.Practical implicationsThe research study of the soliton theory has already played a pioneering role in modern nonlinear science. It is widely used in many natural sciences, including communication, biology, chemistry and mathematics, as well as almost all branches of physics, including nonlinear optics, plasma physics, fluid dynamics, condensed matter physics and field theory, among others. As a result, while constructing possible soliton solutions to a nonlinear NNV model arising from the field of an incompressible fluid is a popular topic, solving nonlinear fluid mechanics problems is significantly more difficult than solving linear ones.Originality/valueTo the best of the authors’ knowledge, for the first time in the literature, this study presents Prof. Ji Huan He's variational algorithm for finding and studying solitary solutions of the fractal NNV model. The reported solutions are novel and present a valuable addition to the literature in soliton theory.

Bäcklund transformations, nonlocal symmetry and exact solutions of a generalized (2+1)-dimensional Korteweg–de Vries equation

In this paper, nonlocal residual symmetry of a generalized (2+1)-dimensional Korteweg–de Vries equation is derived with the aid of truncated Painlevé expansion. Three kinds of non-auto and auto Bäcklund transformations are established. The nonlocal symmetry is localized to a Lie point symmetry of a prolonged system by introducing auxiliary dependent variables. The linear superposed multiple residual symmetries are presented, which give rise to the nth Bäcklund transformation. The consistent Riccati expansion method is employed to derive a Bäcklund transformation. Furthermore, the soliton solutions, fusion-type N-solitary wave solutions and soliton–cnoidal wave solutions are gained through Bäcklund transformations.

Bilinear form, soliton, breather, hybrid and periodic-wave solutions for a (3+1)-dimensional Korteweg–de Vries equation in a fluid

Fluids are studied in such disciplines as atmospheric science, oceanography and astrophysics. In this paper, we investigate a (3+1)-dimensional Korteweg–de Vries equation in a fluid. Bilinear form and N-soliton solutions are obtained, where N is a positive integer. Via the N-soliton solutions, we derive the higher-order breather solutions. We observe the interaction between the two perpendicular first-order breathers on the $$x-y$$ and $$x-z$$ planes and the interaction between the periodic line wave and the first-order breather on the $$y-z$$ plane, where x, y and z are the independent variables in the equation. We discuss the effects of $$\alpha $$ , $$\beta $$ , $$\gamma $$ and $$\delta $$ on the amplitude of the second-order breather, where $$\alpha $$ , $$\beta $$ , $$\gamma $$ and $$\delta $$ are the constant coefficients in the equation: Amplitude of the second-order breather decreases as $$\alpha $$ increases; amplitude of the second-order breather increases as $$\beta $$ increases; amplitude of the second-order breather keeps invariant as $$\gamma $$ or $$\delta $$ increases. Via the N-soliton solutions, hybrid solutions comprising the breathers and solitons are derived. Based on the Riemann theta function, we obtain the periodic-wave solutions, and find that the periodic-wave solutions approach to the one-soliton solutions under a limiting condition.

Bilinear Bäcklund transformation, N‐soliton, and infinite conservation laws for Lax–Kadomtsev–Petviashvili and generalized Korteweg–de Vries equations

In this paper, we obtain the bilinear form for the Lax–Kadomtsev–Petviashvili (Lax–KP) and the generalized (3 + 1)‐dimensional Korteweg–de Vries equations based on the binary Bell polynomials. Accordingly, N‐soliton solutions, bilinear Bäcklund transformation, Lax pair, and infinite conservation laws will be constructed to Lax–KP and the generalized (2 + 1)‐dimensional Korteweg–de Vries equation . At the same time, we get another bilinear Bäcklund transformation. Finally, exact solutions were obtained by using the exchange formulas for Hirota's bilinear operators.

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.

Deformed two-dimensional rogue waves in the (2+1)-dimensional Korteweg-de Vries equation* *Project supported by the National Natural Scinece Foundation of China (Grant Nos. 11671219, 11871446, 12071304, and 12071451).

Within the (2 + 1)-dimensional Korteweg–de Vries equation framework, new bilinear Bäcklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation. By introducing an arbitrary function ϕ(y), a family of deformed soliton and deformed breather solutions are presented with the improved Hirota’s bilinear method. By choosing the appropriate parameters, their interesting dynamic behaviors are shown in three-dimensional plots. Furthermore, novel rational solutions are generated by taking the limit of the obtained solitons. Additionally, two-dimensional (2D) rogue waves (localized in both space and time) on the soliton plane are presented, we refer to them as deformed 2D rogue waves. The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane, and its evolution process is analyzed in detail. The deformed 2D rogue wave solutions are constructed successfully, which are closely related to the arbitrary function ϕ(y). This new idea is also applicable to other nonlinear systems.