Dimensional Korteweg-de Vries 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.

A new analytic approach and its application to new generalized Korteweg‐de Vries and modified Korteweg‐de Vries equations

The current study is important from two perspectives. Firstly, in this article, we suggest a novel analytical technique for creating the exact solutions to nonlinear partial differential equations (NLPDEs). In order to study the dynamical behaviors of various wave phenomena, we can construct the several exact solutions in the form of Jacobi elliptic solutions, hyperbolic solutions, trigonometric solutions, and exponential solutions by using this method. Secondly, we consider a more generalized form of the (2+1)‐dimensional Korteweg–de Vries (KdV) and modified Korteweg‐de Vries (mKdV) equations that plays an important role in describing the shallow water. We successfully extract several soliton solutions for the examined equation. By choosing the appropriate parameters, some graphs of the discovered solutions have been represented in the figures. Every obtained solution has been demonstrated to satisfy the corresponding equation. The findings demonstrate that the method can be applied to solve a number of nonlinear evolution equations. The new solutions and the paper's findings could improve our understanding of how the waves move through shallow water and open up new research avenues.

Interaction solutions of (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation via bilinear method

Using the bilinear neural network method (BNNM) and the symbolic computation system Mathematica, this paper explains how to find an exact solution for the (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani (KdVSKR) equation. In terms of activation function and weight coefficient, BNNM is a more appealing option for users than traditional symbolic computation methods. It is possible to develop a wide range of solutions and expand the classes of exact solutions by modifying the activation function. The activation function’s versatility allows it to generate a wide range of solutions with several theoretical and practical uses. The analytical solution is obtained by using a double layer type, while the rogue wave solution and mixed solutions are obtained by using a single layer type. The evolution of these waves is then illustrated using various 3D graphs, 2D graphs, and density plots.

Painlevé analysis and inelastic interactions of the lumps for a generalized (2+1)-dimensional Korteweg-de Vries system for the shallow-water waves

Water waves, one of the common natural phenomena, are recognized as complex and often turbulent. A generalized (2+1)-dimensional Korteweg-de Vries system for the shallow-water waves is conducted in this paper. We perform the Painlevé analysis and find that the system is Painlevé integrable. We study the inelastic interactions of the lumps for the system. We find that two lumps, which propagate along the curves with the equal amplitude, are symmetric about the x axis before the interaction, where x is a scaled spatial variable. After the interaction, amplitudes of the two lumps are different, but in the process of moving, the lower lump gradually increases, while the higher lump gradually decreases, and the velocities of two lumps at the infinity are equal. We observe two different inelastic interactions of the three lumps: (1) the three lumps are symmetric in time and space, and they slowly contract (before the interaction) and swell (after the interaction); (2) the three lumps slowly fuse and after the interaction they form a straight line forward, and their amplitudes are gradually equal when t → ∞.

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.

Painlevé analysis, infinite dimensional symmetry group and symmetry reductions for the (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation

The (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani (KdVSKR) equation is studied by the singularity structure analysis. It is proven that it admits the Painlevé property. The Lie algebras which depend on three arbitrary functions of time t are obtained by the Lie point symmetry method. It is shown that the KdVSKR equation possesses an infinite-dimensional Kac–Moody–Virasoro symmetry algebra. By selecting first-order polynomials in t, a finite-dimensional subalgebra of physical transformations is studied. The commutation relations of the subalgebra, which have been established by selecting the Laurent polynomials in t, are calculated. This symmetry constitutes a centerless Virasoro algebra which has been widely used in the field of physics. Meanwhile, the similarity reduction solutions of the model are studied by means of the Lie point symmetry theory.

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.

Dark Korteweg–De Vrise System and Its Higher-Dimensional Deformations

The new dimensional deformation approach is proposed to generate higher-dimensional analogues of integrable systems. An arbitrary (K+1)-dimensional integrable Korteweg–de Vries (KdV) system, as an example, exhibiting symmetry, is illustrated to arise from a reconstructed deformation procedure, starting with a general symmetry integrable (1+1)-dimensional dark KdV system and its conservation laws. Physically, the dark equation systems may be related to dark matter physics. To describe nonlinear physics, both linear and nonlinear dispersions should be considered. In the original lower-dimensional integrable systems, only liner or nonlinear dispersion is included. The deformation algorithm naturally makes the model also include the linear dispersion and nonlinear dispersion.

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.

Novel soliton solutions and localized structures of KdV equation

A (1 + 1)-dimensional Korteweg-de Vries (KdV) equation is studied. Using the extended Tanh-function method, the analytical soliton solutions are obtained by symbolic calculation, and the localized coherent structures containing the compacton, the peakon, periodic multi-solitons and fractal soliton structures are constructed. Through the analysis of various images, the properties of soliton solutions excited by different local structures are discussed and the fractal structure of soliton solutions are further discussed.

(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.

Nonlinear superposition between lump soliton and other nonlinear localized waves for the (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation

In the previous studies, the authors have reported the lump soliton collided with other nonlinear localized waves for (2+1)-dimensional Korteweg–de Vries–Sawada-Kotera–Ramani equation. However, this work will introduce a novel restrictive condition to reveal the nonlinear superposition between a lump soliton and other nonlinear localized waves. Under the premise of the acquired N-soliton solutions, the nonlinear superposition structures among lump soliton, stripe solitons, resonance Y-type solitons and breather waves guarantee that the lump soliton does not collide with other localized waves by means of the long wave limit approach, soliton resonance method and symbolic computation. Meanwhile, a variety of soliton molecules are yielded through nonlinear superposition and velocity resonance mechanisms. The obtained solutions in this work will bring about new perception of the interaction behavior between lump soliton and other nonlinear localized waves, and provide more brand new insights into the nonlinear phenomena triggered by the areas of plasma physics, fluid dynamics and nonlinear optics.

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.

Novel y-type and hybrid solutions for the $$(2+1)$$-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation

In this paper, the research object is \((2+1)\)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani (KdVSKR) equation. After adding new constraint, new solutions which contain y-type molecules are obtained. The process of lump molecules and y-type molecules before and after the collision is studied by long-wave limit method, and the kinetic behavior analysis is given. The interactions between y-type molecules and resonant soliton molecules, y-type molecules and breather molecules are obtained by combining velocity resonance method and mode resonance method, respectively. Finally, a novel hybrid solution containing lump molecule, breather molecule and y-type molecule is given, and the kinetic behavior is shown.

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.

Some traveling wave solutions to the generalized (3 + 1)‐dimensional Korteweg–de Vries–Zakharov–Kuznetsov equation in plasma physics

The generalized Korteweg–de Vries–Zakharov–Kuznetsov model is one of the dominant nonlinear complex structures to exhibit the influence of magnetic fields on weak ion‐acoustic waves in plasma made up of cool and hot electrons. In this study, the nonlinear higher dimensional model is subjected to the extended ‐expansion approach and the ‐expansion strategy in order to discover some novel traveling waves solutions and other exact solutions. There have been several solutions discovered, such as bright, dark, periodic, bell‐shaped, and single bell‐shaped wave structures. Furthermore, the stability analysis of the model is also reported along with the dynamical visualization of the charismatic behavior of various solitons with the aid of 3D, 2D, and contour plots. The proposed techniques can also be implemented to analyze a wide range of nonlinear evolution models emerging in diverse disciplines of science and technology including mathematical physics and plasma physics.

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.