Higher Dimensional Nonlinear Evolution Equations Research Articles

The applications of symbolic computation to exact wave solutions of two HSI-like equations in (2+1)-dimensional

It is renowned that Hirota–Satsuma–Ito (HSI) equation is widely used to study wave dynamics of shallow water. In this work, two new HSI-like equations are investigated which could be extended to diversify problems in natural phenomena and give admirable contributions by applying the generalized exponential rational function method (GERFM). With the aid of symbolic calculations, various constraints on the free parameters are given, while classes of wave solutions are explicitly constructed from the coefficients of the combined non-linear and dissipative terms. After specifying values for free parameters, singular, periodic singular and anti-kink waves are demonstrated in 3D figures to exhibit different kinds of wave propagations. The fact that parameters directly influence the wave amplitude and speed of traveling waves is illustrated. The derived results are innovative and have important applications in the current field of mathematical physics research. Eventually, we show that generalized exponential rational function method is effective and straightforward to solve higher-order and high-dimensional non-linear evolution equations.

Higher-order rogue waves with controllable fission and asymmetry localized in a (3 + 1)-dimensional generalized Boussinesq equation

The purpose of this paper is to report the feasibility of constructing high-order rogue waves with controllable fission and asymmetry for high-dimensional nonlinear evolution equations. Such a nonlinear model considered in this paper as the concrete example is the (3 + 1)-dimensional generalized Boussinesq (gB) equation, and the corresponding method is Zhaqilao’s symbolic computation approach containing two embedded parameters. It is indicated by the (3 + 1)-dimensional gB equation that the embedded parameters can not only control the center of the first-order rogue wave, but also control the number of the wave peaks split from higher-order rogue waves and the asymmetry of higher-order rogue waves about the coordinate axes. The main novelty of this paper is that the obtained results and findings can provide useful supplements to the method used and the controllability of higher-order rogue waves.

Various breathers, Lumps, line solitons and their interaction solutions for the (2+1)-dimensional variable-coefficient Sawada–Kotera equation

The main concern of this paper is to investigate the various of interaction solutions to the (2+1)-dimensional variable-coefficient Sawada–Kotera equation. Analytical one-, two-, three- and four-soliton solutions of this equation are constructed based on its Hirota bilinear form. Through granting appropriate parameters and coefficients, special cases lead to resonance X-type soliton, S-type, U-type or periodic-type soliton solutions. The interaction solutions of multiple line solitons, between S-type line soliton and breather, U-type or S-type line soliton and lump wave, two U-type or S-type or periodic-type breathers, two S-type lump waves, among U-type or S-type two line solitons and one lump are obtained analytically, and some figures are provided with a better understanding of the dynamic behavior. We are confident that the results obtained in this paper are novel, which may be helpful to study other higher-dimensional nonlinear evolution equations

Integrability and high-order localized waves of the (4 + 1)-dimensional nonlinear evolution equation

Constructing the exact solutions of high-dimensional nonlinear evolution equations and exploring their dynamics have always been important and open problems in real-world applications. The celebrated Korteweg-de Vries equation [KdV] and Kadomtsev-Petviashvili [KP] equation are typical examples of one-dimensional and two-dimensional integrable equations respectively. A natural idea is to investigate the integrable analogues of these equations in higher dimensional space. In this paper, an integrable extension of the Kadomtsev-Petviashvili equation, the (4 + 1)-dimensional nonlinear evolution equation (NLEE) [Physical Review Letters 96, (2006) 190201], is investigated. By dimensionality reduction we obtain three new nonlinear equations, namely (3 + 1)-dimensional NLEE, (2 + 1)-dimensional NLEE and (1 + 1)-dimensional NLEE. Bäcklund transformations and multi-soliton solutions are important characteristics of integrable equations. Based on the binary Bell polynomials and Hirota bilinear method we derive bilinear Bäcklund transformations and N-soliton solutions of these new equations, which show that these equations are integrable. We also give the multiple rational solutions of these new equations. These new integrable NLEEs enrich the models of integrable systems and help understand the new characteristics of nonlinear dynamics in real-world applications.

Multiple rational rogue waves for higher dimensional nonlinear evolution equations via symbolic computation approach

The paper investigates the multiple rogue wave solutions associated with the generalized Hirota–Satsuma–Ito (HSI) equation and the newly proposed extended (3 + 1)-dimensional Jimbo–Miwa (JM) equation with the help of a symbolic computation technique. By incorporating a direct variable transformation and utilizing Hirota’s bilinear form, multiple rogue wave structures of different orders are obtained for both generalized HSI and JM equation. The obtained bilinear forms of the proposed equations successfully investigate the 1st, 2nd and 3rd-order rogue waves. The constructed solutions are verified by inserting them into original equations. The computations are assisted with 3D graphs to analyze the propagation dynamics of these rogue waves. Physical properties of these waves are governed by different parameters that are discussed in details.

Analytical soliton solutions to the generalized (3+1)-dimensional shallow water wave equation

In this paper, the soliton solutions and dynamical wave structures for the generalized (3+1)-dimensional shallow water wave (SWW) equation, which is an important physical property in ocean engineering and hydrodynamics, are presented. The generalized exponential rational function (GERF) method is used to investigate the closed-form wave solutions of the generalized SWW equation, which is used to describe the evolutionary dynamics of SWW. We successfully archive a variety of soliton solutions such as exponential solutions, kink wave solutions, non-topological solutions, periodic singular solutions, and topological solutions. These newly established results are also important for understanding the wave-propagation and dynamics of exact solutions of the equation, which is of great significance in physical oceanography and chemical oceanography. Eventually, it is shown that the proposed GERF technique is effective, robust, and straightforward and is also used to solve other types of higher-dimensional nonlinear evolution equations. In our work, we have used Mathematica extensively for such complicated algebraic calculations.

Wave profile analysis of a couple of (3+1)-dimensional nonlinear evolution equations by sine-Gordon expansion approach

The (3+1)-dimensional Kadomtsev-Petviashvili and the modified KdV-Zakharov-Kuznetsov equations have a significant impact in modern science for their widespread applications in the theory of long-wave propagation, dynamics of shallow water wave, plasma fluid model, chemical kinematics, chemical engineering, geochemistry, and many other topics. In this article, we have assessed the effects of wave speed and physical parameters on the wave contours and confirmed that waveform changes with the variety of the free factors in it. As a result, wave solutions are extensively analyzed by using the balancing condition on the linear and nonlinear terms of the highest order and extracted different standard wave configurations, containing kink, breather soliton, bell-shaped soliton, and periodic waves. To extract the soliton solutions of the high-dimensional nonlinear evolution equations, a recently developed approach of the sine-Gordon expansion method is used to derive the wave solutions directly. The sine-Gordon expansion approach is a potent and strategic mathematical tool for instituting ample of new traveling wave solutions of nonlinear equations. This study established the efficiency of the described method in solving evolution equations which are nonlinear and with higher dimension (HNEEs). Closed-form solutions are carefully illustrated and discussed through diagrams.

Symmetry analysis, closed-form invariant solutions and dynamical wave structures of the generalized (3+1)-dimensional breaking soliton equation using optimal system of Lie subalgebra

Nonlinear evolution equations (NLEEs) are primarily relevant to nonlinear complex physical systems in a wide range of fields, including ocean physics, plasma physics, chemical physics, optical fibers, fluid dynamics, biology physics, solid-state physics, and marine engineering. This paper investigates the Lie symmetry analysis of a generalized (3+1)-dimensional breaking soliton equation depending on five nonzero real parameters. We derive the Lie infinitesimal generators, one-dimensional optimal system, and geometric vector fields via the Lie symmetry technique. First, using the three stages of symmetry reductions, we converted the generalized breaking soliton (GBS) equation into various nonlinear ordinary differential equations (NLODEs), which have the advantage of yielding a large number of exact closed-form solutions. All established closed-form wave solutions include special functional parameter solutions, as well as hyperbolic trigonometric function solutions, trigonometric function solutions, dark-bright solitons, bell-shaped profiles, periodic oscillating wave profiles, combo solitons, singular solitons, wave-wave interaction profiles, and various dynamical wave structures, which we present for the first time in this research. Eventually, the dynamical analysis of some established solutions is revealed through three-dimensional sketches via numerical simulations. Some of the new solutions are often useful and helpful for studying the nonlinear wave propagation and wave-wave interactions of shallow water waves in many new high-dimensional nonlinear evolution equations.

Extension of the sine-Gordon expansion scheme and parametric effect analysis for higher-dimensional nonlinear evolution equations

Different wave solutions assist to interpret phenomena in different aspects of optics, physics, plasma physics, engineering, and other related subjects. The higher dimensional generalized Boussinesq equation (gBE) and the Klein-Gordon (KG) equation have remarkable applications in the field of quantum mechanics, recession flow analysis, fluid mechanics etc. In this article, the soliton solutions of the higher-dimensional nonlinear evolution equations (NLEEs) have been extracted through extending the sine-Gordon expansion method and we analyze the effect of the associated parameters and the phenomena establishing the lump, kink, rogue, bright-dark, spiked, periodic wave, anti-bell wave, singular soliton etc. Formerly, the sine-Gordon expansion (sGE) method was used only to search for lower-dimensional NLEEs. In order to illustrate the latency, we have portrayed diagrams for different values of parameters and it is noteworthy that the properties of the features change as the parameters change.

The sine-Gordon expansion method for higher-dimensional NLEEs and parametric analysis

The Estevez-Mansfield-Clarkson (EMC) equation and the (2+1)-dimensional Riemann wave (RW) equation are important mathematical models in nonlinear science, engineering and mathematical physics which have remarkable applications in the field of plasma physics, fluid dynamics, optics, image processing etc. Generally, through the sine-Gordon expansion (SGE) method only the lower-dimensional nonlinear evolution equations (NLEEs) are examined. However, the method has not yet been extended of finding solutions to the higher-dimensional NLEEs. In this article, the SGE method has been developed to rummage the higher-dimensional NLEEs and established steady soliton solutions to the earlier stated NLEEs by putting in use the extended higher-dimensional sine-Gordon expansion method. Scores of soliton solutions are figure out which confirms the compatibility of the extended SGE method. The solutions are analyzed for both lower and higher-dimensional nonlinear evolution equations through sketching graphs for alternative values of the associated parameters. From the figures it is notable to perceive that the characteristic of the solutions depend upon the choice of the parameters. This study might play an impactful role in analyzing higher-dimensional NLEEs through the extended SGE approach.

Linear and nonlinear effects analysis on wave profiles in optics and quantum physics

In the field of quantum mechanics and fluid physics, especially in the study of nonlinear geometric optics and superconductivity, the Landau-Ginzburg-Higgs (LGH) and the (2 + 1)-dimensional Novikov-Veselov (NV) equations are two significant models. In this article, we have affirmed that wave profile changes with the change of the free parameters associated with them and are mainly dominated by linear effects. The effects of nonlinearity and wave speed on the wave contours have also been analyzed. On account of this, wave solutions are computed concerning rational, hyperbolic, and trigonometric structures balancing the exponents of linear and nonlinear terms of the highest order, from which scores of typical wave profiles including kink, bell-shape soliton, lump, and periodic waves have been extracted. The wave solutions are designated through extending the typical concept of the sine-Gordon expansion (SGE) method from the lower dimensional to the higher dimensional nonlinear evolution equations. This study establishes the capability of the stated method in solving both lower and higher dimensional nonlinear evolution equations. The solutions are analyzed by sketching figures for different values of related variables, and it is observed that the attributes of these solutions are pivotal in the selection of parameters.

Traveling, periodic and localized solitary waves solutions of the (4+1)-dimensional nonlinear Fokas equation

By utilizing the three distinctive approaches specifically, the extended Fan sub-equation method, the exp[ $$-G(\xi )$$ ]-function expansion method, and the fractional transformation method, the traveling waves and soliton solutions of the (4+1)-dimensional nonlinear Fokas equation are extracted. Meanwhile, some parametric constraint conditions are described. The acquired solutions are singular and nonsingular soliton solutions, periodic solutions, breather solution, rational solutions, trigonometric periodic wave solutions, hyperbolic solutions, Weierstrass, and Jacobi elliptic doubly periodic wave solutions. The dynamics of some of the obtained solutions are investigated and described in 2-dimensional figures by choosing appropriate parameter values. The comparison of our obtained results with the other solutions in literature shows that the obtained solutions of this paper are new and have not been formulated before by other techniques. We believe that all these results are useful to enrich the knowledge of the important physical phenomenon characterized by the Fokas equation. The reported solutions illustrate the straightforwardness, reliability, and effectiveness of the used techniques that can be further employed to higher-dimensional nonlinear evolution equations.

Localized waves and interactions for the high dimensional nonlinear evolution equations

In this paper, we construct some localized waves for the Bogoyavlenskii–Kadomtsev–Petviashili (BKP) and (3+1)-dimensional KP equations with the aid of the Hirota bilinear method and long wave limit. The BKP equation, as a modification of the KP equation, describes the shallow long waves and the (3+1)-dimensional KP equation can be applied to describe the propagation of nonlinear waves in fluids. Breather and breather-type kink soliton for the BKP equation are obtained from the two-soliton solutions; interactions between breather and kink soliton or between two breathers are obtained from the three-soliton solutions; interactions among one breather and two kink solitons, among two breathers and one kink soliton, or among three breathers are obtained from the four-soliton solutions. Furthermore, based on the explicit solutions with appropriate parameters, breathers of the (3+1)-dimensional KP equation have different behaviors in the different planes, and lump solutions are constructed via the long wave limit method. Results obtained in this paper may be helpful for understanding the propagation of localized waves.

Analytic study of solutions for a (3 + 1) -dimensional generalized KP equation

The (3 + 1) -dimensional generalized KP (gKP) equation is an important nonlinear partial differential equation in theoretical and mathematical physics which can be used to describe nonlinear wave motion. Through the Hirota bilinear method, one-solition, two-solition and N-solition solutions are derived via symbolic computation. Two classes of lump solutions, rationally localized in all directions in space, to the dimensionally reduced cases in (2 + 1)-dimensions, are constructed by using a direct method based on the Hirota bilinear form of the equation. It implies that we can derive the lump solutions of the reduced gKP equation from positive quadratic function solutions to the aforementioned bilinear equation. Meanwhile, we get interaction solutions between a lump and a kink of the gKP equation. The lump appears from a kink and is swallowed by it with the change of time. This work offers a possibility which can enrich the variety of the dynamical features of solutions for higher-dimensional nonlinear evolution equations.

Symmetry reduction and exact solutions of two higher-dimensional nonlinear evolution equations

In this paper, symmetries and symmetry reduction of two higher-dimensional nonlinear evolution equations (NLEEs) are obtained by Lie group method. These NLEEs play an important role in nonlinear sciences. We derive exact solutions to these NLEEs via the exp(-phi (z))-expansion method and complex method. Five types of explicit function solutions are constructed, which are rational, exponential, trigonometric, hyperbolic and elliptic function solutions of the variables in the considered equations.

An ansatz for solving nonlinear partial differential equations in mathematical physics.

In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. We apply this ansatz to examine new and further general traveling wave solutions to the (1+1)-dimensional modified Benjamin–Bona–Mahony equation. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general solitary wave solutions. The solutions emphasize the nobility of this ansatz in providing distinct solutions to various tangible phenomena in nonlinear science and engineering. The ansatz could be more efficient tool to deal with higher dimensional nonlinear evolution equations which frequently arise in many real world physical problems.

Cross-kink multi-soliton solutions for the (3+1)-D Jimbo-Miwa equation

Purpose – The purpose of this paper is to reveal the dynamical behavior of higher dimensional nonlinear wave by searching for the multi-wave solutions to the (3+1)-D Jimbo-Miwa equation. Design/methodology/approach – The authors apply bilinear form and extended homoclinic test approach to the (3+1)-D Jimbo-Miwa equation. Findings – In this paper, by using bilinear form and extended homoclinic test approach, the authors obtain new cross-kink multi-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equation, including the periodic breathertype of kink three-soliton solutions, the cross-kink four-soliton solutions, the doubly periodic breather-type of soliton solutions and the doubly periodic breather-type of cross-kink two-soliton solutions. It is shown that the extended homoclinic test approach, with the help of symbolic computation, provides an effective and powerful mathematical tool for solving higher dimensional nonlinear evolution equations in mathematical physics. Research limitations/implications – The research manifests that the structures of the solution to higher dimensional nonlinear equations are diversified and complicated. Originality/value – The methods used in this paper can be widely applied to the research of spatial and temporal characteristics of nonlinear equations in physics and engineering technology. These methods are also conducive for people to know objective laws and grasp the essential features of the development of the world.

A symmetry-preserving difference scheme for high dimensional nonlinear evolution equations

In this paper, a procedure for constructing discrete models of the high dimensional nonlinear evolution equations is presented. In order to construct the difference model, with the aid of the potential system of the original equation and compatibility condition, the difference equations which preserve all Lie point symmetries can be obtained. As an example, invariant difference models of the (2+1)-dimensional Burgers equation are presented.

The Using of Conservation Laws in Symmetry-Preserving Difference Scheme

In this paper, by means of the potential systems of the given nonlinear evolution equations, a procedure of symmetry preserving discretization of differential equations is presented. The specific process will be given detailed in section 2. This extended method is effective for discreting the high-order (high-dimensional) nonlinear evolution equations. As examples, the invariant difference models of the mKdV equation and the Boussinesq equation are constructed.

New infinite sequence soliton-like solutions of (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation

This paper will study in detail homogeneous linear ordinary differential equation with constant coefficients of second order and draw new conclusion to construct new infinite sequence soliton-like solutions of high-dimensional nonlinear evolution equations. Step one: the solving of a homogeneous linear ordinary differential equation with constant coefficients of second order is changed into the solving of the quadratic equation with one unknown and the Riccati equation. Based on this, new infinite sequence solutions of homogeneous linear ordinary differential equation with constant coefficients of second order are found by using nonlinear superposition formula for the solutions to Riccati equation. Step two: new infinite sequence soliton-like solutions to (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation are constructed using the above conclusion and the symbolic computation system Mathematica.