Dimensional Generalized Breaking Soliton Equation Research Articles

Bäcklund transformation, Wronskian solutions and interaction solutions to the (3+1)-dimensional generalized breaking soliton equation

Bäcklund transformation, Wronskian solutions and interaction solutions to the (3+1)-dimensional generalized breaking soliton equation

EVOLUTIONARY BEHAVIOR OF THE INTERACTION SOLUTIONS FOR A (3+1)-DIMENSIONAL GENERALIZED BREAKING SOLITON EQUATION

The interaction solutions have attracted the attention of many scholars because of they are valuable in analyzing the nonlinear dynamics of waves in shallow water and can be used for forecasting the appearance of rogue waves. In this paper, we investigate the interaction and rational solutions of a (3+1)-dimensional generalized breaking soliton equation by employing the Hirota bilinear and parameter limit methods along with symbolic computations. By studying the Hirota bilinear form of the equation, abundant interaction and rational solutions are derived by choosing appropriate parameters of the test function. The evolutionary behavior of the interaction solutions is also analyzed theoretically and graphically. Compare with the published literatures, we get some completely new results of the equation in this paper.

BREATHER-WAVE, MULTI-WAVE AND INTERACTION SOLUTIONS FOR THE (3+1)-DIMENSIONAL GENERALIZED BREAKING SOLITON EQUATION

In this paper, a (3+1)-dimensional generalized breaking soliton equation in nonlinear media is investigated. The interaction solution between lump wave and N-soliton (<i>N</i>=2, 3, 4) are derived. The interaction solution between lump wave and periodic waves is also studied. Breather-wave and multi-wave solutions are obtained. The dynamical behavior is demonstrated by some 3D graphics and density plots. Via means of mathematical induction, we also obtain the exact solution containing three arbitrary functions.

Dynamics of D’Alembert wave and soliton molecule for a ([formula omitted])-dimensional generalized breaking soliton equation

The D’Alembert solution is an important basic formula in linear partial differential theory due to that it can be considered as a general solution of the wave motion equation. However, the study of the D’Alembert wave is few works in nonlinear partial differential systems. In this paper, one construct the D’Alembert solution of a (2+1)-dimensional generalized breaking soliton equation which possesses the nonlinear terms. This D’Alembert wave has one arbitrary function in the traveling wave variable. We investigate the dynamics of the three soliton molecule, the soliton molecule by bound as an asymmetry soliton and one-soliton, the interaction between the half periodic wave and two-kink, and the interaction among the half periodic wave, one-kink and a kink soliton molecule of the (2+1)-dimensional generalized breaking soliton equation by selecting the appropriate parameters.

Mechanisms of nonlinear wave transitions in the (2+1)-dimensional generalized breaking soliton equation

Mechanisms of nonlinear wave transitions in the (2+1)-dimensional generalized breaking soliton equation

Periodic-wave solutions and asymptotic properties for a (3+1)-dimensional generalized breaking soliton equation in fluids and plasmas

In this paper, a (3+1)-dimensional generalized breaking soliton equation is investigated. Based on the one- and two-dimensional Riemann theta functions, one- and two-periodic-wave solutions are derived. We observe that the one-periodic wave is one-dimensional and is viewed as a superposition of the overlapping waves, placed one period apart. With certain parameters, the symmetric feature appears in the two-periodic wave, and the two-periodic wave degenerates to the one-periodic wave. With the series expansions, we explore the relations between the soliton and periodic-wave solutions. According to those relations, asymptotic properties for the periodic-wave solutions to approach to the soliton solutions under certain amplitude conditions are derived.

Closed-Form Solutions and Conserved Vectors of a Generalized (3+1)-Dimensional Breaking Soliton Equation of Engineering and Nonlinear Science

In this article, we examine a (3+1)-dimensional generalized breaking soliton equation which is highly applicable in the fields of engineering and nonlinear sciences. Closed-form solutions in the form of Jacobi elliptic functions of the underlying equation are derived by the method of Lie symmetry reductions together with direct integration. Moreover, the (G′/G)-expansion technique is engaged, which consequently guarantees closed-form solutions of the equation structured in the form of trigonometric and hyperbolic functions. In addition, we secure a power series analytical solution of the underlying equation. Finally, we construct local conserved vectors of the aforementioned equation by employing two approaches: the general multiplier method and Ibragimov’s theorem.

Bäcklund transformations and Riemann–Bäcklund method to a (3 + 1)-dimensional generalized breaking soliton equation

In this paper, the bilinear method is employed to investigate the N-soliton solutions of a (3 + 1)-dimensional generalized breaking soliton equation. Three sets of bilinear Backlund transformations are obtained by means of gauge transformation. The Riemann–Backlund method is further extended to the (3 + 1)-dimensional nonlinear integrable systems. The quasiperiodic wave solutions of the (3 + 1)-dimensional generalized breaking soliton equation are systematically analyzed. The asymptotic properties of the quasiperiodic solutions are discussed by using a limiting procedure. The one-periodic and two-periodic waves tend to the 1-soliton and 2-soliton under a small amplitude limit, respectively. The dynamical characteristics of the one- and two-periodic waves are summarized by selecting different parameters. Furthermore, we obtain some new types of the quasiperiodic wave solutions of the variable coefficient (3 + 1)-dimensional generalized breaking soliton equation. These solutions present the dynamical behaviors of C-type, anti-C-type and Z-type periodic waves moving on the background of the periodic waves of bell type.

Integrability, multiple-cosh , lumps and lump-soliton solutions to a (2+1)-dimensional generalized breaking soliton equation

In this paper, we focus on investigating a (2+1)-dimensional generalized breaking soliton (gBS) equation with five model parameters, which contains a lot of important nonlinear partial differential equations (PDEs) as its special cases. Firstly, the integrability features of two special cases of the gBS equation are clarified. Secondly, a general method is established to construct solutions formed by a combination of n−cosh and n−cos expressions. The similar results can be generalized to other PDEs which possess the Hirota bilinear forms. Thirdly, by introducing the nonzero seed solution, we obtain the real non-static lumps, lump-soliton solutions and other relevant exact solutions. The results expand the understanding of lump, freak wave and their interaction solutions in soliton theory. Moreover, various graphical analyses on the presented solutions are made to reveal the dynamic behaviors, which gives an essential improvement in the physical realizing of higher-dimensional lump waves in oceanography and nonlinear optics.

Multiwave, multicomplexiton, and positive multicomplexiton solutions to a (3 + 1)-dimensional generalized breaking soliton equation

There are a lot of physical phenomena which their mathematical models are decided by nonlinear evolution (NLE) equations. Our concern in the present work is to study a special type of NLE equations called the (3 + 1)-dimensional generalized breaking soliton (3D-GBS) equation. To this end, the linear superposition (LS) method along with a series of specific techniques are utilized and as an achievement, multiwave, multicomplexiton, and positive multicomplexiton solutions to the 3D-GBS equation are formally constructed. The study confirms the efficiency of the methods in handling a wide variety of nonlinear evolution equations.

Lump and interactional solutions of the (2+1)-dimensional generalized breaking soliton equation

In this paper, by using the long wave limit method, we study lump solution and interactional solution of the (2[Formula: see text]+[Formula: see text]1)-dimensional generalized breaking soliton equation without using bilinear form. The moving properties of the lump solution, and the interactional properties of a lump and a solitary wave, are analyzed theoretically and graphically with asymptotic analysis.

Lump-type solutions, rogue wave type solutions and periodic lump-stripe interaction phenomena to a (3 + 1)-dimensional generalized breaking soliton equation

In this paper, we present a bilinear form for an extended (2+1)-dimensional generalized breaking soliton equation, namely, a (3+1)-dimensional generalized breaking soliton (GBS) equation, through using the Hirota direct method. Abundant lump-type solutions, rogue wave type solutions, breather lump wave solutions and interaction solutions are constructed, based on the Hirota bilinear form. Particularly, we generate a type of new interaction solutions in terms of a new combination of quadratic function, trigonometric function and exponential function, namely, a kind of periodic lump-stripe solitons, which is a sort of mixed type solutions of periodic lump-type solitons and stripe solitons. We show that the periodic lump-type solitons will be swallowed by the stripe solitons after they collide. Finally, dynamics characteristics and evolution behaviors are exhibited for the obtained solution waves through particular plots with proper choices of different values for the parameters.

Characteristics of integrability, bidirectional solitons and localized solutions for a ( $$3+1$$ 3 + 1 )-dimensional generalized breaking soliton equation

The characteristics of integrability, bidirectional solitons and localized solutions are investigated for a ( $$3+1$$ )-dimensional breaking soliton (GBS) equation with general forms. Firstly, starting from the GBS equation, we perform the singularity manifold analysis and obtain a new integrable model in the sense of Painleve property. Secondly, taking advantage of the Bell polynomial approach, we construct the Backlund transformation, Lax pair and an infinite sequence of conservation laws. Subsequently, this new equation is also found to allow bidirectional soliton solutions, and the head-on and overtaking collisions between solitons are illustrated by some illustrative graphs. Finally, some localized excitations, such as lump solution, multi-dromions, periodic solitary waves solution, are obtained.

Characteristics of solitary wave, homoclinic breather wave and rogue wave solutions in a (2+1)-dimensional generalized breaking soliton equation

We consider a (2+1)-dimensional generalized breaking soliton (gBS) equation, which describes the interaction of the Riemann wave propagated along the y-axis with a long wave propagated along the x-axis. By using Bell’s polynomials, we derive a bilinear form of the gBS equation. Based on the resulting Hirota’s bilinear equation, we explicitly construct its soliton solutions. Furthermore, by using the extended homoclinic test theory, its homoclinic breather waves and rogue waves are well derived, respectively. It is hoped that our results can enrich the dynamical behavior of the gBS-type equations.

A New Integrable (2+1)-Dimensional Generalized Breaking Soliton Equation: N-Soliton Solutions and Traveling Wave Solutions

In this work, we study a new (2+1)-dimensional generalized breaking soliton equation which admits the Painleve property for one special set of parameters. We derive multiple soliton solutions, traveling wave solutions, and periodic solutions as well. We use the simplified Hirotas method and a variety of ansatze to achieve our goal.

Quasiperiodic wave solutions of a (2 + 1)-dimensional generalized breaking soliton equation via bilinear Bäcklund transformation

In this paper, we focus on a \( (2+1)\)-dimensional generalized breaking soliton equation, which describes the \( (2+1)\)-dimensional interaction of a Riemann wave propagating along the y -direction with a long wave along the x-direction. Based on a multidimensional Riemann theta function, the quasiperiodic wave solutions of a \( (2+1)\)-dimensional generalized breaking soliton equation are investigated by means of the bilinear Backlund transformation. The relations between the quasiperiodic wave solutions and the soliton solutions are rigorously established by a limiting procedure. The dynamical behaviors of the quasiperiodic wave solutions are discussed by presenting the numerical figures.

Integrability of a (2+1)-dimensional generalized breaking soliton equation

Under investigation in this letter is a (2+1)-dimensional generalized breaking soliton equation, which describes the (2+1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis. A singularity analysis is carried out and it is shown that this generalized equation admits the Painlevé property for one set of parametric choices. Some integrable properties of the corresponding Painlevé integrable equation, such as its bilinear form, N-soliton solution, bilinear Bäcklund transformation, Lax pair and infinite conservation laws are derived with the binary Bell polynomials.

Group Analysis and Nonlinear Self-Adjointness for a Generalized Breaking Soliton Equation

In this paper, a variable coefficient (2 + 1)-dimensional generalized breaking soliton equation is considered by means of the Lie group method. Having written an equation as a system of two dependent variables, we perform a complete group classification for the system. Consequently, for arbitrary functions, solutions of the generalized breaking soliton equation are connected with the ones of a variable coefficient Korteweg–de Vries equation by a transformation. For other cases, the reduced equations and exact solutions are constructed. Meanwhile, we prove that the system is nonlinearly self-adjoint and construct the general conservation law formulae.

Solitonic interaction of a variable-coefficient (2 + 1)-dimensional generalized breaking soliton equation

In fluids, Korteweg–de Vries-type equations are used to describe certain nonlinear phenomena. Studied in this paper is a variable-coefficient (2 + 1)-dimensional generalized breaking soliton equation, which models the interactions of Riemann waves with long waves. By virtue of the Bell-polynomial approach, bilinear forms of such an equation are obtained. N-soliton solutions are constructed in terms of the exponential functions and Wronskian determinant, respectively. Solitonic propagation and interaction are discussed with the following conclusions: (i) the appearance of characteristic lines such as the periodic and parabolic shapes depends on the form of the variable coefficients; and (ii) interactions of two solitons and three solitons are shown to be elastic.