Generalized Breaking Soliton Research Articles

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.

Interactions between soliton and rogue wave for a (2+1)-dimensional generalized breaking soliton system: Hidden rogue wave and hidden soliton

A (2+1)-dimensional generalized breaking soliton system is under investigation, which can be used to describe the interaction between Riemann wave and long wave along with two space directions in nonlinear media. Rogue wave solutions, two types of mixed soliton and rogue wave solutions are constructed through appropriately choosing the functions in the bilinear forms. The rogue wave structures and the interaction characteristics between the soliton and rogue wave are studied in details. Novel hidden rogue wave and hidden soliton, which are invisible but work, are observed during the interactions.

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.

Multiple soliton solutions for the Bogoyavlenskii’s generalized breaking soliton equations and its extension form

In this work, two generalized breaking soliton equations, namely, the Bogoyavlenskii’s breaking soliton equation and its extended form, are examined. The complete integrability of these equation are justified. Multiple soliton solutions and multiple singular soliton solutions are formally derived for each equation. The additional terms of these equations do not kill the integrability of the typical breaking soliton equation. The Cole–Hopf transformation method and the simplified Hereman’s method are applied to conduct this analysis.

Generalized Riccati equation expansion method and its application to the Bogoyavlenskii's generalized breaking soliton equation

Based on the computerized symbolic system Maple and a Riccati equation, a Riccati equation expansion method is presented by a general ansatz. Compared with most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method, the proposed method is more powerful. By use of the method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some nonlinear differential equations. Making use of the method, we study the Bogoyavlenskii's generalized breaking soliton equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions.

New Exact Solutions for a Generalized Breaking Soliton Equation

Abstract The breaking soliton equations are a class of nonlinear evolution equations of broad interest in physical and mathematical sciences. In this paper, the application of the generalized tanh method with symbolic computation leads to new exact solutions for a generalized breaking soliton equation, of which the previously-obtained solutions are the special cases.

On a generalized breaking soliton equation

We find a new class of exact solutions for a generalized breaking soliton equation, which is linear with respect to y and independent of the Bogoyavlenskii parameter. Sample solutions are given.

Using symbolic computation to exactly solve for the Bogoyavlenskii's generalized breaking soliton equation

In this paper, the symbolic computation systems help us find some exact solutions for the Bogoyavlenskii's generalized breaking soliton equation, which are the sums of the arbitrary functions of time and parameterized solitary waves.