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A new (2 + 1)-dimensional breaking soliton equation with the help of the nonisospectral Lax pair is presented. It is shown that the compatible solutions of the first two nontrivial equations in the (1 + 1)-dimensional Kaup–Newell soliton hierarchy provide solutions of the new breaking soliton equation. Then, the new breaking soliton equation is decomposed into the systems of solvable ordinary differential equations. Finally, a hyperelliptic Riemann surface and Abel–Jacobi coordinates are introduced to straighten the associated flow, from which the algebro-geometric solutions of the new (2 + 1)-dimensional integrable equation are constructed by means of the Riemann θ functions.

Based on the positive quadratic function method, the rich nonautonomous solutions of a generalized (2+1)-dimensional variable-coefficient breaking soliton equation with different wave structures are given. In this case, due to the influence of nonlinearity and dispersion, the characteristics, amplitude and velocity of nonautonomous wave will change with time. The breather wave and the interaction among lump wave, solitary wave and periodic wave solutions are studied. For different choices of arbitrary functions in these solutions, the corresponding dynamic properties are demonstrated.

In this article, the dynamics of modified nonlinear waves for the generalized breaking soliton (gBS) equation have been explored by using analytical and numerical approaches. Firstly, the Tanh method is used to derive the solitonic solutions of the considered equation and their graphical illustrations for different values of parameters have been discussed. By using the proposed strategy, a variety of solutions can be found. Furthermore, the Galilean transformation is used to investigate the bifurcation behavior of the model. All possible phase portraits have been drawn and examined in detail. More specifically, it is shown that under defined parametric restrictions, the model has a straightforward specific value. At each fixed point, the existence of periodic points and bifurcation analysis is also confirmed. To clarify theoretical results, detailed numerical simulations are offered. Then by introducing perturbed term, the periodic and quasi-periodic behaviors of the discussed equation for various initial values are reported. Different periodic solutions can also be found in this parameter zone. Furthermore, it is reported that discussed equation is multi-stable for different values of parameters. Finally, sensitivity analysis for different initial value problems has been elaborated.

In this paper, we have introduced the analytical solutions of the Benjamin-Bona-Mahony equation and the (2+1) dimensional breaking soliton equations with the help of a new Algorithm of first integral method formula two (AFIM), by depending on mathematical software's. New and more general variety of families of exact solutions have been represented by different structures of 3rd dimension plotting and contouring plotting with different parameters. So, the solution in this research is unique, new and more general. We can apply in computer sciences, mathematical physics, with a different vision, general and Programmable. [ABSTRACT FROM AUTHOR]

We have employed the exp(-φ(ξ))-expansion method to derive traveling waves solutions of breaking solition (BS), Zakharov-Kuznetsov-Burgers (ZKB), Ablowitz-Kaup-Newell-Segur (AKNS) water wave, Unstable nonlinear Schrödinger (UNLS) and Dodd-Bullough-Mikhailov (DBM) equations. These models have valuable applications in mathematical physics. The results of the constructed model, along with some graphical representations provide the basic knowlegde about these models. The derived results have various applications in applied science.

The present article deals with the symmetry reductions and invariant solutions of breaking soliton equation by virtue of similarity transformation method. The equation represents the collision of a Riemann wave propagating along the y-axis with a long wave along the x-axis. The infinitesimal transformations under one parameter for the governing system have been derived by exploiting the invariance property of Lie group theory. Consequently, the number of independent variables is reduced by one and the system remains invariant. A repeated application transforms the governing system into systems of ordinary differential equations. These systems degenerate well-known soliton solutions under some limiting conditions. The obtained solutions are extended with numerical simulation resulting in dark solitons, lumps, compactons, multisolitons, stationary and parabolic profiles and are shown graphically. [ABSTRACT FROM AUTHOR]

In this paper, some new invariant solutions of breaking soliton (BS) equation have been derived by using similarity transformations method. The system represents the interaction of Riemann waves propagating along y -axis and long waves along x -axis. The commutative relation and symmetry analysis of BS equation are derived using Lie group theory. Meanwhile, the method reduces the number of independent variables by one in each step. A repeated application of similarity transformations method reduces the BS equation into overdetermined equations, which provide invariant solutions. The derived results are more general than previous findings. The obtained solutions are supplemented by numerical simulation, which makes this research physically meaningful. Eventually, doubly soliton, multisoliton and asymptotic profiles of solutions are analyzed in the analysis and discussion section. [ABSTRACT FROM AUTHOR]

Under inquisition in this paper is a ( 2 + 1 ) -dimensional Breaking Soliton equation, which can describe various nonlinear scenarios in fluid dynamics. Using the Bell polynomials, some proficient auxiliary functions are offered to apparently construct its bilinear form and corresponding soliton solutions which are different from the previous literatures. Moreover, a direct method is used to construct its rogue wave and solitary wave solutions using particular auxiliary function with the assist of bilinear formalism. Finally, the interactions between solitary waves and rogue waves are offered with a complete derivation. These results enhance the variety of the dynamics of higher dimensional nonlinear wave fields related to mathematical physics and engineering. [ABSTRACT FROM AUTHOR]

We have employed the exp(-φ(ξ))-expansion method to derive traveling waves solutions of breaking so- lition (BS), Zakharov-Kuznetsov-Burgers (ZKB), Ablowitz-Kaup-Newell-Segur (AKNS) water wave, Unstable nonlinear Schrodinger (UNLS) and Dodd-Bullough-Mikhailov (DBM) equations. These models have valuable applications in mathematical physics. The results of the constructed model, along with some graphical representations provide the basic knowlegde about these models. The derived results have various applications in applied science. [ABSTRACT FROM AUTHOR]

In this paper, we apply the multiple exp-function method to obtain single-wave, double-wave and multi-wave solutions for the generalized (2+1)-dimensional nonlinear evolution equations. This method is executed to find the exact solutions for the (2+1)-dimensional breaking soliton equation, the (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation and the (2+1)-dimensional Bogoyavlenskii's breaking soliton equation. It has been shown that the proposed method is direct, straightforward and alternative, and can be used for some nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

The Boussinesq equation with dual dispersion and modified Korteweg–de Vries–Kadomtsev–Petviashvili equations describe weakly dispersive and small amplitude waves propagating in a quasi three-dimensional media. In this article, we study the analytical Bright-Dark solitary wave solutions for (3 + 1)-dimensional Breaking soliton equation, Boussinesq equation with dual dispersion and modified Korteweg–de Vries–Kadomtsev–Petviashvili equation have been extracted. These results hold numerous travelling wave solutions that are of key importance in elucidating some physical circumstance. The technique can also be functional to other sorts of nonlinear evolution equations in contemporary areas of research. [ABSTRACT FROM AUTHOR]

By combining the generalized variable separation method with the extended homoclinic test approach (EHTA) explicit exact non-traveling wave solutions of the (2+1)-dimensional breaking soliton equation are constructed. With the aid of symbolic computation, a series of new non-traveling wave solutions of the (2+1)-dimensional breaking soliton equation are expressed explicitly. These non-traveling wave solutions are new solutions with three arbitrary functions, which have a more general form than that in the previous literatures. The result obtained here reveals the complex structure of the solutions of the (2+1)-dimensional breaking soliton equation. The previous results obtained in literatures can be regarded as special cases here. When some arbitrary functions included in these solutions are taken as some special functions, exact periodic solitary wave, cross soliton-like wave, periodic cross-kink wave, periodic two-solitary wave are presented. [ABSTRACT FROM AUTHOR]

In this article, the dynamics of modified nonlinear waves for the generalized breaking soliton (gBS) equation have been explored by using analytical and numerical approaches. Firstly, the Tanh method is used to derive the solitonic solutions of the considered equation and their graphical illustrations for different values of parameters have been discussed. By using the proposed strategy, a variety of solutions can be found. Furthermore, the Galilean transformation is used to investigate the bifurcation behavior of the model. All possible phase portraits have been drawn and examined in detail. More specifically, it is shown that under defined parametric restrictions, the model has a straightforward specific value. At each fixed point, the existence of periodic points and bifurcation analysis is also confirmed. To clarify theoretical results, detailed numerical simulations are offered. Then by introducing perturbed term, the periodic and quasi-periodic behaviors of the discussed equation for various initial values are reported. Different periodic solutions can also be found in this parameter zone. Furthermore, it is reported that discussed equation is multi-stable for different values of parameters. Finally, sensitivity analysis for different initial value problems has been elaborated. • Solitonic structures of generalized breaking soliton. • Transformation of the considered model in the planer dynamical system. • Investigate the bifurcation behavior of the model by Galilean transformation. • Periodic and quasi-periodic behaviors of the (gBS) equation. • Multistability, Dynamic behavior of non-linear wave solutions of perturbed term with (gBS) equation. [ABSTRACT FROM AUTHOR]

Abstract: Using the compatibility method, we obtain three types of symmetry of the (3+1)-dimensional breaking soliton equation. By solving the corresponding characteristic equations associated with symmetry equations, symmetry reductions of the (3+1)-dimensional breaking soliton equation are given first, then some special types of similarity solutions are constructed. [Copyright &y& Elsevier]

Starting from the standard truncated Painleve´ expansion, by making use of the variable separation procedure, a general variable separation solution of the (2+1)-dimensional breaking soliton system is derived. Two classes of novel localized coherent structures like both multi-peakon–antipeakon solution and multi-compacton–anticompacton solution are found by selecting appropriate functions. These new structures exhibit some novel interaction features which are different from one of the known results. [Copyright &y& Elsevier]

The application of computer algebra to science has a bright future. In this paper, using computerized symbolic computation, new families of soliton-like solutions are obtained for (2+1)-dimensional breaking soliton equations using an ansatz. These solutions contain traveling wave solutions that are of important significance in explaining some physical phenomena. The method can also be applied to other types of nonlinear evolution equations in mathematical physics. [Copyright &y& Elsevier]

The possible high dimensional integrable models are studied in three different aspects: (i) starting from a strong symmetry operator of a known (1+1) -dimensional integrable model, we can construct a type of (n+1)-dimensional integrable models, high dimensional breaking soliton equations; (ii) from every concrete realization of the generalized Virasoro algebra, we can get many high dimensional integrable models in the meaning that the models possess generalized Virasoro symmetry algebra; (iii) starting from the Schwartz equations which possess conformal invariance, we can also get various high dimensional integrable models in the meaning that they possess Painlevé property. [ABSTRACT FROM AUTHOR]

By employing the Hirota's bilinear method and different test functions, we investigate different forms of breather solitons and some new lump solutions of the (2+1)-dimensional Breaking Soliton system. During this process, some new nonlinear phenomena, such as degeneration of breather solitons, evolution of double breathers and emergence of lump solutions, are studied and shown. Besides, some interaction solutions between high-order lump solutions and N -soliton solutions (N → ∞) are studied. We construct the existence theorem of the interaction solutions and give a detailed proof process for the first time. Some different types of interaction solutions are used as concrete examples to illustrate the effectiveness of the described theorem, such as rational-exponential type, rational-cosh-cos type, rational-logarithmic type, and some three-dimensional spatial structure figures are simulated and displayed to reflect the evolutionary behavior of the interaction solutions with the increase of soliton number N. • Some different forms of breathers and lump solutions are obtained. • The existence theory of interaction between lump and N solitons is proved. • Some new nonlinear phenomena and spatial structures are reported. • Some different types of interaction solutions are simulated and displayed. [ABSTRACT FROM AUTHOR]