The simplified Hirota’s method for studying three extended higher-order KdV-type equations

In this work we study three extended higher-order KdV-type equations.The Lax-type equation,the Sawada-Kotera-type equation and the CDG-type equation are derived from the extended KdV equation.We use the simplified Hirota’s direct method to derive multiple soliton solutions for each equation.We show that each model gives multiple soliton solutions,where the structures of the obtained solutions differ from the solutions of the canonical form of these equations.

Soliton fusion and fission for the high-order coupled nonlinear Schr?dinger system in fiber lasers

With the rapid development of communication technology,optical fiber communication has become a key research area in communications.When there are two signals in the optical fiber,the transmission of them can be abstracted as a high-order coupled nonlinear Schr¨odinger system.In this paper,by using the Hirota’s method,we construct the bilinear forms,and study the analytical solution of three solitons in the case of focusing interactions.In addition,by adjusting different wave numbers for phase control,we further discuss the influence of wave numbers on soliton transmissions.It is verified that wave numbers k_(11),k_(21),k_(31),k_(22),and k_(32)can control the fusion and fission of solitons.The results are beneficial to the study of all-optical switches and fiber lasers in nonlinear optics.

N-soliton Solution for Two Multidimensional Analogues of the m-KdV Equation

Using the Hirota's bilinear method,some new N-soliton solution are presented for two multidimensional analogues of the m-KdV equation wt+wxxx-6w 2 wx+3 2( w x -1 wy+w-x -1 wz)x=0 and wt+wxxx?6w 2 wx+3 2( wwy+wx-x-1 wy)=0 in view of a different treatment.

Covariant Prolongation Structure, Conservation Laws and Soliton Solutions of the Gross-Pitaevskii Equation in the Bose-Einstein Condensate

In this paper, we investigate the Gross-Pitaevskii (GP) equation which describes the propagation of an electron plasma wave packet with a large wavelength and small amplitude in a medium with a parabolic density and constant interactional damping by the Covariant Prolongation Structure Theory. As a result, we obtain general forms of Lax-Pair representations. In addition, some hidden structural symmetries that govern the dynamics of the GP equation such as SL(2,R), SL(2,C), Virasoro algebra, SU(1,1) and SU(2) are unearthed. Using the Riccati form of the linear eigenvalue problem, infinite number of conservation laws of the GP equation is explicitly constructed and the exact analytical soliton solutions are obtained by employing the simple and straightforward Hirota’s bilinear method.

The Lump Solutions of the (1 + 1)-Dimensional Ito-Equation

In this paper, several kinds of lump solutions for the (1 + 1)-dimensional Ito-equation are introduced. The proposed method in this work is based on a Hirota bilinear differential equation. The form of the solutions to the equation is constructed and the solutions are improved through analysis and symbolic computations with Maple. Finally, figure of the solution is made for specific examples for the lump solutions.

Integrability Tests and Some New Soliton Solutions of an Extended Potential Boiti-Leon-Manna-Pempinelli Equation

This paper is devoted to the study of a (2 + 1)-dimensional extended Potential Boiti-Leon-Manna-Pempinelli equation. Firstly, By means of the standard Weiss Tabor Carnevale approach and Kruskal’s simplification, we prove the painlevé non integrability of the equation. Secondly, A new breather solution and lump type solution are obtained based on the parameter limit method and Hirota’s bilinear method. Besides, some interaction behavior between lump type solution and N-soliton solutions (N is any positive integer) are studied. We construct the existence theorem of the interaction solution and give the process of calculation and proof. We also give a concrete example to illustrate the effectiveness of the theorem, and some spatial structure figures are displayed to reflect the evolutionary behavior of the interaction solutions with the change of soliton number N and time t.

Nonisospectral Lotka–Volterra Systems as a Candidate Model for Food Chain

In this paper,we derive a generalized nonisospectral semi-infinite Lotka-Volterra equation,which possesses a determinant solution.We also give its a Lax pair expressed in terms of symmetric orthogonal polynomials.In addition,if the simplified case of the moment evolution relation is considered,that is,without the convolution term,we also give a generalized nonisospectral finite Lotka-Volterra equation with an explicit determinant solution.Finally,an application of the generalized nonisospectral continuous-time Lotka-Volterra equation in the food chain is investigated by numerical simulation.Our approach is mainly based on Hirota’s bilinear method and determinant techniques.

On the interaction phenomena to the nonlinear generalized Hietarinta-type equation

In this paper,we describe the nonlinear behavior of a generalized fourth-order Hietarinta-type equa-tion for dispersive waves in(2+1)dimension.The various wave formations are retrieved by using Hirota’s bilinear method(HBM)and various test function perspectives.The Hirota method is a widely used and robust mathematical tool for finding soliton solutions of nonlinear partial differential equa-tions(NLPDEs)in a variety of disciplines like mathematical physics,nonlinear dynamics,oceanography,engineering sciences,and others requires bilinearization of nonlinear PDEs.The different wave structures in the forms of new breather,lump-periodic,rogue waves,and two-wave solutions are recovered.In addi-tion,the physical behavior of the acquired solutions is illustrated in three-dimensional,two-dimensional,density,and contour profiles by the assistance of suitable parameters.Based on the obtained results,we can assert that the employed methodology is straightforward,dynamic,highly efficient,and will serve as a valuable tool for discussing complex issues in a diversity of domains specifically ocean and coastal engineering.We have also made an important first step in understanding the structure and physical be-havior of complex structures with our findings here.We believe this research is timely and relevant to a wide range of engineering modelers.The results obtained are useful for comprehending the fundamental scenarios of nonlinear sciences.

Dynamical analysis of diversity lump solutions to the(2+1)-dimensional dissipative Ablowitz-Kaup-Newell-Segure equation

The lump solution is one of the exact solutions of the nonlinear evolution equation.In this paper,we study the lump solution and lump-type solutions of(2+1)-dimensional dissipative Ablowitz-Kaup-Newell-Segure(AKNS)equation by the Hirota bilinear method and test function method.With the help of Maple,we draw three-dimensional plots of the lump solution and lump-type solutions,and by observing the plots,we analyze the dynamic behavior of the(2+1)-dimensional dissipative AKNS equation.We find that the interaction solutions come in a variety of interesting forms.

Multiple-pole solutions and degeneration of breather solutions to the focusing nonlinear Schr?dinger equation

Based on the Hirota’s method,the multiple-pole solutions of the focusing Schr?dinger equation are derived directly by introducing some new ingenious limit methods.We have carefully investigated these multi-pole solutions from three perspectives:rigorous mathematical expressions,vivid images,and asymptotic behavior.Moreover,there are two kinds of interactions between multiple-pole solutions:when two multiple-pole solutions have different velocities,they will collide for a short time;when two multiple-pole solutions have very close velocities,a long time coupling will occur.The last important point is that this method of obtaining multiple-pole solutions can also be used to derive the degeneration of N-breather solutions.The method mentioned in this paper can be extended to the derivative Schr?dinger equation,Sine-Gorden equation,mKdV equation and so on.

Various exact wave solutions for KdV equation with time-variable coefficients

In this study,we investigate the(2+1)-dimensional Korteweg-De Vries(KdV)equation with the extension of time-dependent coefficients.A symbolic computational method,the simplified Hirota’s method,and a long-wave method are utilized to create various exact solutions to the suggested equation.The symbolic computational method is applied to create the Lump solutions and periodic lump waves.Hirota’s method and a long-wave method are implemented to explore single-,double-and triple-M-lump waves,and interaction physical phenomena such as an interaction of single-M-lump with one-,twosoliton solutions,as well as a collision of double-M-lump with single-soliton waves.Furthermore,the simplified Hirota’s method is employed to explore complex multi-soliton solutions.To realize dynamics,the gained solutions are drawn via utilizing different arbitrary variable coefficients.