Bilinear Backlund Transformation Research Articles

The soliton solutions and combined solutions of a high-dimensional wave soliton equation

In this paper, a high-dimensional wave soliton equation is considered and the simple Hirota method and bilinear backlund transformation are applied to construct the new soliton solutions, lump soliton solutions, breather solutions and their combined solutions. Then, through long-wave limit method and appropriate parameter constraints, one-soliton, one-breather, combined soliton-lump and lump-breather and soliton-breather solutions to nonlinear wave equation are obtained. Moreover, the physical interaction and frontal collision phenomena to the solutions of this equation are explored. In addition, a kind of hyperbolic and trigonometric ansatz is employed to derive other solutions such as kink soliton solution, periodic solutions for the high-dimensional wave equation.The obtained results verify the proposed solutions.

Bilinear Bäcklund transformation, kink and breather-wave solutions for a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation in fluid mechanics

Fluid mechanics studies are applied in the turbines, airplanes, ships, rivers, windmills, pipes, etc. Under investigation is a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation in fluid mechanics. Via the Hirota bilinear method, bilinear Backlund transformation is obtained, based on which the kink solutions are constructed. Breather-wave solutions are constructed via the extended homoclinic test approach. When the periods of the breather-wave solutions tend to infinity, the lump solutions are derived. It is found that the amplitudes and shapes of the breather waves keep unchanged during the propagation. Effects of the coefficients in the equation on the amplitudes and velocities for the breather waves are analyzed graphically.

Bilinear Bäcklund transformation, Lax pair and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physics

In this paper, outcomes of the study on the Backlund transformation, Lax pair, and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics, fluid mechanics, and plasma physics are presented. Via the Hirota bilinear method, a bilinear Backlund transformation is obtained, based on which a Lax pair is constructed. Via the symbolic computation, mixed rogue–solitary and rogue–periodic wave solutions are derived. Interactions between the rogue waves and solitary waves, and interactions between the rogue waves and periodic waves, are studied. It is found that (1) the one rogue wave appears between the two solitary waves and then merges with the two solitary waves; (2) the interaction between the one rogue wave and one periodic wave is periodic; and (3) the periodic lump waves with the amplitudes invariant are depicted. Furthermore, effects of the noise perturbations on the obtained solutions will be investigated.

Bäcklund transformation, Pfaffian, Wronskian and Grammian solutions to the $$(3+1)$$-dimensional generalized Kadomtsev–Petviashvili equation

With the Hirota bilinear method and symbolic computation, we investigate the $$(3+1)$$ -dimensional generalized Kadomtsev–Petviashvili equation. Based on its bilinear form, the bilinear Backlund transformation is constructed, which consists of four equations and five free parameters. The Pfaffian, Wronskian and Grammian form solutions are derived by using the properties of determinant. As an example, the one-, two- and three-soliton solutions are constructed in the context of the Pfaffian, Wronskian and Grammian forms. Moreover, the triangle function solutions are given based on the Pfaffian form solution. A few particular solutions are plotted by choosing the appropriate parameters.

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.

Lump solutions and bilinear Bäcklund transformation for the $$(4+1)$$-dimensional Fokas equation

The $$(4+1)$$ -dimensional Fokas (4DFK) equation is explicitly written out by means of the linearized operator of the 4DFK equation after introducing an additional auxiliary variable. Based on the bilinear form, bilinear Backlund transformations consisting of three bilinear equations are provided. A new class of lump solutions is given and analyzed using the extremum theory of multivariate function, of which the peculiar dynamic property is demonstrated graphically.

Bilinear representations and lump-type waves for a fifth-order nonlinear wave equation

In this work, we study a fifth-order nonlinear wave equation. By using the proper transformation and auxiliary parameter, two different bilinear representations are presented. We also derive a bilinear Backlund transformation, a Lax pair and lump-type wave solutions. The dynamics of lump-type wave solutions are investigated and exhibited mathematically and graphically. We further investigate the interaction between a kink-type solitary wave and a lump-type wave. Absorb and emit interactions between two types of solitary wave solutions are presented.

Integrability aspects and localized wave solutions for a new $$\mathbf (4+1) $$-dimensional Boiti–Leon–Manna–Pempinelli equation

In this paper, we introduce a new integrable nonlinear evolution equation in $$4+1$$ dimensions, which is an extension of Boiti–Leon–Manna–Pempinelli equation. We prove that this new equation has the Painleve property. By using the Bell polynomial approach, we obtain the bilinear representation, bilinear Backlund transformation, Lax pair and infinite conservation laws. Furthermore, several types of new exact solutions are also constructed based on the Hirota bilinear method, including the N-soliton solutions, periodic soliton solutions and mixed lump–kink wave solutions. The dynamics and interactions of localized wave solutions are illustrated by some graphs.

Lax pair and lump solutions for the (2+1)-dimensional DJKM equation associated with bilinear Bäcklund transformations

We aim to explore exact solutions and integrable properties to the (2+1)-dimensional DJKM equation. Based on the bilinear Backlund transformation, we first furnish Lax pair and complex exponential wave function solutions, and then give complexitons or hyperbolic function solutions. Moreover, via the nonlinear superposition formula, the construction procedure for presenting rational solutions is improved. The key step is that all the involved parameters are extended to the complex field. In particular, we show that the (2+1)-dimensional DJKM equation possesses a general class of lump solutions when $${\sigma }^2=-1$$.

On the integrability and Riemann theta functions periodic wave solutions of the Benjamin Ono equation

In this paper, the complete integrability of the Benjamin Ono equation is systematically studied. Its bilinear equation, soliton solutions, bilinear Backlund transformation and Lax pair are successfully obtained, by virtue of generalized Bell’s polynomials scheme. Moreover, by using multidimensional Riemann theta functions, the periodic wave solutions of the Benjamin Ono equation are constructed. Further, the asymptotic behaviors of the periodic wave solutions are presented with a limiting procedure, which shows the relations between the periodic wave solutions and soliton solutions.

Solitonic interaction and Bäcklund transformation for a generalized inhomogeneous coupled nonlinear Schrödinger system

A generalized inhomogeneous coupled nonlinear Schrodinger system, which has certain applications in nonlinear optics or Bose–Einstein condensation, is investigated. The bilinear form and bilinear Backlund transformation are obtained via the Hirota method. N-soliton solutions are derived, and we graphically investigate the effects of the time-dependent coefficients on the solitons and their interactions. Physical quantities, such as the amplitude, width, velocity, and energy, are also derived. We see that the time-dependent coefficients have no influence on the amplitude of the soliton. With the changes in the amplitudes, the two solitons can, respectively, be enhanced and suppressed after the interaction. Interactions between the two cubic and two periodic solitons are also derived. The interactions are inelastic.

A direct Bäcklund transformation for a (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation

In this paper, a Kadomtsev–Petviashvili–Boussinesq-like equation in (3+1)-dimensions is firstly introduced by using the combination of the Hirota bilinear Kadomtsev–Petviashvili equation and Boussinesq equation in terms of function f. And then a direct bilinear Backlund transformation of this new model is constructed, which consists of seven bilinear equations and ten arbitrary parameters. Based on this constructed bilinear Backlund transformation, some classes of exponential and rational traveling wave solutions with arbitrary wave numbers are presented.

Bäcklund transformation, multiple wave solutions and lump solutions to a (3 + 1)-dimensional nonlinear evolution equation

In this paper, a $$(3+1)$$ -dimensional nonlinear evolution equation is cast into Hirota bilinear form with a dependent variable transformation. A bilinear Backlund transformation is then presented, which consists of six bilinear equations and involves nine arbitrary parameters. With multiple exponential function method and symbolic computation, nonresonant-typed one-, two-, and three-wave solutions are obtained. Furthermore, two classes of lump solutions to the dimensionally reduced cases with $$y=x$$ and $$y=z$$ are both derived. Finally, some figures are given to reveal the propagation of multiple wave solutions and lump solutions.

Bilinear Bäcklund transformations and explicit solutions of equation in (3+1)-dimension

Based on Hirota's technique we have obtained two forms of bilinear Backlund transformations for (3+1)-dimensional generalised Kadomtsev-Petviashvili equation which are not reported before. The first form involves five arbitrary parameters and second form involves eight arbitrary parameters. In addition to this, the exponential and stationary rational wave solutions are also obtained by proposed bilinear Backlund transformations.

Bilinear Bäcklund transformations and explicit solutions of a ( 3 + 1 ) $(3+1)$ -dimensional nonlinear equation

Binary Bell polynomials are applied to construct two kinds of bilinear derivative equations of a $(3+1)$ -dimensional nonlinear equation. Based on one of the bilinear forms, we derive a Backlund transformation, the corresponding Lax pair, infinite conservation laws, and explicit solutions with an arbitrary function in y. In the meantime, from the other bilinear form, we get another bilinear Backlund transformation and exact solutions by utilizing the exchange formulas for Hirota’s bilinear operators.

Lump solution and integrability for the associated Hirota bilinear equation

This paper studies lump solution and integrability for the associated Hirota bilinear equation. The integrability in the sense of Lax pair and the bilinear Backlund transformations is presented by the binary Bell polynomial method. The lump solution is derived when the period of complexiton solution goes to infinite. Conversely, complexiton solution can also be derived from the lump solution. Complexiton solution is a superposition structure of lump solutions. The dynamics of the lump solution are investigated and exhibited mathematically and graphically. These results further supplement and enrich the theories for the associated Hirota bilinear equation. It is hoped that these results might provide us with useful information on the dynamics of the relevant fields in nonlinear science.

Painlevé analysis, integrability and exact solutions for a (2 + 1)-dimensional generalized Nizhnik-Novikov-Veselov equation

Under investigation in this paper is a (2 + 1)-dimensional generalized NNV equation, which includes as many important nonlinear models as its particular cases. First, we perform the Painleve test for the generalized NNV equation with the help of symbolic computation, and it is shown that this generalized equation admits the Painleve property for one set of parametric choices. For the newly obtained integrable equation, we then employ the binary Bell polynomial method to construct the bilinear form, N-soliton solution, bilinear Backlund transformation and Lax pair in a systematic way. In addition, some new doubly periodic wave solutions with two arbitrary functions are obtained by means of truncated Painleve expansions. Finally, the collisions of multiple solitons and periodic waves are interesting and shown by some graphs.

Quasi-periodic wave solutions and asymptotic behavior for an extended ( 2 + 1 ) $(2+1)$ -dimensional shallow water wave equation

Based on Riemann theta function and bilinear Backlund transformation, quasi-periodic wave solutions are constructed for an extended $(2+1)$ -dimensional shallow water wave equation. A detail asymptotic analysis procedure to the one- and two-periodic wave solutions are presented, and the asymptotic properties of this type of solutions are proved. It is shown that the quasi-periodic wave solutions converge to the soliton solutions under small amplitude limits.

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.

On Lie symmetries, exact solutions and integrability to the KdV-Sawada-Kotera-Ramani equation

In this paper, the KdV-Sawada-Kotera-Ramani equation is investigated, which is used to describe the resonances of solitons in one-dimensional space. By using the Lie symmetry analysis method, the vector field and optimal system of the equation are derived, respectively. The optimal system is further used to study the symmetry reductions and exact solutions. Furthermore, the exact analytic solutions of the equation can be obtained by considering the power series theory. Finally, the complete integrability of the equation is systematically presented by using binary Bell’s polynomials, which includes the bilinear representation, bilinear Backlund transformation, Lax pair and infinite conservation laws. Based on its bilinear representation, the N-soliton solutions of the equation are also constructed with exact analytic expression.