Soliton Cnoidal Wave Solutions Research Articles

Nonlocal symmetries, soliton-cnoidal wave solution and soliton molecules to a (2+1)-dimensional modified KdV system

A (2+1)-dimensional modified KdV (2DmKdV) system is considered from several perspectives. Firstly, residue symmetry, a type of nonlocal symmetry, and the Bäcklund transformation are obtained via the truncated Painlevé expansion method. Subsequently, the residue symmetry is localized to a Lie point symmetry of a prolonged system, from which the finite transformation group is derived. Secondly, the integrability of the 2DmKdV system is examined under the sense of consistent tanh expansion solvability. Simultaneously, explicit soliton-cnoidal wave solutions are provided. Finally, abundant patterns of soliton molecules are presented by imposing the velocity resonance condition on the multiple-soliton solution.

Consistent Riccati expansion solvability and soliton–cnoidal wave solutions of a coupled KdV system

The consistent Riccati expansion (CRE) solvability of a coupled KdV system is investigated with the aid of the Riccati equation. Differing from the complete Painlevé integrability for three branches, this coupled KdV system is shown to be CRE solvable for one branch and non-CRE solvable for two other branches. Based on the last consistent differential equation under the CRE solvable case, a kind of soliton–cnoidal wave interaction solution is derived explicitly.

On the similarity reduction solutions of the Bogoyavlenskii equation

In the present paper, we attempt to investigate the nonlocal symmetries and similarity reduction solutions for the Bogoyavlenskii equation. The nonlocal symmetries are obtained by the truncated Painlevé expansion and localized to a prolonged system by introducing four auxiliary variables. Then we construct two types similarity reduction solutions. It is observed that some interesting soliton-cnoidal wave solutions can be generated. We further discuss their asymptotic behaviors both in analytical and in graphical ways.

Bäcklund transformations, nonlocal symmetry and exact solutions of a generalized (2+1)-dimensional Korteweg–de Vries equation

In this paper, nonlocal residual symmetry of a generalized (2+1)-dimensional Korteweg–de Vries equation is derived with the aid of truncated Painlevé expansion. Three kinds of non-auto and auto Bäcklund transformations are established. The nonlocal symmetry is localized to a Lie point symmetry of a prolonged system by introducing auxiliary dependent variables. The linear superposed multiple residual symmetries are presented, which give rise to the nth Bäcklund transformation. The consistent Riccati expansion method is employed to derive a Bäcklund transformation. Furthermore, the soliton solutions, fusion-type N-solitary wave solutions and soliton–cnoidal wave solutions are gained through Bäcklund transformations.

Bäcklund transformations, rational solutions and soliton–cnoidal wave solutions of the modified Kadomtsev–Petviashvili equation

In this paper, the truncated Painlevé expansion is employed to derive the Bäcklund transformation and the nth Bäcklund transformation of the modified Kadomtsev–Petviashvili equation. The method for constructing the rational solutions with the aid of the Bäcklund transformation is presented. It is proved that the modified Kadomtsev–Petviashvili equation is CRE solvable. Then the soliton–cnoidal wave solutions are explicitly presented. These results are useful in explaining nonlinear wave phenomena in oceanography.

Nonlocal symmetries, consistent tanh expansion solvability and interaction solutions for a new fifth-order nonlinear integrable equation

ABSTRACTThe nonlocal symmetries for a new fifth-order integrable equation are obtained with the truncated Painlevé method. The nonlocal symmetries can be localized to the Lie point symmetries by introducing auxiliary dependent variables and the corresponding finite transformations are computed directly. New exact solutions of the new fifth-order integrable equation are also proved to have the consistent tanh expansion form. New exact interaction excitations such as soliton–cnoidal wave solutions and soliton–periodic wave solutions are given out analytically and graphically.

High order nonlocal symmetries and exact interaction solutions of the variable coefficient KdV equation

In this paper, the nonlocal symmetries and exact interaction solutions of the variable coefficient Korteweg–de Vries (KdV) equation are studied. With the help of pseudo-potential, we construct the high order nonlocal symmetries of the time-dependent coefficient KdV equation for the first time. In order to construct the new exact interaction solutions, two auxiliary variables are introduced, which can transform nonlocal symmetries into Lie point symmetries. Furthermore, using the Lie point symmetries of the closed system, some exact interaction solutions are obtained. For some interesting solutions, such as the soliton–cnoidal wave solutions are discussed in detail, and the corresponding 2D and 3D figures are given to illustrate their dynamic behavior.

Residual symmetry, Bäcklund transformation and CRE solvability of a ( $$\mathbf{2}{\varvec{+}}{} \mathbf{1}$$ 2 + 1 )-dimensional nonlinear system

In this paper, the truncated Painleve expansion is employed to derive a Backlund transformation of a (\(2+1\))-dimensional nonlinear system. This system can be considered as a generalization of the sine-Gordon equation to \(2+1\) dimensions. The residual symmetry is presented, which can be localized to the Lie point symmetry by introducing a prolonged system. The multiple residual symmetries and the nth Backlund transformation in terms of determinant are obtained. Based on the Backlund transformation from the truncated Painleve expansion, lump and lump-type solutions of this system are constructed. Lump wave can be regarded as one kind of rogue wave. It is proved that this system is integrable in the sense of the consistent Riccati expansion (CRE) method. The solitary wave and soliton–cnoidal wave solutions are explicitly given by means of the Backlund transformation derived from the CRE method. The dynamical characteristics of lump solutions, lump-type solutions and soliton–cnoidal wave solutions are discussed through the graphical analysis.

Nonlocal symmetry and explicit solutions from the CRE method of the Boussinesq equation

In this paper, we analyze the integrability of the Boussinesq equation by using the truncated Painleve expansion and the CRE method. Based on the truncated Painleve expansion, the nonlocal symmetry and Backlund transformation of this equation are obtained. A prolonged system is introduced to localize the nonlocal symmetry to the local Lie point symmetry. It is proved that the Boussinesq equation is CRE solvable. The two-solitary-wave fusion solutions, single soliton solutions and soliton-cnoidal wave solutions are presented by means of the Backlund transformations.

A coupled KdV system: Consistent tanh expansion, soliton-cnoidal wave solutions and nonlocal symmetries

In this paper, a coupled KdV system is shown to be solvable, having a consistent tanh expansion solution. From the consistent tanh expansion solution, a kind of soliton-cnoidal interaction solution is obtained explicitly. Moreover, the nonlocal symmetries related to the truncated Painlevé expansion and the corresponding transformation groups are also given.

Residual symmetry, interaction solutions and consistent tanh expansion solvability for the third-order Burgers equation

The residual symmetries for the third-order Burgers equation are obtained with the truncated Painlevé method. The nonlocal symmetries can be localized to the Lie point symmetries by introducing auxiliary dependent variables and the corresponding finite transformations are computed directly. New exact solutions of the third-order Burgers equation is also proved to have the consistent tanh expansion form. New exact interaction excitations such as soliton-cnoidal wave solutions and soliton-periodic wave solutions are given out analytically and graphically.

Soliton–cnoidal interactional wave solutions for the reduced Maxwell–Bloch equations**Project supported by the Global Change Research Program of China (Grant No. 2015CB953904), the National Natural Science Foundation of China (Grant Nos. 11675054 and 11435005), the Outstanding Doctoral Dissertation Cultivation Plan of Action (Grant No. YB2016039), and Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213). Some

In this paper, we study soliton–cnoidal wave solutions for the reduced Maxwell–Bloch equations. The truncated Painlevé analysis is utilized to generate a consistent Riccati expansion, which leads to solving the reduced Maxwell–Bloch equations with solitary wave, cnoidal periodic wave, and soliton–cnoidal interactional wave solutions in an explicit form. Particularly, the soliton–cnoidal interactional wave solution is obtained for the first time for the reduced Maxwell–Bloch equations. Finally, we present some figures to show properties of the explicit soliton–cnoidal interactional wave solutions as well as some new dynamical phenomena.

Exact Interaction Solutions of an Extended (2+1)-Dimensional Shallow Water Wave Equation*

Applying the consistent Riccati expansion method, the extended (2+1)-dimensional shallow water wave equation is proved consistent Riccati solvable and the exact interaction solutions including soliton-cnoidal wave solutions, solitoff-typed solutions are obtained. With the help of the truncated Painlevé expansion, the corresponding nonlocal symmetry is also given, and furthermore, the nonlocal symmetry is localized by prolonging the related enlarged system.

Residual symmetry, interaction solutions, and conservation laws of the (2+1)-dimensional dispersive long-wave system**Project supported by the National Natural Science Foundation of China (Grant Nos. 11371293 and 11505090), the Natural Science Foundation of Shaanxi Province, China (Grant No. 2014JM2-1009), the Research Award Foundation for Outstanding Young Scientists of Shandong Province, China (Grant No. BS2015SF009), and the Science and Technology Innovation Foundation

We explore the (2+1)-dimensional dispersive long-wave (DLW) system. From the standard truncated Painlevé expansion, the Bäcklund transformation (BT) and residual symmetries of this system are derived. The introduction to an appropriate auxiliary dependent variable successfully localizes the residual symmetries to Lie point symmetries. In particular, it is verified that the (2+1)-dimensional DLW system is consistent Riccati expansion (CRE) solvable. If the special form of (CRE)-consistent tanh-function expansion (CTE) is taken, the soliton-cnoidal wave solutions and corresponding images can be explicitly given. Furthermore, the conservation laws of the DLW system are investigated with symmetries and Ibragimov theorem.

Nonlocal symmetries and interaction solutions for the KdV-type [formula omitted] equation

The nonlocal symmetries for the special K(m,n) equation, which is called KdV-type K(3,2) equation, are obtained by means of the truncated Painlevé method. The nonlocal symmetries can be localized to the Lie point symmetries by introducing auxiliary dependent variables and the corresponding finite symmetry transformations are computed directly. The KdV-type K(3,2) equation is also proved to be consistent tanh expansion solvable. New exact interaction excitations such as soliton–cnoidal wave solutions are given out analytically and graphically.

New interaction solutions to the combined KdV–mKdV equation from CTE method

The consistent tanh expansion (CTE) method is developed for the combined KdV–mKdV equation. The combined KdV–mKdV equation is proved to be CTE solvable. New exact interaction solutions such as soliton–cnoidal wave solutions, soliton–periodic wave solutions for the combined KdV–mKdV equation are given out analytically and graphically.

Nonlocal Symmetry and Consistent Tanh Expansion Method for the Coupled Integrable Dispersionless Equation

Abstract Nonlocal symmetries are obtained for the coupled integrable dispersionless (CID) equation. The CID equation is proved to be consistent, tanh-expansion solvable. New, exact interaction excitations such as soliton–cnoidal wave solutions, soliton–periodic wave solutions, and multiple resonant soliton solutions are discussed analytically and shown graphically.

Nonlocal symmetry and soliton–cnoidal wave solutions of the Bogoyavlenskii coupled KdV system

The truncated Painlevé expansion is developed to construct Bäcklund transformations and nonlocal symmetries of the Bogoyavlenskii coupled KdV (BcKdV) system. The Schwarzian form of BcKdV system is found while the nonlocal symmetry is localized to offer the corresponding nonlocal group. Furthermore, the BcKdV system is verified to have a consistent Riccati expansion (CRE). Stemming from the consistent tan-function expansion (CTE), which is a special form of CRE, the soliton–cnoidal wave solutions are explicitly studied.

Oblique propagation of ion acoustic soliton-cnoidal waves in a magnetized electron-positron-ion plasma with superthermal electrons

The oblique propagation of ion-acoustic soliton-cnoidal waves in a magnetized electron-positron-ion plasma with superthermal electrons is studied. Linear dispersion relations of the fast and slow ion-acoustic modes are discussed under the weak and strong magnetic field situations. By means of the reductive perturbation approach, Korteweg-de Vries equations governing ion-acoustic waves of fast and slow modes are derived, respectively. Explicit interacting soliton-cnoidal wave solutions are obtained by the generalized truncated Painlevé expansion. It is found that every peak of a cnoidal wave elastically interacts with a usual soliton except for some phase shifts. The influence of the electron superthermality, positron concentration, and magnetic field obliqueness on the soliton-cnoidal wave are investigated in detail.