Soliton-cnoidal Wave Interaction Solutions Research Articles

Non-local residual symmetry and soliton-cnoidal periodic wave interaction solutions of the KdV6 equation

The residual symmetry of the KdV6 equation is obtained by the Painlevé truncate expansion. Since the residual symmetry is non-local, five field quantities are introduced to localize it into the local one. Besides, the interaction solutions between solitons and cnoidal periodic waves of the KdV6 equation are constructed by making use of the consistent tanh expansion method. As an illustration, a specific interaction solution in the form of tanh function and Jacobian elliptic function is discussed both analytically and graphically.

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.

Multiple residual symmetries and soliton-cnoidal wave interaction solution of the $$(2+1)$$-dimensional negative-order modified Calogero–Bogoyavlenskii–Schiff equation

The residual symmetry for the \((2+1)\)-dimensional negative-order modified Calogero–Bogoyavlenskii–Schiff (nmCBS) equation is derived from the truncated Painleve expansion, and is extended to the multiple residual symmetries, which can be transformed to Lie point symmetries by introducing a suitable prolonged system. The nth Backlund transformation (BT) related to multiple residual symmetries is given in terms of determinant. More importantly, we obtain the explicit soliton-cnoidal wave interaction solution from a consistent differential equation.

Investigation of Interaction Solutions for Modified Korteweg-de Vries Equation by Consistent Riccati Expansion Method

A consistent Riccati expansion (CRE) method is proposed for obtaining interaction solutions to the modified Korteweg-de Vries (mKdV) equation. Using the CRE method, it is shown that interaction solutions such as the soliton-tangent (or soliton-cotangent) wave cannot be constructed for the mKdV equation. More importantly, exact soliton-cnoidal periodic wave interaction solutions are presented. While soliton-cnoidal interaction solutions were found to degenerate to special resonant soliton solutions for the values of modulus (n) closer to one (upper bound of modulus) in the Jacobi elliptic function, a normal kink-shaped soliton was observed for values of n closer to zero (lower bound).

Consistent Riccati expansion solvable classification and soliton-cnoidal wave interaction solutions for an extended Korteweg-de Vries equation

In this paper, we consider an extended Korteweg-de Vries (KdV) equation. Using the consistent Riccati expansion (CRE) method of Lou, the extended KdV equation is proved to be CRE solvable in only two distinct cases. These two CRE solvable models are the KdV-Lax and KdV-Sawada-Kotera (KdV-SK) equations. In addition, applying the nonauto-Bäcklund transformations which are provided by the CRE method, we present the explicit construction for soliton-cnoidal wave interaction solutions which represent a soliton propagating on a cnoidal periodic wave background in the KdV-Lax and KdV-SK equations, respectively.

Explicit soliton–cnoidal wave interaction solutions for the (2+1)-dimensional negative-order breaking soliton equation

ABSTRACTBy using the truncated Painlevé expansion, we construct the residual symmetry for the (2+1)-dimensional negative-order breaking soliton equation. It is found that such nonlocal symmetry can be transformed to local Lie point symmetries by introducing auxiliary variables. The multiple residual symmetries are presented via the linear superposition of the single one. Based on the localized multiple residual symmetries, the n-th Bäcklund transformation is obtained in terms of determinant. Via the aid of the link between the truncated Painlevé expansion and the CTE method, we derive the explicit soliton–cnoidal wave interaction solutions for the (2+1)-dimensional NBS equation.

Exact PsTd invariant and PsTd symmetric breaking solutions, symmetry reductions and Bäcklund transformations for an AB–KdV system

In natural and social science, many events happened at different space–times may be closely correlated. Two events, A (Alice) and B (Bob) are defined as correlated if one event is determined by another, say, B=fˆA for suitable fˆ operators. A nonlocal AB–KdV system with shifted-parity (Ps, parity with a shift), delayed time reversal (Td, time reversal with a delay) symmetry where B=PsˆTdˆA is constructed directly from the normal KdV equation to describe two-area physical event. The exact solutions of the AB–KdV system, including PsTd invariant and PsTd symmetric breaking solutions are shown by different methods. The PsTd invariant solution show that the event happened at A will happen also at B. These solutions, such as single soliton solutions, infinitely many singular soliton solutions, soliton–cnoidal wave interaction solutions, and symmetry reduction solutions etc., show the AB–KdV system possesses rich structures. Also, a special Bäcklund transformation related to residual symmetry is presented via the localization of the residual symmetry to find interaction solutions between the solitons and other types of the AB–KdV system.

Nonlocal symmetry, Darboux transformation and soliton–cnoidal wave interaction solution for the shallow water wave equation

In classical shallow water wave (SWW) theory, there exist two integrable one-dimensional SWW equation [Hirota–Satsuma (HS) type and Ablowitz–Kaup–Newell–Segur (AKNS) type] in the Boussinesq approximation. In this paper, we mainly focus on the integrable SWW equation of AKNS type. The nonlocal symmetry in form of square spectral function is derived starting from its Lax pair. Infinitely many nonlocal symmetries are presented by introducing the arbitrary spectrum parameter. These nonlocal symmetries can be localized and the SWW equation is extended to enlarged system with auxiliary dependent variables. Then Darboux transformation for the prolonged system is found by solving the initial value problem. Similarity reductions related to the nonlocal symmetry and explicit group invariant solutions are obtained. It is shown that the soliton–cnoidal wave interaction solution, which represents soliton lying on a cnoidal periodic wave background, can be obtained analytically. Interesting characteristics of the interaction solution between soliton and cnoidal periodic wave are displayed graphically.

Bäcklund transformation and soliton–cnoidal wave interaction solution for the coupled Klein–Gordon equations

In this paper, Backlund transformation and interaction solutions of the generalized coupled Klein–Gordon (cKG) equations with the variable coefficients are studied via the residual symmetry and the consistent Riccati expansion (CRE) method. From the truncated Painleve expansion, we drive the residual symmetry which can be transformed into the local Lie point symmetry. The multiple residual symmetries are established and nth Backlund transformation in terms of determinant is presented. The cKG equations are proved to be CRE solvable by using the CRE method. More importantly, we construct exact interaction solutions between soliton and cnoidal periodic wave. It is shown that when the modulus of the Jacobi elliptic function tends to one, the interaction solutions degenerate to the special resonant soliton solutions. Moreover, these reduced resonant soliton solutions exhibit the interesting properties which differ from the usual resonant soliton solutions.

CRE Solvability, Nonlocal Symmetry and Exact Interaction Solutions of the Fifth-Order Modified Korteweg-de Vries Equation**Supported by National Natural Science Foundation of China under Grant No. 11505090, and Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No. BS2015SF009

This paper is concerned with the fifth-order modified Korteweg-de Vries (fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion (CRE) solvable. Three special form of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically. Furthermore, based on the consistent tanh expansion (CTE) method, the nonlocal symmetry related to the consistent tanh expansion (CTE) is investigated, we also give the relationship between this kind of nonlocal symmetry and the residual symmetry which can be obtained with the truncated Painlevé method. We further study the spectral function symmetry and derive the Lax pair of the fmKdV equation. The residual symmetry can be localized to the Lie point symmetry of an enlarged system and the corresponding finite transformation group is computed.

Residual symmetries and soliton-cnoidal wave interaction solutions for the negative-order Korteweg–de Vries equation

The residual symmetry is derived for the negative-order Korteweg–de Vries equation from the truncated Painlevé expansion. This nonlocal symmetry is transformed into the Lie point symmetry and the finite symmetry transformation is presented. The multiple residual symmetries are constructed and localized by introducing new auxiliary variables, and then n th Bäcklund transformation in terms of determinant is provided. With the help of the consistent tanh expansion (CTE) method, the explicit soliton-cnoidal wave interaction solutions are obtained from the last consistent differential equation.

Nonlocal Symmetries and Consistent Riccati Expansions of the (2+1)-Dimensional Dispersive Long Wave Equation

Abstract In this article, the (2+1)-dimensional dispersive long wave equation (DLWE) is investigated, which is derived in the context of a water wave propagating in narrow infinitely long channels of finite constant depth. By using of the truncated Painlevé expansion, we construct its nonlocal symmetry and Bäcklund transformation. After implanting the equation into an enlarged one, then the residual symmetry is localised. Meanwhile, the symmetry group transformation can be computed from the prolonged system. Furthermore, the equation is verified to be consistent Riccati expansion (CRE) solvable. Outing from the CRE, the soliton-cnoidal wave interaction solution in terms of Jacobi elliptic functions and the third type of incomplete elliptic integral are studied, respectively.

Nonlocal symmetry, CRE solvability and soliton–cnoidal solutions of the ( $$2+1$$ 2 + 1 )-dimensional modified KdV-Calogero–Bogoyavlenkskii–Schiff equation

In this paper, a modified KdV-CBS equation is investigated by using the truncated Painleve expansion and consistent Riccati expansion method, respectively. It is shown that the modified KdV-CBS equation has a nonlocal symmetry related to the residue of its truncated Painleve expansion. It is also proved that the modified KdV-CBS equation is consistent Riccati expansion solvable. Furthermore, with the help of the consistent Riccati expansion method, the soliton–cnoidal wave interaction solutions are explicitly given.

CRE solvability and soliton-cnoidal wave interaction solutions of the dissipative (2+1)-dimensional AKNS equation

By using the truncated Painlevé expansion and consistent Riccati expansion (CRE), we investigate a dissipative ( 2 + 1 )-dimensional Ablowitz–Kaup–Newell–Segur (AKNS) equation. Through the truncated Painlevé expansion, its nonlocal symmetry and Bäcklund transformation (BT) are presented. Then the nonlocal symmetry is localized to the corresponding nonlocal group by an enlarged system. Based on the CRE method proposed by Lou (2013), the AKNS equation is proved CRE solvable, and the soliton-cnoidal wave interaction solutions are explicitly given.

Nonlocal symmetry and consistent Riccati expansion integrability of the (1+1)-dimensional integrable nonlinear dispersive-wave system

Under investigation in this paper is nonlocal symmetry, consistent Riccati expansion (CRE) integrability of the (1+1)-dimensional integrable nonlinear dispersive-wave system, which can be used to describes a bidirectional soliton for wave propagation. We construct the Bäcklund transformation and consider the truncated Painlevé expansion of the system. It’s Schwarzian form is derived, whose nonlocal symmetry is localized to provide the corresponding nonlocal group. Furthermore, we verify that the system is solvable via the CRE. Based on the CRE, we further present its soliton-cnoidal wave interaction solution in terms of Jacobi elliptic functions and the third type of incomplete elliptic integral.

Nonlinear Self-Adjointness, Conservation Laws and Soliton-Cnoidal Wave Interaction Solutions of (2+1)-Dimensional Modified Dispersive Water-Wave System**Supported by National Natural Science Foundation of China under Grant Nos. 11371293, 11505090, the Natural Science Foundation of Shaanxi Province under Grant No. 2014JM2-1009, Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No. BS2015SF009 and the Science and Technology Innovation Foundation of

This paper mainly discusses the (2+1)-dimensional modified dispersive water-wave (MDWW) system which will be proved nonlinear self-adjointness. This property is applied to construct conservation laws corresponding to the symmetries of the system. Moreover, via the truncated Painlevé analysis and consistent tanh-function expansion (CTE) method, the soliton-cnoidal periodic wave interaction solutions and corresponding images will be eventually achieved.

Painlevé Integrability, Consistent Riccati Expansion Solvability and Interaction Solution for the Coupled mKdV-BLMP System**Supported by the Natural Science Foundation of Zhejiang Province of China under Grant No LY14A010005.

The integrability of the coupled, modified KdV equation and the potential Boiti–Leon–Manna–Pempinelli (mKdV-BLMP) system is investigated using the Painlevé analysis approach. It is shown that this coupled system possesses the Painlevé property in both the principal and secondary branches. Then, the consistent Riccati expansion (CRE) method is applied to the coupled mKdV-BLMP system. As a result, it is CRE solvable for the principal branch while non-CRE solvable for the secondary branch. Finally, starting from the last consistent differential equation in the CRE solvable case, soliton, multiple resonant soliton solutions and soliton-cnoidal wave interaction solutions are constructed explicitly.

Consistent Riccati expansion solvability and soliton–cnoidal wave interaction solution of a (2+1)-dimensional Korteweg–de Vries equation

In this paper, a ( 2 + 1 ) -dimensional KdV equation is investigated by using the consistent Riccati expansion (CRE) method proposed by Lou (2015). It is proved that the ( 2 + 1 ) -dimensional KdV equation is CRE solvable. Furthermore, soliton, cnoidal wave and soliton–cnoidal wave interaction solutions are obtained explicitly from different special solutions of the modified Schwarzian equation.

Painlevé analysis, nonlocal symmetry and explicit interaction solutions for supersymmetric mKdVB equation

The N=1 supersymmetric mKdVB system is transformed to a coupled bosonic system by using the bosonization approach. By a singularity structure analysis, the bosonized supersymmetric mKdVB (BSmKdVB) equation admits the Painlevé property. Starting from the standard truncated Painlevé method, the nonlocal symmetry for the BSmKdVB equation is obtained. To solve the first Lie’s principle related with the nonlocal symmetry, the nonlocal symmetry is localized to the Lie point symmetry by introducing multiple new fields. Thanks to localization processes, similarity reductions for the prolonged systems are studied by the Lie point symmetry method. The interaction solutions among solitons and other complicated waves including Painlevé II waves and periodic cnoidal waves are given through the reduction theorems. The soliton-cnoidal wave interaction solutions are explicitly given by using the mapping and deformation method. The concrete soliton-cnoidal interaction solutions are displayed both in analytical and graphical ways.

Interaction Behaviours Between Solitons and Cnoidal Periodic Waves for (2+1)-Dimensional Caudrey—Dodd—Gibbon—Kotera—Sawada Equation**Supported by the National Natural Science Foundation of China under Grant No. 11505154, the Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ16A010003, and the Scientific Research Foundation for Doctoral Program of Zhejiang Ocean University under Grant No. Q1511

The consistent tanh expansion (CTE) method is employed to the (2+1)-dimensional Caudrey—Dodd—Gibbon-Kotera—Sawada (CDGKS) equation. The interaction solutions between solitons and the cnoidal periodic waves are explicitly obtained. Concretely, we discuss a special kind of interaction solution in the form of tanh functions and Jacobian elliptic functions in both analytical and graphical ways. The results show that the profiles of the soliton-cnoidal periodic wave interaction solutions can be designed by choosing different values of wave parameters.