Symmetry Reduction Solutions Research Articles

Residual symmetry and interactions between solitons and periodic waves of the Drinfeld-Sokolov-Satsuma-Hirota system

Abstract In this paper, the Drinfeld-Sokolov-Satsuma-Hirota (DSSH) system is studied by using residual symmetry and consistent Riccati expansion (CRE) method, respectively. The residual symmetry of the DSSH system is localized to a Lie point symmetry in a properly prolonged system, based on which we get a new Bäcklund transformation for this system. New symmetry reduction solutions of the DSSH system are obtained by applying the classical Lie group approach on the prolonged system. Moreover, the DSSH system is proved to be CRE integrable and new interesting interaction solutions between solitons and periodic waves are generated and analyzed.

Residual Symmetry and Interaction Solutions of the (2+1)-Dimensional Generalized Calogero–Bogoyavlenskii–Schiff Equation

In this paper, we investigate the (2+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff (gCBS) equation by using the residual symmetry analysis and consistent Riccati expansion (CRE) method, respectively. The residual symmetry of the gCBS equation is localized into a Lie point symmetry in a prolonged system and a new Bäcklund transformation of this equation is obtained. By applying the standard Lie symmetry method to the prolonged gCBS system, new symmetry reduction solutions of the gCBS equation are obtained. The gCBS equation is proved to be CRE integrable and new Bäcklund transformations of it are obtained, from which interaction solutions between solitons and periodic waves are generated and analyzed.

(3+1)-Dimensional Gardner Equation Deformed from (1+1)-Dimensional Gardner Equation and its Conservation Laws

Through the application of the deformation algorithm, a novel (3+1)-dimensional Gardner equation and its associated Lax pair are derived from the (1+1)-dimensional Gardner equation and its conservation laws. As soon as the (3+1)-dimensional Gardner equation is set to be y or z independent, the Gardner equations in (2+1)-dimension are also obtained. To seek the exact solutions for these higher dimensional equations, the traveling wave method and the symmetry theory are introduced. Hence, the implicit expressions of traveling wave solutions to the (3+1)-dimensional and (2+1)-dimensional Gardner equations, the Lie point symmetry and the group invariant solutions to the (3+1)-dimensional Gardner equation are well investigated. In particular, after selecting some specific parameters, both the traveling wave solutions and the symmetry reduction solutions of hyperbolic function form are given.

Residual symmetry, CRE integrability and interaction solutions of two higher-dimensional shallow water wave equations

Two (3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion (CRE) method. Through localization of residual symmetries, symmetry reduction solutions of the two equations are obtained. The CRE method is applied to the two equations to obtain new Bäcklund transformations from which a type of interesting interaction solution between solitons and periodic waves is generated.

Multiple soliton solutions and symmetry analysis of a nonlocal coupled KP system

A nonlocal coupled Kadomtsev–Petviashivili (ncKP) system with shifted parity () and delayed time reversal () symmetries is generated from the local coupled Kadomtsev–Petviashivili (cKP) system. By introducing new dependent variables which have determined parities under the action of , the ncKP is transformed to a local system. Through this way, multiple even number of soliton solutions of the ncKPI system are generated from N-soliton solutions of the cKP system, which become breathers by choosing appropriate parameters. The standard Lie symmetry method is also applied on the ncKPII system to get its symmetry reduction solutions.

Symmetry reduction and exact solutions of the (3+1)-dimensional nKdV-nCBS equation

In this paper, the exact solutions of the (3+1)-dimensional nKdV-nCBS equation are investigated by various approaches. Firstly, according to Lie group theory, the (3+1)-dimensional nKdV-nCBS equation is reduced to two (1+1)-dimensional partial differential equations. Secondly, three different types of exact solutions are obtained by the homoclinic test approach, including the singular kinked-type periodic solitary wave solutions, the kinked solitary wave solutions and the kinked-cross periodic wave solutions. The diagrams of the solutions for the specified parameters are given. Furthermore, taking one reduced equation as an example, the N-soliton solutions are derived. And under some constraints, higher-order soliton solution can degenerate into lower-order soliton solution, which is a specific degeneration phenomenon of this equation.

Lie symmetries, invariant solutions and phenomena dynamics of Boiti–Leon–Pempinelli system

The present work deals with highly nonlinear dispersive water equation named Boiti–Leon–Pempinelli (BLP) system, which is a well known two dimensional generalization of sinh-Gordon equation. The BLP system describes interaction of horizontal wave component propagating in plane in infinite narrow channel with constant depth. The Lie symmetry method is employed to derive infinitesimal generators under invariance conditions of one-parameter transformations. The invariance conditions ensure the reducibility of independent variables and preserve the characteristics of system. The commutative relations of infinitesimal vectors show infinite dimensional algebra of symmetries. Thereafter, similarity variables are derived using infinitesimal generators, which lead to first symmetry reductions. Thus, a repeated process of symmetry reductions provide equivalent system of ODEs. These ODEs are integrated and thus some invariant solutions of BLP system are constructed under parametric constraints. The derived solutions are general than previous researches [J Q Yu et al 2010 Exact solutions and conservation laws of (2 + 1)-dimensional Boiti–Leon–Pempinelli equation, Appl. Math. Comput. 216 2293–2300; M Kumar, R Kumar, 2014 On new similarity solutions of the boiti–leon–pempinelli system, Commun. Theor. Phys. 61 121–126; M Kumar et al 2015 Some more similarity solutions of the (2 + 1)-dimensional BLP system, Comput. Math. Appl. 70 212–221; J Fei et al 2015 Symmetry reduction and explicit solutions of the (2 + 1)-dimensional boiti–leon–pempinelli system, Appl. Math. Comput. 268 432–438; M Kumar et al 2021 Some more invariant solutions of (2 + 1)-water waves, Int. J. Appl. Comput. Math. 7 18] as they contain all four arbitrary functions involved in infinitesimals and several free parameters as well. The deduction of previous results obtained by taking particular choices of arbitrary functions and constants, ensure novelty and importance of these solutions as well as the authenticity and effectiveness of Lie symmetry method. To demonstrate the distinct physical structures of phenomena, these solutions are expanded via numerical simulation. Thus, doubly soliton, line soliton, multi soliton, soliton fusion and fission nature have been analyzed. The graphical representations are seemed useful to understand the dynamics of BLP system and propagation of dispersive water waves in infinite narrow channel.

Solitons and symmetry reduction solutions of a nonlocal two‐mode Korteweg–de Vries equation

The two‐mode Korteweg–de Vries (TMKdV) equation describes propagation of two different waves modes simultaneously. By using consistent correlated bang (CCB) method, a nonlocal form of the TMKdV equation is constructed and converted into two local equations with its dependent variables having definite parity. Multiple soliton solutions of the nonlocal TMKdV equation are obtained by using known multiple soliton solutions of the TMKdV equation, and some singular travelling wave solutions are also generated in a similar way. The classical Lie symmetry method is also carried on the nonlocal TMKdV equation to get its symmetry reduction solutions.

Symmetry analysis and conservation laws of the modified Korteweg–de Vries equation with a quartic nonlinear term

In this paper, a modified Korteweg–de Vries equation with a quartic nonlinear term in two-electron temperature plasmas is studied. Firstly, the symmetries are constructed using the generalized symmetry method. In addition, the potential equation is studied, it can been found that this equation does not posses potential symmetries. Meanwhile, the symmetry reduction and group-invariant solutions are derived rely on the one-dimensional optimal system of subalgebras. Finally, conservation laws are showed resort by the multipliers method, three polynomial type conserved quantities are presented, also the Hamiltonian form is obtained. In particular, reciprocal Bäcklund transformations for conservation laws are obtained for the first time.

A nonlocal Boussinesq equation: Multiple-soliton solutions and symmetry analysis

A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N = 2,3,4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from the known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.

Perturbation, symmetry analysis, Bäcklund and reciprocal transformation for the extended Boussinesq equation in fluid mechanics

In this work, we study a generalized double dispersion Boussinesq equation that plays a significant role in fluid mechanics, scientific fields, and ocean engineering. This equation will be reduced to the Korteweg–de Vries equation via using the perturbation analysis. We derive the corresponding vectors, symmetry reduction and explicit solutions for this equation. We readily obtain Bäcklund transformation associated with truncated Painlevé expansion. We also examine the related conservation laws of this equation via using the multiplier method. Moreover, we investigate the reciprocal Bäcklund transformations of the derived conservation laws for the first time.

Symmetry Reduction and Exact Solutions of a Class of Wave Equations

In this paper, the Lie symmetries and similarity reduction of a class of wave equations are investigated. First, Lie algorithm is used to get the determining equations of symmetry for the given equations which are complicated and difficult to be solved. Next, differential form of Wu’s method is used to solve this problem. Moreover, a special case of differential invariant method is used to get similarity reduction of the given equations.

Symmetry reduction, conservation laws and acoustic wave solutions for the extended Zakharov–Kuznetsov dynamical model arising in a dust plasma

In this article, we consider the extended Zakharov–Kuznetsov (EZK) equation, which describes the nonlinear plasma dust acoustic waves (DAWs) in a magnetized dusty plasma. Dusty plasmas consist of three components: electrons, highly negatively charged dust grains, and two-temperature ions (low-temperature ions and high-temperature ions). We study the Lie symmetries, reductions, conservation laws and new exact solutions of EZK equations. Conservation laws for EZK equation is derived by applying the new conservation theorem of Ibragimov. Similarity solution for EZK equation will be obtained using Lie symmetry method. We find the Lie symmetries group of EZK equation, using similarity variables, get reduction equation, solving the reduction equations and then get the similarity solution. Solitary wave solutions of the EZK equation are derived from the reduction equation. Thus, some new exact explicit solutions of the EZK equation are obtained.

Symmetry reduction and new interaction solutions for the negative-order potential KdV equation

New soliton–cnoidal interaction solutions for the negative-order potential KdV equation are studied with the help of the consistent tanh expansion method. The non-local symmetry for the negative-order potential KdV equation is derived from the truncated Painlevé expansion method. The non-local symmetry is transformed into the standard Lie point symmetry by introducing new dependent variables and the finite symmetry transformation is also presented to construct new exact solutions with non-zero seed solution. The similarity solutions of the enlarged negative-order potential KdV system are obtained through different constant selections.

Symmetry Analysis, Bäcklund Transformations, and Interaction Solutions for an AB Modified KdV System

An AB modified KdV (AB-mKdV) system which can be used to describe two-place event is studied in this manuscript. Because the AB-mKdV system is considered as a special reduction of the famous AKNS system, the properties of the AKNS system are first revealed by using symmetry analysis. The nonlocal symmetries related to truncated Painlevé expansion, the finite transformation, and the symmetry reduction solutions of the AKNS system are presented. The corresponding Bäcklund transformations and the interaction solutions of the AB-mKdV system are constructed based on the special reduction. The results demonstrate that the AB-mKdV system possesses many kinds of interaction solutions, such as the interactions between kink and soliton and kink and cnoidal waves. The soliton can be changed from bright to dark during propagation.

Symmetry Reduction and Periodic Solutions in Hamiltonian Vlasov Systems

In this paper, we discuss a general approach to find periodic solutions bifurcating from equilibrium points of classical Vlasov systems. The main access to the problem is chosen through the Hamiltonian representation of any Vlasov system, firstly put forward by Fr\"ohlich, Knowles, and Schwarz, and generalized more recently by the author. The method transforms the problem into a setup of complex valued $\mathcal{L}^2$ functions with phase equivariant Hamiltonian. Through Marsden-Weinstein symmetry reduction, the problem is mapped on a Hamiltonian system on the quotient manifold $\mathbb{S}^{\mathcal{L}^2}/\mathbb{S}^1$ which actually proves to be necessary to close many trajectories of the dynamics. As a toy model to apply the method we use the Harmonic Vlasov system, a non-relativistic Vlasov equation with attractive harmonic two-body interaction potential. The simple structure of this model allows to compute all of its solutions directly and therefore test the benefits of the Hamiltonian formalism and symmetry reduction in Vlasov systems.

A nonlocal nonlinear Schrödinger equation derived from a two-layer fluid model

By applying a simple symmetry reduction on a two-layer liquid model, a nonlocal counterpart of it is obtained. Then, a general form of nonlocal nonlinear Schrodinger (NNLS) equation with shifted parity, charge conjugate and delayed time reversal is obtained by using multi-scale expansion method. Some kinds of elliptic periodic wave solutions of the NNLS equation, which become soliton solutions and kink solutions when the modulus is taken as unity, are obtained by using elliptic function expansion method. Some representative figures of these solutions are given and analyzed in detail. In addition, by carrying out the classical symmetry method on the NNLS equation, not only the Lie symmetry group but also the related symmetry reduction solutions are given.

Residual Symmetry Reduction and Consistent Riccati Expansion to a Nonlinear Evolution Equation

The residual symmetry of a (1 + 1)‐dimensional nonlinear evolution equation (NLEE) ut + uxxx − 6u2ux + 6λux = 0 is obtained through Painlevé expansion. By introducing a new dependent variable, the residual symmetry is localized into Lie point symmetry in an enlarged system, and the related symmetry reduction solutions are obtained using the standard Lie symmetry method. Furthermore, the (1 + 1)‐dimensional NLEE equation is proved to be integrable in the sense of having a consistent Riccati expansion (CRE), and some new Bäcklund transformations (BTs) are given. In addition, some explicitly expressed solutions including interaction solutions between soliton and cnoidal waves are derived from these BTs.

A nonlocal Burgers equation in atmospheric dynamical system and its exact solutions**Project supported by the National Natural Science Foundation of China (Grant Nos. 11405110, 11275129, and 11472177) and the Natural Science Foundationof Zhejiang Province of China (Grant No. LY18A050001).

From a two-vortex interaction model in atmospheric and oceanic systems, a nonlocal counterpart with shifted parity and delayed time reversal is derived by a simple AB reduction. To obtain some approximate analytic solutions of this nonlocal system, the multi-scale expansion method is applied to get an AB-Burgers system. Various exact solutions of the AB-Burgers equation, including elliptic periodic waves, kink waves and solitary waves, are obtained and shown graphically. To show the applications of these solutions in describing correlated events, a simple approximate solution for the two-vortex interaction model is given to show two correlated dipole blocking events at two different places. Furthermore, symmetry reduction solutions of the nonlocal AB-Burgers equation are also given by using the standard Lie symmetry method.

Residual symmetry analysis and CRE integrability of the (3 + 1) -dimensional Burgers system

The residual symmetry of the (3 + 1) -dimensional Burgers system is localized to a Lie point symmetry in a prolonged system and the corresponding finite transformation is obtained by using Lie’s first theorem. By further localize the linear superposition of multiple residual symmetries, the N -th Backlund transformations (BT) of the (3 + 1) -dimensional Burgers system are also got. By applying the standard Lie symmetry method to the prolonged system, not only the Lie symmetry group but also the symmetry reduction solutions are obtained, which include abundant interaction solutions between solitons and nonlinear waves. Furthermore, the (3 + 1) -dimensional Burgers system is proved to have consistent Riccati expansion (CRE) property, based on which some new BTs are given.