Different wave patterns for two-coupled Maccari's system with complex structure via truncated Painlevéapproach

We focused on the two-coupled Maccari's system.With the help of truncated Painlevéapproach(TPA),we express local solution in the form of arbitrary functions.From the solution obtained,using its appropriate arbitrary functions,we have generated the rogue wave pattern solutions,rogue wave solutions,and lump solutions.In addition,by controlling the values of the parameters in the solutions,we show the dynamic behaviors of the rogue wave pattern solutions,rogue wave solutions,and lump solutions with the aid of Maple tool.The results of this study will contribute to the understanding of nonlinear wave dynamics in higher dimensional Maccari's systems.

A connection between the(G'/G)-expansion method and the truncated Painlevé expansion method and its application to the mKdV equation

Recently the （G′/G）-expansion method was proposed to find the traveling wave solutions of nonlinear evolution equations. This paper shows that the （G′/G）-expansion method is a special form of the truncated Painlev＇e expansion method by introducing an intermediate expansion method. Then the generalized （G′/G）-（G/G′） expansion method is naturally derived from the standpoint of the nonstandard truncated Painlev＇e expansion. The application of the generalized method to the mKdV equation shows that it extends the range of exact solutions obtained by using the （ G′/ G）-expansion method.

Residual symmetry, interaction solutions, and conservation laws of the(2+1)-dimensional dispersive long-wave system

We explore the （2＋l）-dimensional dispersive long-wave （DLW） system. From the standard truncated Painleve expansion, the Baicklund 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＋l）-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,consistent Riccati expansion integrability,and their applications of the (2+1)-dimensional Broer–Kaup–Kupershmidt system

For the （2＋1）-dimensional Broer–Kaup–Kupershmidt（BKK） system, the nonlocal symmetries related to the Schwarzian variable and the corresponding transformation group are found. Moreover, the integrability of the BKK system in the sense of having a consistent Riccati expansion（CRE） is investigated. The interaction solutions between soliton and cnoidal periodic wave are explicitly studied.

Finite symmetry transformation group and localized structures of the (2+1)-dimensional coupled Burgers equation

In this paper, the finite symmetry transformation group of the （2＋1）-dimensional coupled Burgers equation is studied by the modified direct method, and with the help of the truncated Painleve′ expansion approach, some special localized structures for the （2＋1）-dimensional coupled Burgers equation are obtained, in particular, the dromion-like and solitoff-like structures.

Analytic Solutions of a Predator-Prey System

A predator prey interaction of both populations was treated by the Painlevé analysis method. The analytic solutions for the related equations was obtained using the truncated Painlevé expansion.

The residual symmetry,Bäcklund transformations,CRE integrability and interaction solutions:(2+1)-dimensional Chaffee–Infante equation

In this paper,we consider the(2+1)-dimensional Chaffee-Infante equation,which occurs in the fields of fluid dynamics,high-energy physics,electronic science etc.We build Bäcklund transformations and residual symmetries in nonlocal structure using the Painlevétruncated expansion approach.We use a prolonged system to localize these symmetries and establish the associated one-parameter Lie transformation group.In this transformation group,we deliver new exact solution profiles via the combination of various simple(seed and tangent hyperbolic form)exact solution structures.In this manner,we acquire an infinite amount of exact solution forms methodically.Furthermore,we demonstrate that the model may be integrated in terms of consistent Riccati expansion.Using the Maple symbolic program,we derive the exact solution forms of solitary-wave and soliton-cnoidal interaction.Through 3D and 2D illustrations,we observe the dynamic analysis of the acquired solution forms.

Nonlinear Self-Adjointness, Conservation Laws and Soliton-Cnoidal Wave Interaction Solutions of(2+1)-Dimensional Modified Dispersive Water-Wave System

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 Painlev′e analysis and consistent tanh-function expansion(CTE)method, the soliton-cnoidal periodic wave interaction solutions and corresponding images will be eventually achieved.