Nonlinear Folded Solitary Wave And Periodic Propagating Wave Patterns

Exploring nonlinear excitation patterns and complex nonlinear waves in high dimension nonlinear systems is an important topic in the research fields of nonlinear soliton theory,wave theory and nonlinear mechanics.There are various types of nonlinear folded phenomena and periodic waves in nature,which attract us to seek reasonable theoretical explanations in the study of nonlinear wave models.This paper focus on 2+1 dimensional nonlinear wave models (equations) originated from practical water and other nonlinear physical problems,seeking their exact solutions,revealing abundant nonlinear folded solitary wave and periodic propagating wave patterns and investigating their nonlinear interaction properties analytically and graphically.The main works and our achievements are outlined in three parts in the follows.PartⅠ,the approaches multi-linear variable separation method,Painlevétruncated expansion method and generalized deformation mapping method that presented in recent years are extended to solve many 2+1 dimensional nonlinear physical models including 2+1 dimensional long wave short wave resonance interaction equation,Maccari system,generalized NNV equation,2+1 dimensional KdV equation,HBK equation and 2+1 dimensional variable coefficients BK equation,the corresponding exact variable separation solutions of them are obtained respectively.Based on 2+1 dimensional variable separation solutions,introducing proper single valued and multivalued functions,abundant nonlinear single valued and multivalued localized excitation patterns especially folded solitary wave excitations are derived.Selecting proper single valued variable separation functions in the 2+1 variable separation solutions,we construct many types of single valued coherent soliton structures such as dromion,dromion lattice,solitoff,2-solitoff,static and kinetic breather,instanton,compacton, peakon,and so on.The interactions among these coherent localized excitations including the collisions of compaction and compacton,peakon and peakon,line soliton and y-periodic soliton,dromion and dromion,dromion and solitoff,et al,are investigated analytically and graphically,and found that can be elastic and nonelastic respectively,which reveal various exotic and important nonlinear characters and interaction properties.Introducing proper multivalued functions in 2+1 dimensional variable separation solutions, we construct many types of mulitivalued localized solitary waves such as "worm" shape folded solitary wave,"rope knot" shape folded solitary wave,"tent" shape folded solitary wave,worm-solitoff folded solitary wave and worm-dromion folded solitary wave,et al. The folded solitary wave excitation is multivalued localized and folded in all directions and the interaction between foldons is elastic.We consider the asymptotic behaviors of the localized excitations produced on two families of variable separation solutions and give out the phase shifting conditions that lead to completely elastic interaction for the folded solitary waves,and we investigate the completely elastic interactions between two folded solitary waves(foldons) analytically and graphically.Introducing single valued and multivalued functions properly in the 2+1 dimensional variable separation solutions,we derive various semifolded solitary waves such as bell-like semifolded soliton,compaction-like semifolded soliton and peakon-like semifolded solitons,et al.The semifolded solitary wave excitations are multivalued loop soliton folded in one direction and single valued localized in other direction. The interactions between two semifolded solitons are elastic,while the interactions among other nonlinear multivalued and single valued excitation patterns such as the interactions between semifolded soliton and single valued localized soliton,semifolded soliton and folded soliton,and the interaction among semifolded soltion,folded soliton and single valued localized solitons,are nonelastic generally,which reveal different exotic and important nonlinear mechanical properties.Our research work expose various interesting and important localized multivalued folded excitations,which help us to further understand and explore complex nonlinear folded phenomena in nature.PartⅡ,Applying the developed and extended direct algebraic methods that presented in recent years and based on the 2+1 dimensional variable separation solutions,abundant Jacobi elliptic function doubly periodic waves and periodic propagating wave patterns for many 2+1 dimensional nonlinear wave models are obtained,and in long wave limit solitary waves are derived as well.We develop and apply the Hirota bilinear-θfunction method,Jacobi elliptic function expansion method,linear superposition method and F-expansion method respectively to solve many 2+1 dimensional nonlinear wave models including 2+1 dimensional 2DsG equation,the coupled ZK equation,2+1 dimensional KdV equation,2+1 dimensional long wave short wave resonance interaction equation and 2+1 dimensional dispersive long wave equation,abundant Jacobi elliptic function doubly periodic solutions are derived.These solutions show various periodic wave shapes and special periodic characters.In long wave limit,i.e,the modulus of the Jacobi elliptic function m→1,some elliptic function waves may degenerate into solitary wave solutions such as bell-like solitary wave,kink solitary wave,saddle-like solitary wave and line soliton as well.Further more,we also extend the Jacobi elliptic function expansion method to discrete differential-difference nonlinear systems.The periodic wave solutions and discrete solitons for discrete AL equation are obtained,and important nonlinear properties of discrete bright soliton and dark solitons are revealed.Based on 2+1 dimensional variable separation solutions,introducing proper Jacobi elliptic functions such as sn,cn,dn and nd,et al.,we obtain many types of periodic propagating wave patterns for 2+1 dimensional HBK equation and the soliton equation presented by Maccari respectively,which denote the characters and nonlinear properties of periodic propagating waves for two classes of nonlinear wave models that possess complex variables or not.These elliptic function waves shows different forms as the modulus changes,which may be periodic oscillation waves,dromion lattic and multi-peakons,et al.In long wave limit, these elliptic function wave patterns may degenerate into single valued localized excitations dromion and multi-lumps.The interactions of elliptic function waves are nonelastic,while the interaction of the degenerated coherent soliton structures dromions and lumps may be elastic and nonelastic as well.Moreover,we also construct a new kind of combined wave possessing dromion solitons in the background of Jacobi elliptic periodic waves,which is very similar in the forms to the freak extreme waves that arise sometime in ocean.PartⅢ,we present simple and effective technics to find new nonlinear waves the periodic folded waves based on 2+1 dimensional variable separation solutions,and the periodic folded waves of 2+1 dimensional HBK equation and 2+1 dimensional KdV equation are constructed respectively.Introducing suitable multivalued functions and Jacobi elliptic functions such as sn and cn appropriately,many types of complex nonlinear periodic folded waves such as wormlike periodic folded waves and tent-like periodic folded waves are derived.The periodic folded wave is multivalued loop soilton folded in one direction and single periodic oscillated in other direction,which may be view as a special type of semifolded wave in mathematical analytic. In long wave limit,the periodic folded waves may degenerate as single folded solitary waves. The interactions of the periodic folded waves and of their degenerated single solitary waves are completely elastic,i.e,there are no changes of the shapes and velocities before and after the interaction besides phases shifts.This work help us to further understand and acquaint such special periodic folded phenomena in nature.