A large member of lump chain solutions of the(2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili(BKP)equation are constructed by means of theτ-function in the form of Grammian.The lump chains are formed by period...A large member of lump chain solutions of the(2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili(BKP)equation are constructed by means of theτ-function in the form of Grammian.The lump chains are formed by periodic arrangement of individual lumps and travel with distinct group and velocities.An analytical method related dominant regions of polygon is developed to analyze the interaction dynamics of the multiple lump chains.The degenerate structures of parallel,superimposed,and molecular lump chains are presented.The interaction solutions between lump chains and kink-solitons are investigated,where the kink-solitons lie on the boundaries of dominant region determined by the constant term in theτ-function.Furthermore,the hybrid solutions consisting of lump chains and individual lumps controlled by the parameter with high rank and depth are investigated.The analytical method presented in this paper can be further extended to other integrable systems to explore complex wave structures.展开更多
In this paper,we mainly focus on proving the existence of lump solutions to a generalized(3+1)-dimensional nonlinear differential equation.Hirota’s bilinear method and a quadratic function method are employed to deri...In this paper,we mainly focus on proving the existence of lump solutions to a generalized(3+1)-dimensional nonlinear differential equation.Hirota’s bilinear method and a quadratic function method are employed to derive the lump solutions localized in the whole plane for a(3+1)-dimensional nonlinear differential equation.Three examples of such a nonlinear equation are presented to investigate the exact expressions of the lump solutions.Moreover,the 3d plots and corresponding density plots of the solutions are given to show the space structures of the lump waves.In addition,the breath-wave solutions and several interaction solutions of the(3+1)-dimensional nonlinear differential equation are obtained and their dynamics are analyzed.展开更多
In this paper,a modified version of the solution in form of a Gramian formula is employed to investigate a new type of multiple lump molecule solution of the Kadomtsev–Petviashvili I equation.The high-order multiple ...In this paper,a modified version of the solution in form of a Gramian formula is employed to investigate a new type of multiple lump molecule solution of the Kadomtsev–Petviashvili I equation.The high-order multiple lump molecules consisting of M N-lump molecules are constructed by means of the Mth-order determinant and the non-homogeneous polynomial in the degree of 2N.The interaction solutions describing P line solitons radiating P of the M N-lump molecules are constructed.The dynamic behaviors of some specific solutions are analyzed through numerical simulation.All the results will enrich our understanding of the multiple lump waves of the Kadomtsev–Petviashvili I equation.展开更多
Some two-component extensions of the modifiedμ-Camassa-Holm equation are proposed.We show that these systems admit Lax pairs and bi-Hamiltonian structures.Furthermore,we consider the blow-up phenomena for one of thes...Some two-component extensions of the modifiedμ-Camassa-Holm equation are proposed.We show that these systems admit Lax pairs and bi-Hamiltonian structures.Furthermore,we consider the blow-up phenomena for one of these extensions(2μmCH),and the periodic peakons of this system are derived.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant No.12101572)the Research Project Supported by Shanxi Scholarship Council of China(Grant No.2020-105)。
文摘A large member of lump chain solutions of the(2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili(BKP)equation are constructed by means of theτ-function in the form of Grammian.The lump chains are formed by periodic arrangement of individual lumps and travel with distinct group and velocities.An analytical method related dominant regions of polygon is developed to analyze the interaction dynamics of the multiple lump chains.The degenerate structures of parallel,superimposed,and molecular lump chains are presented.The interaction solutions between lump chains and kink-solitons are investigated,where the kink-solitons lie on the boundaries of dominant region determined by the constant term in theτ-function.Furthermore,the hybrid solutions consisting of lump chains and individual lumps controlled by the parameter with high rank and depth are investigated.The analytical method presented in this paper can be further extended to other integrable systems to explore complex wave structures.
基金supported by the National Natural Science Foundation of China(Nos.12101572,12371256)2023 Shanxi Province Graduate Innovation Project(No.2023KY614)the 19th Graduate Science and Technology Project of North University of China(No.20231943)。
文摘In this paper,we mainly focus on proving the existence of lump solutions to a generalized(3+1)-dimensional nonlinear differential equation.Hirota’s bilinear method and a quadratic function method are employed to derive the lump solutions localized in the whole plane for a(3+1)-dimensional nonlinear differential equation.Three examples of such a nonlinear equation are presented to investigate the exact expressions of the lump solutions.Moreover,the 3d plots and corresponding density plots of the solutions are given to show the space structures of the lump waves.In addition,the breath-wave solutions and several interaction solutions of the(3+1)-dimensional nonlinear differential equation are obtained and their dynamics are analyzed.
基金This work is supported by the National Natural Science Foundation of China(No.12 101 572)the Research Project Supported by the Shanxi Scholarship Council of China(No.2020-105).
文摘In this paper,a modified version of the solution in form of a Gramian formula is employed to investigate a new type of multiple lump molecule solution of the Kadomtsev–Petviashvili I equation.The high-order multiple lump molecules consisting of M N-lump molecules are constructed by means of the Mth-order determinant and the non-homogeneous polynomial in the degree of 2N.The interaction solutions describing P line solitons radiating P of the M N-lump molecules are constructed.The dynamic behaviors of some specific solutions are analyzed through numerical simulation.All the results will enrich our understanding of the multiple lump waves of the Kadomtsev–Petviashvili I equation.
基金the National Nature Science Foundation of China(Grant Nos.11871471,11931017)Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi(Grant No.2019L0531)+1 种基金Shanxi Province Science Foundation for Youths(Grant No.201901D211274)the Fund for Shanxi‘1331KIRT’。
文摘Some two-component extensions of the modifiedμ-Camassa-Holm equation are proposed.We show that these systems admit Lax pairs and bi-Hamiltonian structures.Furthermore,we consider the blow-up phenomena for one of these extensions(2μmCH),and the periodic peakons of this system are derived.