This paper explores the rich structure of peakon and pseudo-peakon solutions for a class of higher-order b-family equations,referred to as the J-th b-family(J-bF)equations.We propose several conjectures concerning the...This paper explores the rich structure of peakon and pseudo-peakon solutions for a class of higher-order b-family equations,referred to as the J-th b-family(J-bF)equations.We propose several conjectures concerning the weak solutions of these equations,including a b-independent pseudo-peakon solution,a b-independent peakon solution,and a b-dependent peakon solution.These conjectures are analytically verified for J≤14 and/or J≤9 using the symbolic computation system MAPLE,which includes a built-in definition of the higher-order derivatives of the sign function.The b-independent pseudo-peakon solution is a 3rd-order pseudo-peakon for general arbitrary constants,with higher-order pseudo-peakons derived under specific parameter constraints.Additionally,we identify both b-independent and b-dependent peakon solutions,highlighting their distinct properties and the nuanced relationship between the parameters b and J.The existence of these solutions underscores the rich dynamical structure of the J-bF equations and generalizes previous results for lower-order equations.Future research directions include higher-order generalizations,rigorous proofs of the conjectures,interactions between different types of peakons and pseudo-peakons,stability analysis,and potential physical applications.These advancements significantly contribute to the understanding of peakon systems and their broader implications in mathematics and physics.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
A new three-component Camassa-Holm equation is introduced. This system is endowed with a structuresimilar to the Camassa-Holm equation. It has peakon solitons and conserves H^1-norm conservation law.
Shallow water waves and a host of long wave phenomena are commonly investigated by various models of nonlinear evolution equations. Examples include the Korteweg-de Vries, the Camassa-Holm, and the Whitham-Broer-Kaup ...Shallow water waves and a host of long wave phenomena are commonly investigated by various models of nonlinear evolution equations. Examples include the Korteweg-de Vries, the Camassa-Holm, and the Whitham-Broer-Kaup （WBK） equations. Here a generalized WBK system is studied via the multi-linear variable separation approach. A special class of wave profiles with discontinuous derivatives （＂peakons＂） is developed. Peakons of various features, e.g. periodic, pulsating or fractal, are investigated and interactions of such entities are studied.展开更多
Starting from the known variable separation excitations of a(2 + 1)-dimensional generalized Ablowitz-Kaup-Newell-Segur system,rich coherent structures can be derived.The interactions among different types of solitary ...Starting from the known variable separation excitations of a(2 + 1)-dimensional generalized Ablowitz-Kaup-Newell-Segur system,rich coherent structures can be derived.The interactions among different types of solitary waves like peakons,dromions,and compactons are investigated and some novel features or interesting behaviors are revealed.The results show that the interactions for peakon-dromion,compacton-dromion,and peakon-compacton may be completely nonelastic or completely elastic.展开更多
We investigate the orbital stability of the peakons for a generalized Camassa-Holm equation (gCH). Using variable transformation, a planar dynamical system is obtained from the gCH equation. It is shown that the plana...We investigate the orbital stability of the peakons for a generalized Camassa-Holm equation (gCH). Using variable transformation, a planar dynamical system is obtained from the gCH equation. It is shown that the planar system has two heteroclinic cycles which correspond two peakon solutions. We then prove that the peakons for the gCH equation are orbitally stable by using the method of Constantin and Strauss.展开更多
In this paper,we focus on peaked traveling wave solutions of the modified highly nonlinear Novikov equation by dynamical systems approach.We obtain a traveling wave system which is a singular planar dynamical system w...In this paper,we focus on peaked traveling wave solutions of the modified highly nonlinear Novikov equation by dynamical systems approach.We obtain a traveling wave system which is a singular planar dynamical system with three singular straight lines,and derive all possible phase portraits under corresponding parameter conditions.Then we show the existence and dynamics of two types of peaked traveling wave solutions including peakons and periodic cusp wave solutions.The exact explicit expressions of two peakons are given.Besides,we also derive smooth solitary wave solutions,periodic wave solutions,compacton solutions,and kink-like(antikink-like)solutions.Numerical simulations are further performed to verify the correctness of the results.Most importantly,peakons and periodic cusp wave solutions are newly found for the equation,which extends the previous results.展开更多
In this paper,we delve into a generalized higher order Camassa-Holm type equation,(or,an ghmCH equation for short).We establish local well-posedness for this equation under the condition that the initial data uo belon...In this paper,we delve into a generalized higher order Camassa-Holm type equation,(or,an ghmCH equation for short).We establish local well-posedness for this equation under the condition that the initial data uo belongs to the Sobolev space H'(R)for some s>2.In addition,we obtain the weak formulation of this equation and prove the existence of both single peakon solution and a multi-peakon dynamic system.展开更多
We prove that the two-component peakon solutions are orbitally stable in the energy space.The system concerned here is a two-component Novikov system,which is an integrable multi-component extension of the integrable ...We prove that the two-component peakon solutions are orbitally stable in the energy space.The system concerned here is a two-component Novikov system,which is an integrable multi-component extension of the integrable Novikov equation.We improve the method for the scalar peakons to the two-component case with genuine nonlinear interactions by establishing optimal inequalities for the conserved quantities involving the coupled structures.Moreover,we also establish the orbital stability for the train-profiles of these two-component peakons by using the refined analysis based on monotonicity of the local energy and an induction method.展开更多
By the use of the extended homogenous balance method,the B(?)cklund transformation for a (2+1)- dimensional integrable model,the(2+1)-dimensional Nizhnik-Novikov-Veselov (NNV) equation,is obtained,and then the NNV equ...By the use of the extended homogenous balance method,the B(?)cklund transformation for a (2+1)- dimensional integrable model,the(2+1)-dimensional Nizhnik-Novikov-Veselov (NNV) equation,is obtained,and then the NNV equation is transformed into three equations of linear,bilinear,and tri-linear forms,respectively.From the above three equations,a rather general variable separation solution of the model is obtained.Three novel class localized structures of the model are founded by the entrance of two variable-separated arbitrary functions.展开更多
By means of variable separation approach, quite a general excitation of the new (2 + 1)-dimensional long dispersive wave system: is derived. Some types of the usual localized excitations such as dromions, lumps, ring...By means of variable separation approach, quite a general excitation of the new (2 + 1)-dimensional long dispersive wave system: is derived. Some types of the usual localized excitations such as dromions, lumps, rings, and oscillating soliton excitations can be easily constructed by selecting the arbitrary functions appropriately. Besides these usual localized structures, some new localized excitations like fractal-dromion, fractal-lump, and multi-peakon excitations of this new system are found by selecting appropriate functions.展开更多
Starting from the variable separation solution obtained by using the extended homogenous balance method,a new class of combined structures, such as multi-peakon and multi-dromion solution, multi-compacton and multidro...Starting from the variable separation solution obtained by using the extended homogenous balance method,a new class of combined structures, such as multi-peakon and multi-dromion solution, multi-compacton and multidromion solution, multi-peakon and multi-compacton solution, for the (2+1)-dimensional Nizhnik-Novikov-Veselov equation are found by selecting appropriate functions. These new structures exhibit novel interaction features. Their interaction behavior is very similar to the completely nonelastic collisions between two classical particles.展开更多
In this paper,the supersymmetric Camassa-Holm equation and Degasperis-Procesi equation are derived from a general superfield equations by choosing different parameters.Their peakon-type solutions are shown in weak sen...In this paper,the supersymmetric Camassa-Holm equation and Degasperis-Procesi equation are derived from a general superfield equations by choosing different parameters.Their peakon-type solutions are shown in weak sense.At the same time,the dynamic behaviors are analyzed particularly when the two peakons collide elastically,and some results are compared with each other between the two equations.展开更多
基金supported by the National Natural Science Foundations of China(Grant Nos.12235007,12271324,and 11975131)。
文摘This paper explores the rich structure of peakon and pseudo-peakon solutions for a class of higher-order b-family equations,referred to as the J-th b-family(J-bF)equations.We propose several conjectures concerning the weak solutions of these equations,including a b-independent pseudo-peakon solution,a b-independent peakon solution,and a b-dependent peakon solution.These conjectures are analytically verified for J≤14 and/or J≤9 using the symbolic computation system MAPLE,which includes a built-in definition of the higher-order derivatives of the sign function.The b-independent pseudo-peakon solution is a 3rd-order pseudo-peakon for general arbitrary constants,with higher-order pseudo-peakons derived under specific parameter constraints.Additionally,we identify both b-independent and b-dependent peakon solutions,highlighting their distinct properties and the nuanced relationship between the parameters b and J.The existence of these solutions underscores the rich dynamical structure of the J-bF equations and generalizes previous results for lower-order equations.Future research directions include higher-order generalizations,rigorous proofs of the conjectures,interactions between different types of peakons and pseudo-peakons,stability analysis,and potential physical applications.These advancements significantly contribute to the understanding of peakon systems and their broader implications in mathematics and physics.
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.
基金Supported by National Natural Science Foundation of China under Grant Nos. 10671156 and 10671153
文摘A new three-component Camassa-Holm equation is introduced. This system is endowed with a structuresimilar to the Camassa-Holm equation. It has peakon solitons and conserves H^1-norm conservation law.
基金the National Natural Science Foundation of China (10475055,10547124 and 90503006)the Hong Kong Research Grant Council Contract HKU 7123/05E.
文摘Shallow water waves and a host of long wave phenomena are commonly investigated by various models of nonlinear evolution equations. Examples include the Korteweg-de Vries, the Camassa-Holm, and the Whitham-Broer-Kaup （WBK） equations. Here a generalized WBK system is studied via the multi-linear variable separation approach. A special class of wave profiles with discontinuous derivatives （＂peakons＂） is developed. Peakons of various features, e.g. periodic, pulsating or fractal, are investigated and interactions of such entities are studied.
文摘Starting from the known variable separation excitations of a(2 + 1)-dimensional generalized Ablowitz-Kaup-Newell-Segur system,rich coherent structures can be derived.The interactions among different types of solitary waves like peakons,dromions,and compactons are investigated and some novel features or interesting behaviors are revealed.The results show that the interactions for peakon-dromion,compacton-dromion,and peakon-compacton may be completely nonelastic or completely elastic.
文摘We investigate the orbital stability of the peakons for a generalized Camassa-Holm equation (gCH). Using variable transformation, a planar dynamical system is obtained from the gCH equation. It is shown that the planar system has two heteroclinic cycles which correspond two peakon solutions. We then prove that the peakons for the gCH equation are orbitally stable by using the method of Constantin and Strauss.
基金Supported by the National Natural Science Foundation of China(12071162)the Natural Science Foundation of Fujian Province(2021J01302)the Fundamental Research Funds for the Central Universities(ZQN-802).
文摘In this paper,we focus on peaked traveling wave solutions of the modified highly nonlinear Novikov equation by dynamical systems approach.We obtain a traveling wave system which is a singular planar dynamical system with three singular straight lines,and derive all possible phase portraits under corresponding parameter conditions.Then we show the existence and dynamics of two types of peaked traveling wave solutions including peakons and periodic cusp wave solutions.The exact explicit expressions of two peakons are given.Besides,we also derive smooth solitary wave solutions,periodic wave solutions,compacton solutions,and kink-like(antikink-like)solutions.Numerical simulations are further performed to verify the correctness of the results.Most importantly,peakons and periodic cusp wave solutions are newly found for the equation,which extends the previous results.
文摘In this paper,we delve into a generalized higher order Camassa-Holm type equation,(or,an ghmCH equation for short).We establish local well-posedness for this equation under the condition that the initial data uo belongs to the Sobolev space H'(R)for some s>2.In addition,we obtain the weak formulation of this equation and prove the existence of both single peakon solution and a multi-peakon dynamic system.
基金National Natural Science Foundation of China(Grants Nos.12271424 and 11871395)National Natural Science Foundation of China(Grants Nos.11971251,11631007 and 12111530003)。
文摘We prove that the two-component peakon solutions are orbitally stable in the energy space.The system concerned here is a two-component Novikov system,which is an integrable multi-component extension of the integrable Novikov equation.We improve the method for the scalar peakons to the two-component case with genuine nonlinear interactions by establishing optimal inequalities for the conserved quantities involving the coupled structures.Moreover,we also establish the orbital stability for the train-profiles of these two-component peakons by using the refined analysis based on monotonicity of the local energy and an induction method.
文摘By the use of the extended homogenous balance method,the B(?)cklund transformation for a (2+1)- dimensional integrable model,the(2+1)-dimensional Nizhnik-Novikov-Veselov (NNV) equation,is obtained,and then the NNV equation is transformed into three equations of linear,bilinear,and tri-linear forms,respectively.From the above three equations,a rather general variable separation solution of the model is obtained.Three novel class localized structures of the model are founded by the entrance of two variable-separated arbitrary functions.
文摘By means of variable separation approach, quite a general excitation of the new (2 + 1)-dimensional long dispersive wave system: is derived. Some types of the usual localized excitations such as dromions, lumps, rings, and oscillating soliton excitations can be easily constructed by selecting the arbitrary functions appropriately. Besides these usual localized structures, some new localized excitations like fractal-dromion, fractal-lump, and multi-peakon excitations of this new system are found by selecting appropriate functions.
文摘Starting from the variable separation solution obtained by using the extended homogenous balance method,a new class of combined structures, such as multi-peakon and multi-dromion solution, multi-compacton and multidromion solution, multi-peakon and multi-compacton solution, for the (2+1)-dimensional Nizhnik-Novikov-Veselov equation are found by selecting appropriate functions. These new structures exhibit novel interaction features. Their interaction behavior is very similar to the completely nonelastic collisions between two classical particles.
基金Supported by the National Science Foundation of China under Grant No. 11071092the Texas Norman Hackerman Advanced Research Program under Grant No. 003599-0001-2009
文摘In this paper,the supersymmetric Camassa-Holm equation and Degasperis-Procesi equation are derived from a general superfield equations by choosing different parameters.Their peakon-type solutions are shown in weak sense.At the same time,the dynamic behaviors are analyzed particularly when the two peakons collide elastically,and some results are compared with each other between the two equations.
