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
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
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
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.
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
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
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
We study the Alice-Bob peakon system generated from an integrable peakon system using the strategy of the so- called Alice-Bob non-local KdV approach [Scientific Reports 7 （2017） 869]. Nonlocal integrable peakon equ...We study the Alice-Bob peakon system generated from an integrable peakon system using the strategy of the so- called Alice-Bob non-local KdV approach [Scientific Reports 7 （2017） 869]. Nonlocal integrable peakon equations are obtained and shown to have peakon solutions.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
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
By means of an extended mapping approach and a linear variable separation approach,a new family ofexact solutions of the general(2+1)-dimensional Korteweg de Vries system(GKdV)are derived.Based on the derivedsolitary ...By means of an extended mapping approach and a linear variable separation approach,a new family ofexact solutions of the general(2+1)-dimensional Korteweg de Vries system(GKdV)are derived.Based on the derivedsolitary wave excitation,we obtain some special peakon excitations and fractal dromions in this short note.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
We study the multi-peakon solutions for two new coupled Camassa–Holm equations, which include twocomponent and three-component Camassa–Holm equations. These multi-peakon solutions are shown in weak sense. In particu...We study the multi-peakon solutions for two new coupled Camassa–Holm equations, which include twocomponent and three-component Camassa–Holm equations. These multi-peakon solutions are shown in weak sense. In particular, the double peakon solutions of both equations are investigated in detail. At the same time, the dynamic behaviors of three types double peakon solutions are analyzed by some figures.展开更多
This paper is contributed to study two new integrable four-component systems reduced from a multi-component generation of Camassa-Holm equation. Some double peakon solutions of both systems are described in an explici...This paper is contributed to study two new integrable four-component systems reduced from a multi-component generation of Camassa-Holm equation. Some double peakon solutions of both systems are described in an explicit formula by the method of variation of constant for ordinary differential equations. These double peakon solutions are established in weak sense. The dynamic behaviors of the obtained double peakon solutions are illustrated by some figures.展开更多
We report two integrable peakon systems that have weak kink and kink-peakon interactional solutions. Both peakon systems are guaranteed integrable through providing their Lax pairs. The peakon and multi-peakon solutio...We report two integrable peakon systems that have weak kink and kink-peakon interactional solutions. Both peakon systems are guaranteed integrable through providing their Lax pairs. The peakon and multi-peakon solutions of both equations are studied. In particular, the two-peakon dynamic systems are explicitly presented and their collisions are investigated. The weak kink solution is studied, and more interesting, the kink-peakon interactional solutions are proposed for the first time.展开更多
In this paper,we study the Camassa-Holm equation and the Degasperis-Procesi equation.The two equations are in the family of integrable peakon equations,and both have very rich geometric properties.Based on these geome...In this paper,we study the Camassa-Holm equation and the Degasperis-Procesi equation.The two equations are in the family of integrable peakon equations,and both have very rich geometric properties.Based on these geometric structures,we construct the geometric numerical integrators for simulating their soliton solutions.The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties,however they also have the significant difference,for example there exist the shock wave solutions for the Degasperis-Procesi equation.By using the symplectic Fourier pseudo-spectral integrator,we simulate the peakon solutions of the two equations.To illustrate the smooth solitons and shock wave solutions of the DP equation,we use the splitting technique and combine the composition methods.In the numerical experiments,comparisons of these two kinds of methods are presented in terms of accuracy,computational cost and invariants preservation.展开更多
在孤立子理论中,寻找新的可积系统是最基础而重要的内容之一.而如何有效的求得一类孤子方程的精确解,并研究该精确解的性质,一直是一个基本而又富有挑战性的课题.本文便是从这两个方面展开,一方面构造两个具有N-peakon的新可积系统,为...在孤立子理论中,寻找新的可积系统是最基础而重要的内容之一.而如何有效的求得一类孤子方程的精确解,并研究该精确解的性质,一直是一个基本而又富有挑战性的课题.本文便是从这两个方面展开,一方面构造两个具有N-peakon的新可积系统,为目前并不丰富的具有尖孤子解的可积非线性家族提供了极为重要的可积动力模型;另一方面,基于超椭圆代数曲线理论,本文对Lax对的有限展开法进行改进,并将其拓广到求解相联系的孤子方程可积形变后的代数几何解,给出著名的KdV(Korteweg de Vries)6方程的解.进一步,通过研究与孤子方程族相应的亚纯函数、Baker-Akhiezer函数和超椭圆曲线的渐近性质和代数几何特征,本文摆脱现有代数几何方法中使用Riemann定理的限制,构造mKdV(modifed Korteweg de Vries)型方程和混合AKNS(Ablowitz Kaup Newell Segur)方程等孤子方程的代数几何解.为构造高阶矩阵谱问题所对应的孤子方程族的代数几何解提供了有力的工具.展开更多
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.
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.
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.
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.
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.
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.
基金Supported by the Global Change Research Program of China under Grant No 2015CB953904the National Natural Science Foundation of China under Grant No 11435005+3 种基金the Shanghai Knowledge Service Platform for Trustworthy Internet of Things under Grant No ZF1213the K.C.Wong Magna Fund at Ningbo Universitythe UTRGV President's Endowed Professorshipthe Seed Grant of the UTRGV College of Science
文摘We study the Alice-Bob peakon system generated from an integrable peakon system using the strategy of the so- called Alice-Bob non-local KdV approach [Scientific Reports 7 （2017） 869]. Nonlocal integrable peakon equations are obtained and shown to have peakon solutions.
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.
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The project supported by the Natural Science Foundation of Zhejiang Province under Grant No.Y604106the Natural Science Foundation of Zhejiang Lishui University under Grant No.KZ05010
文摘By means of an extended mapping approach and a linear variable separation approach,a new family ofexact solutions of the general(2+1)-dimensional Korteweg de Vries system(GKdV)are derived.Based on the derivedsolitary wave excitation,we obtain some special peakon excitations and fractal dromions in this short note.
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 under Grant No.11261037the Natural Science Foundation of Inner Mongolia Autonomous Region under Grant No.2014MS0111+1 种基金the Caoyuan Yingcai Program of Inner Mongolia Autonomous Region under Grant No.CYYC2011050the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant No.NJYT14A04
文摘We study the multi-peakon solutions for two new coupled Camassa–Holm equations, which include twocomponent and three-component Camassa–Holm equations. These multi-peakon solutions are shown in weak sense. In particular, the double peakon solutions of both equations are investigated in detail. At the same time, the dynamic behaviors of three types double peakon solutions are analyzed by some figures.
文摘This paper is contributed to study two new integrable four-component systems reduced from a multi-component generation of Camassa-Holm equation. Some double peakon solutions of both systems are described in an explicit formula by the method of variation of constant for ordinary differential equations. These double peakon solutions are established in weak sense. The dynamic behaviors of the obtained double peakon solutions are illustrated by some figures.
文摘We report two integrable peakon systems that have weak kink and kink-peakon interactional solutions. Both peakon systems are guaranteed integrable through providing their Lax pairs. The peakon and multi-peakon solutions of both equations are studied. In particular, the two-peakon dynamic systems are explicitly presented and their collisions are investigated. The weak kink solution is studied, and more interesting, the kink-peakon interactional solutions are proposed for the first time.
基金This research was supported by the National Natural Science Foundation of China 11271357,11271195 and 41504078by the CSC,the Foundation for Innovative Research Groups of the NNSFC 11321061 and the ITER-China Program 2014GB124005。
文摘In this paper,we study the Camassa-Holm equation and the Degasperis-Procesi equation.The two equations are in the family of integrable peakon equations,and both have very rich geometric properties.Based on these geometric structures,we construct the geometric numerical integrators for simulating their soliton solutions.The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties,however they also have the significant difference,for example there exist the shock wave solutions for the Degasperis-Procesi equation.By using the symplectic Fourier pseudo-spectral integrator,we simulate the peakon solutions of the two equations.To illustrate the smooth solitons and shock wave solutions of the DP equation,we use the splitting technique and combine the composition methods.In the numerical experiments,comparisons of these two kinds of methods are presented in terms of accuracy,computational cost and invariants preservation.
文摘在孤立子理论中,寻找新的可积系统是最基础而重要的内容之一.而如何有效的求得一类孤子方程的精确解,并研究该精确解的性质,一直是一个基本而又富有挑战性的课题.本文便是从这两个方面展开,一方面构造两个具有N-peakon的新可积系统,为目前并不丰富的具有尖孤子解的可积非线性家族提供了极为重要的可积动力模型;另一方面,基于超椭圆代数曲线理论,本文对Lax对的有限展开法进行改进,并将其拓广到求解相联系的孤子方程可积形变后的代数几何解,给出著名的KdV(Korteweg de Vries)6方程的解.进一步,通过研究与孤子方程族相应的亚纯函数、Baker-Akhiezer函数和超椭圆曲线的渐近性质和代数几何特征,本文摆脱现有代数几何方法中使用Riemann定理的限制,构造mKdV(modifed Korteweg de Vries)型方程和混合AKNS(Ablowitz Kaup Newell Segur)方程等孤子方程的代数几何解.为构造高阶矩阵谱问题所对应的孤子方程族的代数几何解提供了有力的工具.
