本文首先阐述了具有4 ×4 Lax对的NLS方程的来历、表示形式,并将其表示为矩阵的分块形式,通过对Lax对、特征函数以及对称性的分析,得出全局关系,并表示出s(k)和S(k)的形式,进而构造黎曼–希尔伯特问题,并通过Mn之间的关系,计算出其...本文首先阐述了具有4 ×4 Lax对的NLS方程的来历、表示形式,并将其表示为矩阵的分块形式,通过对Lax对、特征函数以及对称性的分析,得出全局关系,并表示出s(k)和S(k)的形式,进而构造黎曼–希尔伯特问题,并通过Mn之间的关系,计算出其跳跃矩阵。In this paper, the origin and representation of the square matrix NLS equation with a 4 × 4 Lax pair are first expounded, and it is expressed as the block form of the matrix. Through the analysis of lax pair, eigenfunction and symmetry, we establish the global relation and derive the explicit forms of s(k)and S(k). And then the Riemann-Hilbert problem is constructed, and its jump matrix is calculated through the relationship between Mn.展开更多
The aim of this paper is to study an extended modified Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff(mKdV-CBS)equation and present its Lax pair with a spectral parameter.Meanwhile,a Miura transformation is explored...The aim of this paper is to study an extended modified Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff(mKdV-CBS)equation and present its Lax pair with a spectral parameter.Meanwhile,a Miura transformation is explored,which reveals the relationship between solutions of the extended mKdV-CBS equation and the extended(2+1)-dimensional Korteweg-de Vries(KdV)equation.On the basis of the obtained Lax pair and the existing research results,the Darboux transformation is derived,which plays a crucial role in presenting soliton solutions.In addition,soliton molecules are given by the velocity resonance mechanism.展开更多
In this article,a Generalized Calogero-Bogoyavlenskii-Schiff(CBS)equation is studied,serving as an extended shallow water wave model in higher dimensions.Firstly,utilizing the Bell polynomial method,the bilinear form ...In this article,a Generalized Calogero-Bogoyavlenskii-Schiff(CBS)equation is studied,serving as an extended shallow water wave model in higher dimensions.Firstly,utilizing the Bell polynomial method,the bilinear form of the equation,bilinear Bäcklund transformation,Lax pair and infinite conservation laws are derived,confirming the equation’s complete integrability in the context of the Lax pair.Subsequently,the nonlinear superposition formula of the equation is constructed based on the derived bilinear Bäcklund transformation and an array of infinite superposition soliton solutions of the equation are formulated using this nonlinear superposition formula.Lastly,leveraging the obtained bilinear equation,infinite superposition solutions of various functional types are constructed.Their dynamic characteristics are analyzed through illustrated solution images.It is noteworthy that this paper not only uncovers a multitude of properties through the Bell polynomial method but also derives both infinite linear and nonlinear superposition solutions,enriching the diversity of solutions,these aspects have not been previously explored in existing literature.展开更多
文摘本文首先阐述了具有4 ×4 Lax对的NLS方程的来历、表示形式,并将其表示为矩阵的分块形式,通过对Lax对、特征函数以及对称性的分析,得出全局关系,并表示出s(k)和S(k)的形式,进而构造黎曼–希尔伯特问题,并通过Mn之间的关系,计算出其跳跃矩阵。In this paper, the origin and representation of the square matrix NLS equation with a 4 × 4 Lax pair are first expounded, and it is expressed as the block form of the matrix. Through the analysis of lax pair, eigenfunction and symmetry, we establish the global relation and derive the explicit forms of s(k)and S(k). And then the Riemann-Hilbert problem is constructed, and its jump matrix is calculated through the relationship between Mn.
基金supported by the National Natural Science Foundation of China(Grant No.12271488)。
文摘The aim of this paper is to study an extended modified Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff(mKdV-CBS)equation and present its Lax pair with a spectral parameter.Meanwhile,a Miura transformation is explored,which reveals the relationship between solutions of the extended mKdV-CBS equation and the extended(2+1)-dimensional Korteweg-de Vries(KdV)equation.On the basis of the obtained Lax pair and the existing research results,the Darboux transformation is derived,which plays a crucial role in presenting soliton solutions.In addition,soliton molecules are given by the velocity resonance mechanism.
基金the Natural Science Foundation of Inner Mongolia Autonomous Region(Grant No.2024MS01003)the First-Class Disciplines Project,Inner Mongolia Autonomous Region,China(Grant Nos.YLXKZX-NSD-001 and YLXKZX-NSD-009)the Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region(Grant No.NMGIRT2414).
文摘In this article,a Generalized Calogero-Bogoyavlenskii-Schiff(CBS)equation is studied,serving as an extended shallow water wave model in higher dimensions.Firstly,utilizing the Bell polynomial method,the bilinear form of the equation,bilinear Bäcklund transformation,Lax pair and infinite conservation laws are derived,confirming the equation’s complete integrability in the context of the Lax pair.Subsequently,the nonlinear superposition formula of the equation is constructed based on the derived bilinear Bäcklund transformation and an array of infinite superposition soliton solutions of the equation are formulated using this nonlinear superposition formula.Lastly,leveraging the obtained bilinear equation,infinite superposition solutions of various functional types are constructed.Their dynamic characteristics are analyzed through illustrated solution images.It is noteworthy that this paper not only uncovers a multitude of properties through the Bell polynomial method but also derives both infinite linear and nonlinear superposition solutions,enriching the diversity of solutions,these aspects have not been previously explored in existing literature.
