The third-order flow Gerdjikov–Ivanov(TOFGI)equation is studied,and the Darboux transformation(DT)is used to obtain the determinant expression of the solution of this equation.On this basis,the soliton solution,ratio...The third-order flow Gerdjikov–Ivanov(TOFGI)equation is studied,and the Darboux transformation(DT)is used to obtain the determinant expression of the solution of this equation.On this basis,the soliton solution,rational solution,positon solution,and breather solution of the TOFGI equation are obtained by taking zero seed solution and non-zero seed solution.The exact solutions and dynamic properties of the Gerdjikov–Ivanov(GI)equation and the TOFGI equation are compared in detail under the same conditions,and it is found that there are some differences in the velocities and trajectories of the solutions of the two equations.展开更多
Soliton molecules are firstly obtained by velocity resonance for the Gerdjikov–Ivanov equation, and n-order smooth positon solutions for the Gerdjikov–Ivanov equation are generated by means of the general determinan...Soliton molecules are firstly obtained by velocity resonance for the Gerdjikov–Ivanov equation, and n-order smooth positon solutions for the Gerdjikov–Ivanov equation are generated by means of the general determinant expression of n-soliton solution. The dynamics of the smooth positons of the Gerdjikov–Ivanov equation are discussed using the decomposition of the modulus square, the trajectories and time-dependent "phase shifts" of positons after the collision can be described approximately. Additionally, some novel hybrid solutions consisting solitons and positons are presented and their rather complicated dynamics are revealed.展开更多
Solving nonlinear partial differential equations have attracted intensive attention in the past few decades. In this paper, the Darboux transformation method is used to derive several positon and hybrid solutions for ...Solving nonlinear partial differential equations have attracted intensive attention in the past few decades. In this paper, the Darboux transformation method is used to derive several positon and hybrid solutions for the(2+1)-dimensional complex modified Korteweg–de Vries equations. Based on the zero seed solution, the positon solution and the hybrid solutions of positon and soliton are constructed. The composition of positons is studied, showing that multi-positons of(2+1)-dimensional equations are decomposed into multi-solitons as well as the(1+1)-dimensions. Moreover, the interactions between positon and soliton are analyzed. In addition, the hybrid solutions of b-positon and breather are obtained using the plane wave seed solution, and their evolutions with time are discussed.展开更多
The determinant representation of three-fold Darboux transformation for a variable-coefficient modified KdV equation is displayed based on the technique used to solve Ablowitz-Kaup-Newell-Segur system. Additionally, t...The determinant representation of three-fold Darboux transformation for a variable-coefficient modified KdV equation is displayed based on the technique used to solve Ablowitz-Kaup-Newell-Segur system. Additionally, the nonsingular positon solutions of the variable-coefficient modified KdV equation are firstly discovered analytically and graphically.展开更多
Darboux transformation （DT） provides us with a comprehensive approach to construct the exact and explicit solutions to the negative extended KdV （eKdV） equation, by which some new solutions such as singular solito...Darboux transformation （DT） provides us with a comprehensive approach to construct the exact and explicit solutions to the negative extended KdV （eKdV） equation, by which some new solutions such as singular soliton, negaton, and positon solutions are computed for the eKdV equation. We rediscover the soliton solution with finiteamplitude in [A.V. Slyunyaev and E.N. Pelinovskii, J. Exp. Theor. Phys. 89 （1999） 173] and discuss the difference between this soliton and the singular soliton. We clarify the relationship between the exact solutions of the eKdV equation and the spectral parameter. Moreover, the interactions of singular two solitons, positon and negaton, positon and soliton, and two positons are studied in detail.展开更多
In this paper, negatons, positons, and complexiton solutions of higher order for a non-isospectral KdV equation, the KdV equation with loss and non-uniformity terms are obtained through the bilinear Baicklund transfor...In this paper, negatons, positons, and complexiton solutions of higher order for a non-isospectral KdV equation, the KdV equation with loss and non-uniformity terms are obtained through the bilinear Baicklund transformation. Further, the properties of some solutions are shown by some figures made by using Maple.展开更多
Applicable in fluid dynamics and plasmas, a generalized variable-coefficient Korteweg-de Vries （vcKdV） model is investigated. The bilinear form and analytic N-soliton-like solution for such a model are derived by th...Applicable in fluid dynamics and plasmas, a generalized variable-coefficient Korteweg-de Vries （vcKdV） model is investigated. The bilinear form and analytic N-soliton-like solution for such a model are derived by the Hirota method and Wronskian technique. Additionally, the bilinear auto-Bǎcklund transformation between （N-1）- soliton-like and N-soliton-like solutions is verified.展开更多
Employing the method which can be used to demonstrate the infinite conservation laws for the standard Kortewegde Vries （KdV） equation, we prove that the variable-coeFficient KdV equation under the Painlevé test...Employing the method which can be used to demonstrate the infinite conservation laws for the standard Kortewegde Vries （KdV） equation, we prove that the variable-coeFficient KdV equation under the Painlevé test condition also possesses the formal conservation laws.展开更多
The coupled Korteweg-de Vries （CKdV） equation with self-consistent sources （CKdVESCS） and its Lax representation are derived. We present a generalized binary Darboux transformation （GBDT） with an arbitrary time-...The coupled Korteweg-de Vries （CKdV） equation with self-consistent sources （CKdVESCS） and its Lax representation are derived. We present a generalized binary Darboux transformation （GBDT） with an arbitrary time- dependent function for the CKdVESCS as well as the formula for the N-times repeated GBDT. This GBDT provides non-auto-Biicklund transformation between two CKdVESCSs with different degrees of sources and enables us to construct more generM solutions with N arbitrary t-dependent functions. We obtain positon, negaton, complexiton, and negaton- positon solutions of the CKdVESCS.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant No.12201329)the Zhejiang Provincial Natural Science Foundation of China(Grant No.LY24A010002)the Natural Science Foundation of Ningbo(Grant No.2023J126)。
文摘The third-order flow Gerdjikov–Ivanov(TOFGI)equation is studied,and the Darboux transformation(DT)is used to obtain the determinant expression of the solution of this equation.On this basis,the soliton solution,rational solution,positon solution,and breather solution of the TOFGI equation are obtained by taking zero seed solution and non-zero seed solution.The exact solutions and dynamic properties of the Gerdjikov–Ivanov(GI)equation and the TOFGI equation are compared in detail under the same conditions,and it is found that there are some differences in the velocities and trajectories of the solutions of the two equations.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11775121 and 11435005)the K. C. Wong Magna Fund in Ningbo University.
文摘Soliton molecules are firstly obtained by velocity resonance for the Gerdjikov–Ivanov equation, and n-order smooth positon solutions for the Gerdjikov–Ivanov equation are generated by means of the general determinant expression of n-soliton solution. The dynamics of the smooth positons of the Gerdjikov–Ivanov equation are discussed using the decomposition of the modulus square, the trajectories and time-dependent "phase shifts" of positons after the collision can be described approximately. Additionally, some novel hybrid solutions consisting solitons and positons are presented and their rather complicated dynamics are revealed.
基金Project sponsored by NUPTSF(Grant Nos.NY220161and NY222169)the Foundation of Jiangsu Provincial Double-Innovation Doctor Program(Grant No.JSSCBS20210541)+1 种基金the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province,China(Grant No.22KJB110004)the National Natural Science Foundation of China(Grant No.11871446)。
文摘Solving nonlinear partial differential equations have attracted intensive attention in the past few decades. In this paper, the Darboux transformation method is used to derive several positon and hybrid solutions for the(2+1)-dimensional complex modified Korteweg–de Vries equations. Based on the zero seed solution, the positon solution and the hybrid solutions of positon and soliton are constructed. The composition of positons is studied, showing that multi-positons of(2+1)-dimensional equations are decomposed into multi-solitons as well as the(1+1)-dimensions. Moreover, the interactions between positon and soliton are analyzed. In addition, the hybrid solutions of b-positon and breather are obtained using the plane wave seed solution, and their evolutions with time are discussed.
文摘The determinant representation of three-fold Darboux transformation for a variable-coefficient modified KdV equation is displayed based on the technique used to solve Ablowitz-Kaup-Newell-Segur system. Additionally, the nonsingular positon solutions of the variable-coefficient modified KdV equation are firstly discovered analytically and graphically.
基金supported by National Natural Science Foundation of China under Grant No.10601028
文摘Darboux transformation （DT） provides us with a comprehensive approach to construct the exact and explicit solutions to the negative extended KdV （eKdV） equation, by which some new solutions such as singular soliton, negaton, and positon solutions are computed for the eKdV equation. We rediscover the soliton solution with finiteamplitude in [A.V. Slyunyaev and E.N. Pelinovskii, J. Exp. Theor. Phys. 89 （1999） 173] and discuss the difference between this soliton and the singular soliton. We clarify the relationship between the exact solutions of the eKdV equation and the spectral parameter. Moreover, the interactions of singular two solitons, positon and negaton, positon and soliton, and two positons are studied in detail.
基金The project partially supported by the State Key Basic Research Program of China under Grant No. 2004CB318000
文摘In this paper, negatons, positons, and complexiton solutions of higher order for a non-isospectral KdV equation, the KdV equation with loss and non-uniformity terms are obtained through the bilinear Baicklund transformation. Further, the properties of some solutions are shown by some figures made by using Maple.
基金Supported by the Key Project of the Ministry of Education of China under Grant No 106033, and the National Natural Science Foundation of China under Grant Nos 60372095 and 10272017, the Green Path Programme of Air Force of the Chinese People＇s Liberation Army, the Cheung Kong Scholars Programme of the Ministry of Education of China, and Li Ka Shing Foundation of Hong Kong.
文摘Applicable in fluid dynamics and plasmas, a generalized variable-coefficient Korteweg-de Vries （vcKdV） model is investigated. The bilinear form and analytic N-soliton-like solution for such a model are derived by the Hirota method and Wronskian technique. Additionally, the bilinear auto-Bǎcklund transformation between （N-1）- soliton-like and N-soliton-like solutions is verified.
基金Supported by the Key Project of Chinese Ministry of Education under Grant No 106033, the National Natural Science Foundation of China under Grant Nos 60372095 and 60772023, Open Fund of the State Key Laboratory of Software Development Environment under Grant No SKLSDE-07-001, Beijing University of Aeronautics and Astronautics, the National Basic Research Programme of China under Grant No 2005CB321901, the Green Path Programme of Air Force of the Chinese People＇s Liberation Army, the Cheung Kong Scholars Programme of the Ministry of Education of China and Li Ka Shing Foundation of Hong Kong.
文摘Employing the method which can be used to demonstrate the infinite conservation laws for the standard Kortewegde Vries （KdV） equation, we prove that the variable-coeFficient KdV equation under the Painlevé test condition also possesses the formal conservation laws.
基金The project supported by the National Fundamental Research Program of China(973 Program)under Grant No.2007CB814800National Natural Science Foundation of China under Grant No.10601028
文摘The coupled Korteweg-de Vries （CKdV） equation with self-consistent sources （CKdVESCS） and its Lax representation are derived. We present a generalized binary Darboux transformation （GBDT） with an arbitrary time- dependent function for the CKdVESCS as well as the formula for the N-times repeated GBDT. This GBDT provides non-auto-Biicklund transformation between two CKdVESCSs with different degrees of sources and enables us to construct more generM solutions with N arbitrary t-dependent functions. We obtain positon, negaton, complexiton, and negaton- positon solutions of the CKdVESCS.
