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
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
We construct the soliton solution and smooth positon solution of the second-type derivative nonlinear Schr¨odinger(DNLSII) equation. Additionally, we present a detailed discussion about the decomposition of the p...We construct the soliton solution and smooth positon solution of the second-type derivative nonlinear Schr¨odinger(DNLSII) equation. Additionally, we present a detailed discussion about the decomposition of the positon solution, and display its approximate orbits and variable "phase shift". The second and third order breather-positon solutions are also constructed.展开更多
We discuss a modified Wadati-Konno-Ichikawa(m WKI)equation,which is equivalent to the WKI equation by a hodograph transformation.The explicit formula of degenerated solution of m WKI equation is provided by using dege...We discuss a modified Wadati-Konno-Ichikawa(m WKI)equation,which is equivalent to the WKI equation by a hodograph transformation.The explicit formula of degenerated solution of m WKI equation is provided by using degenerate Darboux transformation with respect to the eigenvalues,which yields two kinds of smooth solutions possessing the vanishing and nonvanishing boundary conditions respectively.In particular,a method for the decomposition of modulus square is operated to the positon solution,and the approximate orbits before and after collision of positon solutions are displayed explicitly.展开更多
According to the N-soliton solution derived from Hirota's bilinear method,higher-order smooth positons and breather positons are obtained efficiently through an ingenious limit approach.This paper takes the Sine-G...According to the N-soliton solution derived from Hirota's bilinear method,higher-order smooth positons and breather positons are obtained efficiently through an ingenious limit approach.This paper takes the Sine-Gordon equation as an example to introduce how to utilize this technique to generate these higher-order smooth positons and breather positons in detail.The dynamical behaviors of smooth positons and breather positons are presented by some figures.During the procedure of deduction,the approach mentioned has the strengths of concision and celerity.In terms of feasibility and practicability,this approach can be exploited widely to study higherorder smooth positons and breather positons of other integrable systems.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
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
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 th...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
In this article, we establish the existence of at least two positive solutions for the semi-positone m-point boundary value problem with a parameter u （t） ＋ λf （t, u） = 0, t ∈ （0, 1）, u （0） = sum （biu （ξ...In this article, we establish the existence of at least two positive solutions for the semi-positone m-point boundary value problem with a parameter u （t） ＋ λf （t, u） = 0, t ∈ （0, 1）, u （0） = sum （biu （ξ i ）） from i=1 to m-2, u（1）= sum （aiu（ξ i ）） from i=1 to m-2, where λ 〉 0 is a parameter, 0 〈 ξ 1 〈 ξ 2 〈 ··· 〈 ξ m 2 〈 1 with 0 〈sum ai from i=1 to m-2 〈 1, sum bi from i=1 to m-2 =1 b i 〈 1, a i , b i ∈ [0, ∞） and f （t, u） ≥ M with M is a positive constant. The method employed is the Leggett-Williams fixed-point theorem. As an application, an example is given to demonstrate the main result.展开更多
In this paper,we derive Darboux transformation of the inhomogeneous Hirota and the Maxwell-Bloch(IH-MB)equations which are governed by femtosecond pulse propagation through inhomogeneous doped fibre.The determinant re...In this paper,we derive Darboux transformation of the inhomogeneous Hirota and the Maxwell-Bloch(IH-MB)equations which are governed by femtosecond pulse propagation through inhomogeneous doped fibre.The determinant representation of Darboux transformation is used to derive soliton solutions,positon solutions to the IH-MB equations.展开更多
In this paper, we study the existence of positive solutions of a sub-linear semi-positone differential boundary value problems system with positive parameter. We prove that the semipositone differential boundary value...In this paper, we study the existence of positive solutions of a sub-linear semi-positone differential boundary value problems system with positive parameter. We prove that the semipositone differential boundary value problems system has at least one positive solution for the parameter sufficiently large.展开更多
In this paper we study a class of semi-positone singular boundary value problem. With prior bounds estimate and topology degree method, some existence results of nonnegative solution will be shown.
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.
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 the National Natural Science Foundation of China under Grant No.11671219the K.C.Wong Magna Fund in Ningbo Universitythe Natural Science Foundation of Zhejiang Province under Grant No.LZ19A010001
文摘We construct the soliton solution and smooth positon solution of the second-type derivative nonlinear Schr¨odinger(DNLSII) equation. Additionally, we present a detailed discussion about the decomposition of the positon solution, and display its approximate orbits and variable "phase shift". The second and third order breather-positon solutions are also constructed.
基金Supported by the National Natural Science Foundation of China under Grant No.11671219the K.C.Wong Magna Fund in Ningbo University
文摘We discuss a modified Wadati-Konno-Ichikawa(m WKI)equation,which is equivalent to the WKI equation by a hodograph transformation.The explicit formula of degenerated solution of m WKI equation is provided by using degenerate Darboux transformation with respect to the eigenvalues,which yields two kinds of smooth solutions possessing the vanishing and nonvanishing boundary conditions respectively.In particular,a method for the decomposition of modulus square is operated to the positon solution,and the approximate orbits before and after collision of positon solutions are displayed explicitly.
基金supported by the National Natural Science Foundation of China under Grant Nos.12175111 and 11975131K C Wong Magna Fund in Ningbo University。
文摘According to the N-soliton solution derived from Hirota's bilinear method,higher-order smooth positons and breather positons are obtained efficiently through an ingenious limit approach.This paper takes the Sine-Gordon equation as an example to introduce how to utilize this technique to generate these higher-order smooth positons and breather positons in detail.The dynamical behaviors of smooth positons and breather positons are presented by some figures.During the procedure of deduction,the approach mentioned has the strengths of concision and celerity.In terms of feasibility and practicability,this approach can be exploited widely to study higherorder smooth positons and breather positons of other integrable systems.
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.
基金Supported by Fund of National Natural Science of China (No. 10371068)Science Foundation of Business College of Shanxi University (No. 2008053)
文摘In this article, we establish the existence of at least two positive solutions for the semi-positone m-point boundary value problem with a parameter u （t） ＋ λf （t, u） = 0, t ∈ （0, 1）, u （0） = sum （biu （ξ i ）） from i=1 to m-2, u（1）= sum （aiu（ξ i ）） from i=1 to m-2, where λ 〉 0 is a parameter, 0 〈 ξ 1 〈 ξ 2 〈 ··· 〈 ξ m 2 〈 1 with 0 〈sum ai from i=1 to m-2 〈 1, sum bi from i=1 to m-2 =1 b i 〈 1, a i , b i ∈ [0, ∞） and f （t, u） ≥ M with M is a positive constant. The method employed is the Leggett-Williams fixed-point theorem. As an application, an example is given to demonstrate the main result.
基金supported by the National Natural Science Foundation of China(Grant Nos.11201251 and 11271210)Zhejiang Provincial Natural Science Foundation of China(Grant No.LY12A01007)+1 种基金the Natural Science Foundation of Ningbo(Grant No.2013A610105)K.C.Wong Magna Fund in Ningbo University
文摘In this paper,we derive Darboux transformation of the inhomogeneous Hirota and the Maxwell-Bloch(IH-MB)equations which are governed by femtosecond pulse propagation through inhomogeneous doped fibre.The determinant representation of Darboux transformation is used to derive soliton solutions,positon solutions to the IH-MB equations.
基金Foundation item： the National Natural Science Foundation of China （No. 10671167）
文摘In this paper, we study the existence of positive solutions of a sub-linear semi-positone differential boundary value problems system with positive parameter. We prove that the semipositone differential boundary value problems system has at least one positive solution for the parameter sufficiently large.
基金the National Natural Sciences Foundation of China (No.19871005).
文摘In this paper we study a class of semi-positone singular boundary value problem. With prior bounds estimate and topology degree method, some existence results of nonnegative solution will be shown.
