The Gerdjikov-Ivanov(GI)hierarchy is derived via recursion operator,in this article,we mainly investigate the third-order flow GI equation.In the framework of the Riemann-Hilbert method,the soliton matrices of the thi...The Gerdjikov-Ivanov(GI)hierarchy is derived via recursion operator,in this article,we mainly investigate the third-order flow GI equation.In the framework of the Riemann-Hilbert method,the soliton matrices of the third-order flow GI equation with simple zeros and elementary high-order zeros of Riemann-Hilbert problem are constructed through the standard dressing process.Taking advantage of this result,some properties and asymptotic analysis of single soliton solution and two soliton solution are discussed,and the simple elastic interaction of two soliton are proved.Compared with soliton solution of the classical second-order flow,we find that the higher-order dispersion term affects the propagation velocity,propagation direction and amplitude of the soliton.Finally,by means of a certain limit technique,the high-order soliton solution matrix for the third-order flow GI equation is derived.展开更多
Bilinear forms of the coupled Gerdjikov–Ivanov equation are derived. The $N$-soliton solutions to the equation are obtained by Hirota＇s method. It is interesting that the two-soliton solutions can generate the rogue...Bilinear forms of the coupled Gerdjikov–Ivanov equation are derived. The $N$-soliton solutions to the equation are obtained by Hirota＇s method. It is interesting that the two-soliton solutions can generate the rogue-wave-like phenomena by selecting special parameters. The equation can be reduced to the Gerdjikov–Ivanov equation as well as its bilinear forms and its solutions.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
The nonlocal nonlinear Gerdjikov-Ivanov(GI)equation is one of the most important integrable equations,which can be reduced from the third generic deformation of the derivative nonlinear Schr?dinger equation.The Darbou...The nonlocal nonlinear Gerdjikov-Ivanov(GI)equation is one of the most important integrable equations,which can be reduced from the third generic deformation of the derivative nonlinear Schr?dinger equation.The Darboux transformation is a successful method in solving many nonlocal equations with the help of symbolic computation.As applications,we obtain the bright-dark soliton,breather,rogue wave,kink,W-shaped soliton and periodic solutions of the nonlocal GI equation by constructing its 2 n-fold Darboux transformation.These solutions show rich wave structures for selections of different parameters.In all these instances we practically show that these solutions have different properties than the ones for local case.展开更多
基金supported by the National Natural Science Foundation of China(No.12175069 and No.12235007)Science and Technology Commission of Shanghai Municipality(No.21JC1402500 and No.22DZ2229014)Natural Science Foundation of Shanghai,China(No.23ZR1418100).
文摘The Gerdjikov-Ivanov(GI)hierarchy is derived via recursion operator,in this article,we mainly investigate the third-order flow GI equation.In the framework of the Riemann-Hilbert method,the soliton matrices of the third-order flow GI equation with simple zeros and elementary high-order zeros of Riemann-Hilbert problem are constructed through the standard dressing process.Taking advantage of this result,some properties and asymptotic analysis of single soliton solution and two soliton solution are discussed,and the simple elastic interaction of two soliton are proved.Compared with soliton solution of the classical second-order flow,we find that the higher-order dispersion term affects the propagation velocity,propagation direction and amplitude of the soliton.Finally,by means of a certain limit technique,the high-order soliton solution matrix for the third-order flow GI equation is derived.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11671177 and 11271168the Jiangsu Qing Lan Project(2014)the Six Talent Peaks Project of Jiangsu Province under Grant No 2016-JY-08
文摘Bilinear forms of the coupled Gerdjikov–Ivanov equation are derived. The $N$-soliton solutions to the equation are obtained by Hirota＇s method. It is interesting that the two-soliton solutions can generate the rogue-wave-like phenomena by selecting special parameters. The equation can be reduced to the Gerdjikov–Ivanov equation as well as its bilinear forms and its solutions.
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 the National Natural Science Foundation of China(Grant No.11371326 and Grant No.11975145)。
文摘The nonlocal nonlinear Gerdjikov-Ivanov(GI)equation is one of the most important integrable equations,which can be reduced from the third generic deformation of the derivative nonlinear Schr?dinger equation.The Darboux transformation is a successful method in solving many nonlocal equations with the help of symbolic computation.As applications,we obtain the bright-dark soliton,breather,rogue wave,kink,W-shaped soliton and periodic solutions of the nonlocal GI equation by constructing its 2 n-fold Darboux transformation.These solutions show rich wave structures for selections of different parameters.In all these instances we practically show that these solutions have different properties than the ones for local case.
基金Supported by grants from the National Science Foundation of China under Grant No.11671095National Science Foundation of China under Grant No.11501365+1 种基金Shanghai Sailing Program supported by Science and Technology Commission of Shanghai Municipality under Grant No 15YF1408100the Hujiang Foundation of China(B14005)
文摘The Fokas unified method is Gerdjikov-Ivanonv equation on the half-line. expressed in terms of the solution of a 3 × 3 through the global relation.
