Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained thro...Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.展开更多
The coupled modified nonlinear Schrodinger equations are under investigation in this work. Starting from analyzing the spectral problem of the Lax pair, a Riemann-Hilbert problem for the coupled modified nonlinear Sch...The coupled modified nonlinear Schrodinger equations are under investigation in this work. Starting from analyzing the spectral problem of the Lax pair, a Riemann-Hilbert problem for the coupled modified nonlinear Schrodinger equations is formulated. And then, through solving the obtained Riemann-Hilbert problem under the conditions of irregularity and reflectionless case, N-soliton solutions for the equations are presented. Furthermore, the localized structures and dynamic behaviors of the one-soliton solution are shown graphically.展开更多
Head-on collisions among the single-and multi-soliton’s heavy ion acoustic waves(HIAWs) of multi-ion plasma are studied. The plasma consists of adiabatic positively charged inertial heavy ions, Boltzmann energy distr...Head-on collisions among the single-and multi-soliton’s heavy ion acoustic waves(HIAWs) of multi-ion plasma are studied. The plasma consists of adiabatic positively charged inertial heavy ions, Boltzmann energy distributed light ions and kappa energy distributed electrons. The extended poincaré-Lighthill-Kuo(e PLK) method is applied for the derivation of two-sided Korteweg–de Vries(KdV) equations(KdVEs). The KdV single-soliton solutions(KdVSSs) are determined using the hyperbolic secant method and the KdV multi-soliton solutions(KdVMSs)are obtained using the Hirota method. The effects of superthermality of electrons, temperature ratios of electron to light ion and heavy ion to electron, and the density ratio of electron to heavy ion on phase shifts are investigated for the head-on collisions between two-soliton HIAWs. The effects of plasma parameters on angular frequency, phase speed, production of multi-soliton structures, and the head-on collisions among single-, double-, triple-, quadruple-, and quintuplesolitons are also studied.展开更多
In this paper, by using bilinear form and extended three-wave type of ans¨atz approach, we obtain new cross-kink multi-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equation, including the periodic breath...In this paper, by using bilinear form and extended three-wave type of ans¨atz approach, we obtain new cross-kink multi-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equation, including the periodic breather-type of kink three-soliton solutions, the cross-kink four-soliton solutions, the doubly periodic breathertype of soliton solutions and the doubly periodic breather-type of cross-kink two-soliton solutions. It is shown that the generalized three-wave method, with the help of symbolic computation, provides an effective and powerful mathematical tool for solving high dimensional nonlinear evolution equations in mathematical physics.展开更多
This paper solves the integrable CH-γequation for analytical multiple soliton solutions with the Darboux transformation method.Some properties of the soliton solutions are different from the CH equation.
In this letter, exact chirped multi-soliton solutions of the nonlinear Schrodinger (NLS) equation with varying coefficients are found. The explicit chirped one- and two-soliton solutions are generated. As an example, ...In this letter, exact chirped multi-soliton solutions of the nonlinear Schrodinger (NLS) equation with varying coefficients are found. The explicit chirped one- and two-soliton solutions are generated. As an example, an exponential distributed control system is considered, and some main features of solutions are shown. The results reveal that chirped soliton can all be nonlinearly compressed cleanly and efficiently in an optical fiber with no loss or gain, with the loss, or with the gain. Furthermore, under the same initial condition, compression of optical soliton in the optical fiber with the loss is the most dramatic. Also, under nonintegrable condition and finite initial perturbations, the evolution of chirped soliton has been demonstrated by simulating numerically.展开更多
We propose and develop another approach to constructing multi-soliton solutions of an integrable two-component Camassa–Holm(CH2)system.With the help of a reciprocal transformation and a gauge transformation,we relate...We propose and develop another approach to constructing multi-soliton solutions of an integrable two-component Camassa–Holm(CH2)system.With the help of a reciprocal transformation and a gauge transformation,we relate the CH2 system to a negative flow of the Broer–Kaup or twoboson hierarchy.The solutions of this negative flow are given in terms of Wronskians via Darboux transformation.Then the multi-soliton solutions of the CH2 system are recovered in parametric form by inverting the reciprocal transformation and the gauge transformation.展开更多
In this paper, we construct the Darboux transformation(DT) for the reverse-time integrable nonlocal nonlinear Schrodinger equation by loop group method. Then we utilize the DT to derive soliton solutions with zero see...In this paper, we construct the Darboux transformation(DT) for the reverse-time integrable nonlocal nonlinear Schrodinger equation by loop group method. Then we utilize the DT to derive soliton solutions with zero seed. We investigate the dynamical properties for those solutions and present a sufficient condition for the non-singularity of multi-soliton solutions.Furthermore, the asymptotic analysis of bounded multi-solutions has also been established by the determinant formula.展开更多
The purpose of the paper is to formulate multi-soliton solutions for the nonlocal Hirota equations via the Riemann-Hilbert(RH)approach.The RH problems are constructed and the zero structures are studied via performing...The purpose of the paper is to formulate multi-soliton solutions for the nonlocal Hirota equations via the Riemann-Hilbert(RH)approach.The RH problems are constructed and the zero structures are studied via performing spectral analysis of the Lax pair.Then we consider three types of nonlocal Hirota equations by discussing different symmetry reductions of the potential matrix.On the basis of the resulting matrix RH problem under the restriction of the reflectionless case,we successfully obtain the multi-soliton solutions of the nonlocal Hirota equations.展开更多
This paper studies multi-solitons of non-integrable generalized Davey-Stewartson system in the elliptic-elliptic case. By extending the method for constructing multi-solitons of non-integrable nonlinear Schr¨odin...This paper studies multi-solitons of non-integrable generalized Davey-Stewartson system in the elliptic-elliptic case. By extending the method for constructing multi-solitons of non-integrable nonlinear Schr¨odinger equations and systems developed by Martel et al. to the present non-integrable generalized DaveyStewartson system and overcoming some new difficulties caused by the nonlocal operator B, we prove the existence of multi-solitons for this system. Furthermore, we also give a generalization of this result to a more general class of equations with nonlocal nonlinearities.展开更多
In this paper, we study a coupled modified Volterra lattice equation which is an integrable semidiscrete version of the coupled KdV and the coupled mKdV equation. By using the Darboux transformation, we obtain its new...In this paper, we study a coupled modified Volterra lattice equation which is an integrable semidiscrete version of the coupled KdV and the coupled mKdV equation. By using the Darboux transformation, we obtain its new explicit solutions including multi-soliton and multi-positon. Furthermore, an integrable discretization of the coupled modified Volterra lattice equation is constructed.展开更多
The authors study the multi-soliton,multi-cuspon solutions to the Camassa-Holm equation and their interaction.According to the solution formula due to Li in 2004 and 2005,the authors give the proper choice of paramete...The authors study the multi-soliton,multi-cuspon solutions to the Camassa-Holm equation and their interaction.According to the solution formula due to Li in 2004 and 2005,the authors give the proper choice of parameters for multi-soliton and multicuspon solutions,especially for n ≥ 3 case.The numerical method (the so-called local discontinuous Galerkin (LDG) method) is also used to simulate the solutions and give the comparison of exact solutions and numerical solutions.The numerical results for the two-soliton and one-cuspon,one-soliton and two-cuspon,three-soliton,three-cuspon,three-soliton and one-cuspon,two-soliton and two-cuspon,one-soliton and three-cuspon,four-soliton and four-cuspon are investigated by the numerical method for the first time,respectively.展开更多
Under consideration in this study is the discrete coupled modified Korteweg-de Vries(mKdV)equation with 4×4 Lax pair.Firstly,through using continuous limit technique,this discrete equation can be mapped to the co...Under consideration in this study is the discrete coupled modified Korteweg-de Vries(mKdV)equation with 4×4 Lax pair.Firstly,through using continuous limit technique,this discrete equation can be mapped to the coupled KdV and mKdV equations,which may depict the development of shallow water waves,the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma.Secondly,the discrete generalized(r,N-r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem,from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background,higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived,and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique.Finally,the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions.These results may be helpful for understanding some physical phenomena in fields of shallow water wave,optics,and plasma physics.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant No. 12161061)the Fundamental Research Funds for the Inner Mongolia University of Finance and Economics (Grant No. NCYWT23036)+2 种基金the Young Innovative and Entrepreneurial Talents of the Inner Mongolia Grassland Talents Project in 2022,Autonomous Region “Five Major Tasks” Research Special Project for the Inner Mongolia University of Finance and Economics in 2024 (Grant No. NCXWD2422)High Quality Research Achievement Cultivation Fund for the Inner Mongolia University of Finance and Economics in 2024 (Grant No. GZCG2426)the Talent Development Fund of Inner Mongolia Autonomous Region, China。
文摘Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61072147 and 11271008)
文摘The coupled modified nonlinear Schrodinger equations are under investigation in this work. Starting from analyzing the spectral problem of the Lax pair, a Riemann-Hilbert problem for the coupled modified nonlinear Schrodinger equations is formulated. And then, through solving the obtained Riemann-Hilbert problem under the conditions of irregularity and reflectionless case, N-soliton solutions for the equations are presented. Furthermore, the localized structures and dynamic behaviors of the one-soliton solution are shown graphically.
文摘Head-on collisions among the single-and multi-soliton’s heavy ion acoustic waves(HIAWs) of multi-ion plasma are studied. The plasma consists of adiabatic positively charged inertial heavy ions, Boltzmann energy distributed light ions and kappa energy distributed electrons. The extended poincaré-Lighthill-Kuo(e PLK) method is applied for the derivation of two-sided Korteweg–de Vries(KdV) equations(KdVEs). The KdV single-soliton solutions(KdVSSs) are determined using the hyperbolic secant method and the KdV multi-soliton solutions(KdVMSs)are obtained using the Hirota method. The effects of superthermality of electrons, temperature ratios of electron to light ion and heavy ion to electron, and the density ratio of electron to heavy ion on phase shifts are investigated for the head-on collisions between two-soliton HIAWs. The effects of plasma parameters on angular frequency, phase speed, production of multi-soliton structures, and the head-on collisions among single-, double-, triple-, quadruple-, and quintuplesolitons are also studied.
文摘In this paper, by using bilinear form and extended three-wave type of ans¨atz approach, we obtain new cross-kink multi-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equation, including the periodic breather-type of kink three-soliton solutions, the cross-kink four-soliton solutions, the doubly periodic breathertype of soliton solutions and the doubly periodic breather-type of cross-kink two-soliton solutions. It is shown that the generalized three-wave method, with the help of symbolic computation, provides an effective and powerful mathematical tool for solving high dimensional nonlinear evolution equations in mathematical physics.
基金the National Natural Science Foundation of China (Grant No.10401022)the Research Grants Council of Hong Kong
文摘This paper solves the integrable CH-γequation for analytical multiple soliton solutions with the Darboux transformation method.Some properties of the soliton solutions are different from the CH equation.
基金This work was supported by the National Natural Science Foundation of China (No. 60477026), the Provincial Youth Science Foundation of Shanxi (No. 20011015).
文摘In this letter, exact chirped multi-soliton solutions of the nonlinear Schrodinger (NLS) equation with varying coefficients are found. The explicit chirped one- and two-soliton solutions are generated. As an example, an exponential distributed control system is considered, and some main features of solutions are shown. The results reveal that chirped soliton can all be nonlinearly compressed cleanly and efficiently in an optical fiber with no loss or gain, with the loss, or with the gain. Furthermore, under the same initial condition, compression of optical soliton in the optical fiber with the loss is the most dramatic. Also, under nonintegrable condition and finite initial perturbations, the evolution of chirped soliton has been demonstrated by simulating numerically.
基金supported in part by the National Natural Science Foundation of China(Grant Nos.11871471,11931017,11505064 and 11805071)Natural Science Foundation of Fujian Province,China(Grant No.2016J05008)+1 种基金the Yue Qi Outstanding Scholar Project,China University of Mining and Technology,Beijing(Grant No.00-800015Z1177)the Fundamental Research Funds for the Central Universities.
文摘We propose and develop another approach to constructing multi-soliton solutions of an integrable two-component Camassa–Holm(CH2)system.With the help of a reciprocal transformation and a gauge transformation,we relate the CH2 system to a negative flow of the Broer–Kaup or twoboson hierarchy.The solutions of this negative flow are given in terms of Wronskians via Darboux transformation.Then the multi-soliton solutions of the CH2 system are recovered in parametric form by inverting the reciprocal transformation and the gauge transformation.
基金supported by the National Natural Science Foundation of China(Grant No.11771151)the Guangzhou Science and Technology Program of China(Grant No.201904010362)the Fundamental Research Funds for the Central Universities of China(Grant No.2019MS110)。
文摘In this paper, we construct the Darboux transformation(DT) for the reverse-time integrable nonlocal nonlinear Schrodinger equation by loop group method. Then we utilize the DT to derive soliton solutions with zero seed. We investigate the dynamical properties for those solutions and present a sufficient condition for the non-singularity of multi-soliton solutions.Furthermore, the asymptotic analysis of bounded multi-solutions has also been established by the determinant formula.
基金supported by the National Natural Science Foundation of China(No.11371326 and No.11975145)。
文摘The purpose of the paper is to formulate multi-soliton solutions for the nonlocal Hirota equations via the Riemann-Hilbert(RH)approach.The RH problems are constructed and the zero structures are studied via performing spectral analysis of the Lax pair.Then we consider three types of nonlocal Hirota equations by discussing different symmetry reductions of the potential matrix.On the basis of the resulting matrix RH problem under the restriction of the reflectionless case,we successfully obtain the multi-soliton solutions of the nonlocal Hirota equations.
基金supported by National Natural Science Foundation of China (Grant No. 11571381)
文摘This paper studies multi-solitons of non-integrable generalized Davey-Stewartson system in the elliptic-elliptic case. By extending the method for constructing multi-solitons of non-integrable nonlinear Schr¨odinger equations and systems developed by Martel et al. to the present non-integrable generalized DaveyStewartson system and overcoming some new difficulties caused by the nonlocal operator B, we prove the existence of multi-solitons for this system. Furthermore, we also give a generalization of this result to a more general class of equations with nonlocal nonlinearities.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10971136also in part by the Ministry of Education and Innovation of Spain under Contract MTM2009-12670ZHQ is supported by Shanghai 085 Project
文摘In this paper, we study a coupled modified Volterra lattice equation which is an integrable semidiscrete version of the coupled KdV and the coupled mKdV equation. By using the Darboux transformation, we obtain its new explicit solutions including multi-soliton and multi-positon. Furthermore, an integrable discretization of the coupled modified Volterra lattice equation is constructed.
基金Project supported by the National Natural Science Foundation of China (Nos.10971211,11031007)the Foundation for the Author of National Excellent Doctoral Dissertation of China (No.200916)+1 种基金the Foundation for the Author of National Excellent Doctoral Dissertation of the Chinese Academy of Sciences,Program for New Century Excellent Talents in University of China (No.09-0922)the Fundamental Research Funds for the Central Universities (No.WK0010000005)
文摘The authors study the multi-soliton,multi-cuspon solutions to the Camassa-Holm equation and their interaction.According to the solution formula due to Li in 2004 and 2005,the authors give the proper choice of parameters for multi-soliton and multicuspon solutions,especially for n ≥ 3 case.The numerical method (the so-called local discontinuous Galerkin (LDG) method) is also used to simulate the solutions and give the comparison of exact solutions and numerical solutions.The numerical results for the two-soliton and one-cuspon,one-soliton and two-cuspon,three-soliton,three-cuspon,three-soliton and one-cuspon,two-soliton and two-cuspon,one-soliton and three-cuspon,four-soliton and four-cuspon are investigated by the numerical method for the first time,respectively.
基金Project supported by the National Natural Science Foundation of China (Grant No.12071042)Beijing Natural Science Foundation (Grant No.1202006)。
文摘Under consideration in this study is the discrete coupled modified Korteweg-de Vries(mKdV)equation with 4×4 Lax pair.Firstly,through using continuous limit technique,this discrete equation can be mapped to the coupled KdV and mKdV equations,which may depict the development of shallow water waves,the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma.Secondly,the discrete generalized(r,N-r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem,from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background,higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived,and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique.Finally,the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions.These results may be helpful for understanding some physical phenomena in fields of shallow water wave,optics,and plasma physics.
基金Supported by the Natural Science Foundation of Henan Province of China(0111050200) Natural Science Foundation of Education Committee of Henan Province of China(2003110003)the Science Foundation of Henan University of Science and Technology(2003ZY03)