In this paper,the N-soliton solutions for the massive Thirring model(MTM)in laboratory coordinates are analyzed via the Riemann-Hilbert(RH)approach.The direct scattering including the analyticity,symmetries,and asympt...In this paper,the N-soliton solutions for the massive Thirring model(MTM)in laboratory coordinates are analyzed via the Riemann-Hilbert(RH)approach.The direct scattering including the analyticity,symmetries,and asymptotic behaviors of the Jost solutions as|λ|→∞andλ→0 are given.Considering that the scattering coefficients have simple zeros,the matrix RH problem,reconstruction formulas and corresponding trace formulas are also derived.Further,the N-soliton solutions in the reflectionless case are obtained explicitly in the form of determinants.The propagation characteristics of one-soliton solutions and interaction properties of two-soliton solutions are discussed.In particular,the asymptotic expressions of two-soliton solutions as|t|→∞are obtained,which show that the velocities and amplitudes of the asymptotic solitons do not change before and after interaction except the position shifts.In addition,three types of bounded states for two-soliton solutions are presented with certain parametric conditions.展开更多
An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrodinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the ...An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrodinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the solutions of the nonlinear Schrfdinger equation are reduced to solving a linear algebraic system, from which a unified and explicit formulation of N-soliton solutions with multiparameters for the nonlinear Schrfdinger equation is given.展开更多
In this work,using the Hirota bilinear method,N-soliton solution is obtained for Hirota-Satsuma nonlinear evolution equation:u_t - u_(xxt) - 3u_xu_t + u_x = 0.
Two Darboux transformations of the(2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawaka(CDGKS)equation and(2+1)-dimensional modified Korteweg-de Vries(mKdV) equation are constructed through the Darboux matrix method...Two Darboux transformations of the(2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawaka(CDGKS)equation and(2+1)-dimensional modified Korteweg-de Vries(mKdV) equation are constructed through the Darboux matrix method, respectively. N-soliton solutions of these two equations are presented by applying the Darboux transformations N times. The right-going bright single-soliton solution and interactions of two and three-soliton overtaking collisions of the(2+1)-dimensional CDGKS equation are studied. By choosing different seed solutions, the right-going bright and left-going dark single-soliton solutions, the interactions of two and three-soliton overtaking collisions, and kink soliton solutions of the(2+1)-dimensional mKdV equation are investigated. The results can be used to illustrate the interactions of water waves in shallow water.展开更多
In fluid dynamics, plasma physics and nonlinear optics, Korteweg-de Vries (KdV)-type equations are used to describe certain phenomena. In this paper, a coupled KdV-modified KdV system is investigated. Based on the Bel...In fluid dynamics, plasma physics and nonlinear optics, Korteweg-de Vries (KdV)-type equations are used to describe certain phenomena. In this paper, a coupled KdV-modified KdV system is investigated. Based on the Bell polynomials and symbolic computation, the bilinear form of such system is derived, and its analytic N-soliton solutions are constructed through the Hirota method. Two types of multi-soliton interactions are found, one with the reverse of solitonic shapes, and the other, without. Both the two types can be considered elastic. For a pair of solutions to such system, u and v, with the number of solitons N even, the soliton shapes of u stay unvaried while those of v reverse after the interaction; with N odd, the soliton shapes of both u and v keep unchanged after the interaction.展开更多
In this paper, a nonlocal two-wave interaction system from the Manakov hierarchy is investigated via the Riemann–Hilbert approach. Based on the spectral analysis of the Lax pair, a Riemann–Hilbert problem for the no...In this paper, a nonlocal two-wave interaction system from the Manakov hierarchy is investigated via the Riemann–Hilbert approach. Based on the spectral analysis of the Lax pair, a Riemann–Hilbert problem for the nonlocal two-wave interaction system is constructed. By discussing the solutions of this Riemann–Hilbert problem in both the regular and nonregular cases, we explicitly present the N-soliton solution formula of the nonlocal two-wave interaction system. Moreover,the dynamical behaviour of the single-soliton solution is shown graphically.展开更多
By means of Hirota method,N-soliton solutions of the modified KdV equation under the Bargmannconstraint are obtained through solving the Bargmann constraint and the related Lax pair and conjugate Lax pair ofthe modifi...By means of Hirota method,N-soliton solutions of the modified KdV equation under the Bargmannconstraint are obtained through solving the Bargmann constraint and the related Lax pair and conjugate Lax pair ofthe modified KdV equation.展开更多
The bilinear form of the （2＋1）-dimensional non-isospectral AKNS system is derived. Its N-soliton solutions are obtained by using the Hirota method. As a reduction, a （2＋1）-dimensional non-isospectral Schrodinger...The bilinear form of the （2＋1）-dimensional non-isospectral AKNS system is derived. Its N-soliton solutions are obtained by using the Hirota method. As a reduction, a （2＋1）-dimensional non-isospectral Schrodinger equation and its N-soliton solutions are constructed.展开更多
Using the Hirota's bilinear method,some new N-soliton solution are presented for two multidimensional analogues of the m-KdV equation wt+wxxx-6w 2 wx+3 2( w x -1 wy+w-x -1 wz)x=0 and wt+wxxx?6w 2 wx+3 2( wwy+wx-x-...Using the Hirota's bilinear method,some new N-soliton solution are presented for two multidimensional analogues of the m-KdV equation wt+wxxx-6w 2 wx+3 2( w x -1 wy+w-x -1 wz)x=0 and wt+wxxx?6w 2 wx+3 2( wwy+wx-x-1 wy)=0 in view of a different treatment.展开更多
By the symbolic computation and Hirota method, the bilinear form and an auto-Backlund transformation for a variable-coemcient Korteweg-de Vries equation with nonuniformities are given. Then, the N-solitonic solution i...By the symbolic computation and Hirota method, the bilinear form and an auto-Backlund transformation for a variable-coemcient Korteweg-de Vries equation with nonuniformities are given. Then, the N-solitonic solution in terms of Wronskian form is obtained and verified. In addition, it is shown that the （N - 1）- and N-solitonic solutions satisfy the auto-Backlund transformation through the Wronskian technique.展开更多
In this paper,the N-soliton solutions to the nonlocal reverse space-time Chen-Lee-Liu equation have been derived.Under the nonlocal symmetry reduction to the matrix spectral problem,the nonlocal reverse space-time Che...In this paper,the N-soliton solutions to the nonlocal reverse space-time Chen-Lee-Liu equation have been derived.Under the nonlocal symmetry reduction to the matrix spectral problem,the nonlocal reverse space-time Chen-Lee-Liu equation can be obtained.Based on the spectral problem,the specific matrix Riemann-Hilbert problem is constructed for this nonlocal equation.Through solving this associated Riemann-Hilbert problem,the N-soliton solutions to this nonlocal equation can be obtained in the case of the jump matrix as an identity matrix.展开更多
Based on the Hirota bilinear form, a simple approach without employing the standard perturbation technique, is presented for constructing a novel N-soliton solution for a （3＋1）-dimensional nonlinear evolution equat...Based on the Hirota bilinear form, a simple approach without employing the standard perturbation technique, is presented for constructing a novel N-soliton solution for a （3＋1）-dimensional nonlinear evolution equation. Moreover, the novel N-soliton solution is shown to have resonant behavior with the aid of Mathematica.展开更多
These rational solutions which can be described a kind of algebraic solitary waves which have great potential in applied value in atmosphere and ocean. It has attracted more and more attention recently. In this paper,...These rational solutions which can be described a kind of algebraic solitary waves which have great potential in applied value in atmosphere and ocean. It has attracted more and more attention recently. In this paper, the generalized bilinear method instead of the Hirota bilinear method is used to obtain the rational solutions to the (2 + 1)-dimensional Boiti-Leon-Manna-Pempinelli-like equation (hereinafter referred to as BLMP equation). Meanwhile, the (2 + 1)-dimensional BLMP-like equation is derived on the basis of the generalized bilinear operators D3,x D3,y and D3,t. And the rational solutions to the (2 + 1)-dimensional BLMP-like equation are obtained successively. Finally, with the help of the N-soliton solutions of the (2 + 1)-dimensional BLMP equation, the interactions of the N-soliton solutions can be derived. The results show that the two soliton still maintained the original waveform after happened collision.展开更多
We present new lemmas,theorem and corollaries to construct interactions among higher-order rogue waves,n-periodic waves and n-solitons solutions(n→∞)to the(2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov(ANNV)eq...We present new lemmas,theorem and corollaries to construct interactions among higher-order rogue waves,n-periodic waves and n-solitons solutions(n→∞)to the(2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov(ANNV)equation.Several examples for theories are given by choosing definite interactions of the wave solutions for the model.In particular,we exhibit dynamical interactions between a rogue and a cross bright-dark bell wave,a rogue and a cross-bright bell wave,a rogue and a one-,two-,three-,four-periodic wave.In addition,we also present multi-types interactions between a rogue and a periodic cross-bright bell wave,a rogue and a periodic cross-bright-bark bell wave.Finally,we physically explain such interaction solutions of the model in the 3D and density plots.展开更多
The general bright-dark mixed N-soliton solution of the two-dimensional Maccari system is obtained with the KP hierarchy reduction method. The dynamics of single and two solitons are discussed in detail. Asymptotic an...The general bright-dark mixed N-soliton solution of the two-dimensional Maccari system is obtained with the KP hierarchy reduction method. The dynamics of single and two solitons are discussed in detail. Asymptotic analysis shows that two solitons undergo elastic collision accompanied by a position shift. Furthermore, our analysis on mixed soliton bound states shows that arbitrary higher-order soliton bound states can take place.展开更多
In this paper,we investigate the integrable fractional coupled Gerdjikov-Ivanov equation and derive its explicit form by employing the completeness relation of squared eigenfunctions.Based on the Riemann-Hilbert metho...In this paper,we investigate the integrable fractional coupled Gerdjikov-Ivanov equation and derive its explicit form by employing the completeness relation of squared eigenfunctions.Based on the Riemann-Hilbert method,we construct the fractional N-soliton solutions.We find that as the powerεof the Riesz fractional derivative increases,the amplitudes of the fractional soliton solutions remain invariant,while their widths decrease and the absolute values of the wave velocity,group velocity,and phase velocity increase.Additionally,we examine the long-time asymptotic behavior of the fractional N-soliton solution.The results show that as t→±∞,the solution can be approximated by the sum of N fractional one-soliton solutions,with each soliton's amplitude and velocity remaining constant,whereas both position and phase shifts are observed.展开更多
In this paper,we investigate the(2+1)-dimensional three-component long-wave-short-wave resonance interaction system,which describes complex systems and nonlinear wave phenomena in physics.By employing the Hirota bilin...In this paper,we investigate the(2+1)-dimensional three-component long-wave-short-wave resonance interaction system,which describes complex systems and nonlinear wave phenomena in physics.By employing the Hirota bilinear method,we derive the general nondegenerate N-soliton solution of the system,where each short-wave component contains N arbitrary functions of the independent variable y.The presence of these arbitrary functions in the analytical solution enables the construction of a wide range of nondegenerate soliton types.Finally,we illustrate the structural features of several novel nondegenerate solitons,including M-shaped,multiple double-hump,and sawtooth double-striped solitons,as well as interactions between nondegenerate solitons,such as dromion-like solitons and solitoffs,with the aid of figures.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.12475003 and11705284)by the Natural Science Foundation of Beijing Municipality(Grant Nos.1232022 and 1212007)。
文摘In this paper,the N-soliton solutions for the massive Thirring model(MTM)in laboratory coordinates are analyzed via the Riemann-Hilbert(RH)approach.The direct scattering including the analyticity,symmetries,and asymptotic behaviors of the Jost solutions as|λ|→∞andλ→0 are given.Considering that the scattering coefficients have simple zeros,the matrix RH problem,reconstruction formulas and corresponding trace formulas are also derived.Further,the N-soliton solutions in the reflectionless case are obtained explicitly in the form of determinants.The propagation characteristics of one-soliton solutions and interaction properties of two-soliton solutions are discussed.In particular,the asymptotic expressions of two-soliton solutions as|t|→∞are obtained,which show that the velocities and amplitudes of the asymptotic solitons do not change before and after interaction except the position shifts.In addition,three types of bounded states for two-soliton solutions are presented with certain parametric conditions.
文摘An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrodinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the solutions of the nonlinear Schrfdinger equation are reduced to solving a linear algebraic system, from which a unified and explicit formulation of N-soliton solutions with multiparameters for the nonlinear Schrfdinger equation is given.
基金Foundation item： Supported by the Natural Science Foundation of China（61072147, 11071159） Supported by the Shanghai Leading Academic Discipline Project（J50101） Supported by the Youth Foundation of Zhoukou Normal University（zknuqn200917）
文摘In this work,using the Hirota bilinear method,N-soliton solution is obtained for Hirota-Satsuma nonlinear evolution equation:u_t - u_(xxt) - 3u_xu_t + u_x = 0.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11075055,11275072Innovative Research Team Program of the National Science Foundation of China under Grant No.61021104+3 种基金National High Technology Research and Development Program under Grant No.2011AA010101Shanghai Knowledge Service Platform for Trustworthy Internet of Things under Grant No.ZF1213Talent FundK.C.Wong Magna Fund in Ningbo University
文摘Two Darboux transformations of the(2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawaka(CDGKS)equation and(2+1)-dimensional modified Korteweg-de Vries(mKdV) equation are constructed through the Darboux matrix method, respectively. N-soliton solutions of these two equations are presented by applying the Darboux transformations N times. The right-going bright single-soliton solution and interactions of two and three-soliton overtaking collisions of the(2+1)-dimensional CDGKS equation are studied. By choosing different seed solutions, the right-going bright and left-going dark single-soliton solutions, the interactions of two and three-soliton overtaking collisions, and kink soliton solutions of the(2+1)-dimensional mKdV equation are investigated. The results can be used to illustrate the interactions of water waves in shallow water.
基金Supported by the National Natural Science Foundation of China under Grant No. 60772023the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02+1 种基金the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications)the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 200800130006, Chinese Ministry of Education
文摘In fluid dynamics, plasma physics and nonlinear optics, Korteweg-de Vries (KdV)-type equations are used to describe certain phenomena. In this paper, a coupled KdV-modified KdV system is investigated. Based on the Bell polynomials and symbolic computation, the bilinear form of such system is derived, and its analytic N-soliton solutions are constructed through the Hirota method. Two types of multi-soliton interactions are found, one with the reverse of solitonic shapes, and the other, without. Both the two types can be considered elastic. For a pair of solutions to such system, u and v, with the number of solitons N even, the soliton shapes of u stay unvaried while those of v reverse after the interaction; with N odd, the soliton shapes of both u and v keep unchanged after the interaction.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11331008 and 11522112)
文摘In this paper, a nonlocal two-wave interaction system from the Manakov hierarchy is investigated via the Riemann–Hilbert approach. Based on the spectral analysis of the Lax pair, a Riemann–Hilbert problem for the nonlocal two-wave interaction system is constructed. By discussing the solutions of this Riemann–Hilbert problem in both the regular and nonregular cases, we explicitly present the N-soliton solution formula of the nonlocal two-wave interaction system. Moreover,the dynamical behaviour of the single-soliton solution is shown graphically.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10371070 and 10671121
文摘By means of Hirota method,N-soliton solutions of the modified KdV equation under the Bargmannconstraint are obtained through solving the Bargmann constraint and the related Lax pair and conjugate Lax pair ofthe modified KdV equation.
基金supported by China Postdoctoral Science Foundation and National Natural Science Foundation of China under Grant No.10771207
文摘The bilinear form of the （2＋1）-dimensional non-isospectral AKNS system is derived. Its N-soliton solutions are obtained by using the Hirota method. As a reduction, a （2＋1）-dimensional non-isospectral Schrodinger equation and its N-soliton solutions are constructed.
基金Supported by the National Natural Science Foundation of China（10871132 11074160） Supported by the National Natura Science Foundation of Henan Province（102300410190 092300410202）
文摘Using the Hirota's bilinear method,some new N-soliton solution are presented for two multidimensional analogues of the m-KdV equation wt+wxxx-6w 2 wx+3 2( w x -1 wy+w-x -1 wz)x=0 and wt+wxxx?6w 2 wx+3 2( wwy+wx-x-1 wy)=0 in view of a different treatment.
基金supported by National Natural Science Foundation of China under Grant Nos.60772023 and 60372095the Key Project of the Ministry of Education under Grant No.106033+2 种基金the Open Fund of the State Key Laboratory of Software Development Environment under Grant No.SKLSDE-07-001Beijing University of Aeronautics and Astronautics,the National Basic Research Program of China(973 Program)under Grant No.2005CB321901the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20060006024,the Ministry of Education
文摘By the symbolic computation and Hirota method, the bilinear form and an auto-Backlund transformation for a variable-coemcient Korteweg-de Vries equation with nonuniformities are given. Then, the N-solitonic solution in terms of Wronskian form is obtained and verified. In addition, it is shown that the （N - 1）- and N-solitonic solutions satisfy the auto-Backlund transformation through the Wronskian technique.
基金supported by the National Natural Science Foundation of China under Grant No.11975145。
文摘In this paper,the N-soliton solutions to the nonlocal reverse space-time Chen-Lee-Liu equation have been derived.Under the nonlocal symmetry reduction to the matrix spectral problem,the nonlocal reverse space-time Chen-Lee-Liu equation can be obtained.Based on the spectral problem,the specific matrix Riemann-Hilbert problem is constructed for this nonlocal equation.Through solving this associated Riemann-Hilbert problem,the N-soliton solutions to this nonlocal equation can be obtained in the case of the jump matrix as an identity matrix.
文摘Based on the Hirota bilinear form, a simple approach without employing the standard perturbation technique, is presented for constructing a novel N-soliton solution for a （3＋1）-dimensional nonlinear evolution equation. Moreover, the novel N-soliton solution is shown to have resonant behavior with the aid of Mathematica.
文摘These rational solutions which can be described a kind of algebraic solitary waves which have great potential in applied value in atmosphere and ocean. It has attracted more and more attention recently. In this paper, the generalized bilinear method instead of the Hirota bilinear method is used to obtain the rational solutions to the (2 + 1)-dimensional Boiti-Leon-Manna-Pempinelli-like equation (hereinafter referred to as BLMP equation). Meanwhile, the (2 + 1)-dimensional BLMP-like equation is derived on the basis of the generalized bilinear operators D3,x D3,y and D3,t. And the rational solutions to the (2 + 1)-dimensional BLMP-like equation are obtained successively. Finally, with the help of the N-soliton solutions of the (2 + 1)-dimensional BLMP equation, the interactions of the N-soliton solutions can be derived. The results show that the two soliton still maintained the original waveform after happened collision.
文摘We present new lemmas,theorem and corollaries to construct interactions among higher-order rogue waves,n-periodic waves and n-solitons solutions(n→∞)to the(2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov(ANNV)equation.Several examples for theories are given by choosing definite interactions of the wave solutions for the model.In particular,we exhibit dynamical interactions between a rogue and a cross bright-dark bell wave,a rogue and a cross-bright bell wave,a rogue and a one-,two-,three-,four-periodic wave.In addition,we also present multi-types interactions between a rogue and a periodic cross-bright bell wave,a rogue and a periodic cross-bright-bark bell wave.Finally,we physically explain such interaction solutions of the model in the 3D and density plots.
基金Supported by the Global Change Research Program of China under Grant No 2015CB953904the National Natural Science Foundation of China under Grant Nos 11675054 and 11435005the Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things under Grant No ZF1213
文摘The general bright-dark mixed N-soliton solution of the two-dimensional Maccari system is obtained with the KP hierarchy reduction method. The dynamics of single and two solitons are discussed in detail. Asymptotic analysis shows that two solitons undergo elastic collision accompanied by a position shift. Furthermore, our analysis on mixed soliton bound states shows that arbitrary higher-order soliton bound states can take place.
基金funded by the National Natural Science Foundation of China(Grant Nos.12305003,12101246,12471237)。
文摘In this paper,we investigate the integrable fractional coupled Gerdjikov-Ivanov equation and derive its explicit form by employing the completeness relation of squared eigenfunctions.Based on the Riemann-Hilbert method,we construct the fractional N-soliton solutions.We find that as the powerεof the Riesz fractional derivative increases,the amplitudes of the fractional soliton solutions remain invariant,while their widths decrease and the absolute values of the wave velocity,group velocity,and phase velocity increase.Additionally,we examine the long-time asymptotic behavior of the fractional N-soliton solution.The results show that as t→±∞,the solution can be approximated by the sum of N fractional one-soliton solutions,with each soliton's amplitude and velocity remaining constant,whereas both position and phase shifts are observed.
基金supported by the National Natural Science Foundation of China,Grant No.12375006。
文摘In this paper,we investigate the(2+1)-dimensional three-component long-wave-short-wave resonance interaction system,which describes complex systems and nonlinear wave phenomena in physics.By employing the Hirota bilinear method,we derive the general nondegenerate N-soliton solution of the system,where each short-wave component contains N arbitrary functions of the independent variable y.The presence of these arbitrary functions in the analytical solution enables the construction of a wide range of nondegenerate soliton types.Finally,we illustrate the structural features of several novel nondegenerate solitons,including M-shaped,multiple double-hump,and sawtooth double-striped solitons,as well as interactions between nondegenerate solitons,such as dromion-like solitons and solitoffs,with the aid of figures.
