Under investigation is an integrable generalization of the Fokas–Lenells equation, which can be derived from the negative power flow of a 2 × 2 matrix spectral problem with three potentials. Based on the gauge t...Under investigation is an integrable generalization of the Fokas–Lenells equation, which can be derived from the negative power flow of a 2 × 2 matrix spectral problem with three potentials. Based on the gauge transformation of the matrix spectral problem, one kind of Darboux transformation with multi-parameters for the three-component coupled Fokas–Lenells system is constructed. As a reduction, the N-fold Darboux transformation for the generalized Fokas–Lenells equation is obtained, from which the N-soliton solution in a compact Vandermonde-like determinant form is given. Particularly,the explicit one-and two-soliton solutions are presented and their dynamical behaviors are shown graphically.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
We study the complex Sharma-Tasso-Olver equation using the Riemann-Hilbert approach.The associated Riemann-Hilbert problem for this integrable equation can be naturally constructed by considering the spectral problem ...We study the complex Sharma-Tasso-Olver equation using the Riemann-Hilbert approach.The associated Riemann-Hilbert problem for this integrable equation can be naturally constructed by considering the spectral problem of the Lax pair.Subsequently,in the case that the Riemann-Hilbert problem is irregular,the N-soliton solutions of the equation can be deduced.In addition,the three-dimensional graphic of the soliton solutions and wave propagation image are graphically depicted and further discussed.展开更多
We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with 3×3 matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce ...We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with 3×3 matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter k into a single-valued parameter z. The analyticity of the Jost eigenfunctions and scattering coefficients of the Lax pair for the Sasa-Satsuma equation are analyzed in detail. According to the analyticity of the eigenfunctions and the scattering coefficients, the z-complex plane is divided into four analytic regions of D_(j) : j = 1, 2, 3, 4. Since the second column of Jost eigenfunctions is analytic in D_(j), but in the upper-half or lowerhalf plane, we introduce certain auxiliary eigenfunctions which are necessary for deriving the analytic eigenfunctions in Dj. We find that the eigenfunctions, the scattering coefficients and the auxiliary eigenfunctions all possess three kinds of symmetries;these characterize the distribution of the discrete spectrum. The asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering coefficients are also systematically derived. Then a matrix Riemann-Hilbert problem with four kinds of jump conditions associated with the problem of nonzero asymptotic boundary conditions is established, from this N-soliton solutions are obtained via the corresponding reconstruction formulae. The reflectionless soliton solutions are explicitly given. As an application of the N-soliton formula, we present three kinds of single-soliton solutions according to the distribution of discrete spectrum.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
The nonlinear interactions between the monochromatic wave have been considered by K. Matsunchi, who also proposed one class of the nonlinear Schrdinger equation system with wave operator. We also obtain the same type ...The nonlinear interactions between the monochromatic wave have been considered by K. Matsunchi, who also proposed one class of the nonlinear Schrdinger equation system with wave operator. We also obtain the same type of equations, which are satisfied by transverse velocity of higher frequency electron, as we study soliton in plasma physics. In this paper, under some condition we study the existence and nonexistence for this equations in the cases possessing different signs in nonlinear term.展开更多
The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrodinger(NLS) hierarchies associated with higher-order matrix spectral problems.The Sokho...The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrodinger(NLS) hierarchies associated with higher-order matrix spectral problems.The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations.A new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix,corresponding to the reflectionless inverse scattering transforms,is proposed and applied to construction of soliton solutions to each system in the considered nonlocal reversetime NLS hierarchies.展开更多
In this paper, we studied N-soliton solutions of a new integrable equation studied by Qiao [J. Math. Phys. 48 082701 (2007)]. Firstly, we employed the Darboux matrix method to construct a Darboux transformation for ...In this paper, we studied N-soliton solutions of a new integrable equation studied by Qiao [J. Math. Phys. 48 082701 (2007)]. Firstly, we employed the Darboux matrix method to construct a Darboux transformation for the modified Korteweg-de Vries equation. Then we use the Darboux transformation and a transformation, introduced by Sakovich [J. Math. Phys. 52 023509 (2011)], to derive N-soliton solutions of the new integrable equation from the seed solution. In particular, the multiple soliton solutions are explicitly obtained and shown through some figures.展开更多
We report the analytical nonantonomous soliton solutions (NSSs) for two-component Bose-Einstein condensates with the presence of a time-dependent potential. These solutions show that the time-dependent potential can...We report the analytical nonantonomous soliton solutions (NSSs) for two-component Bose-Einstein condensates with the presence of a time-dependent potential. These solutions show that the time-dependent potential can affect the velocity of NSS. The velocity shows the characteristic of both increasing and oscillation with time. A detailed analysis for the asymptotic behavior of NSSs demonstrates that the collision of two NSSs of each component is elastic.展开更多
In this paper,we investigate a(2+1)-dimensional variable-coefficient modified dispersive waterwave system in fluid mechanics.We prove the Painlevéintegrability for that system via the Painlevéanalysis.We fin...In this paper,we investigate a(2+1)-dimensional variable-coefficient modified dispersive waterwave system in fluid mechanics.We prove the Painlevéintegrability for that system via the Painlevéanalysis.We find some auto-B?cklund transformations for that system via the truncated Painlevéexpansions.Bilinear forms and N-soliton solutions are constructed,where N is a positive integer.We discuss the inelastic interactions,elastic interactions and soliton resonances for the two solitons.We also graphically demonstrate that the velocities of the solitons are affected by the variable coefficient of that system.展开更多
We make a quantitative study on the soliton interactions in the nonlinear Schro¨dinger equation(NLSE) and its variable–coefficient(vc) counterpart. For the regular two-soliton and double-pole solutions of the NL...We make a quantitative study on the soliton interactions in the nonlinear Schro¨dinger equation(NLSE) and its variable–coefficient(vc) counterpart. For the regular two-soliton and double-pole solutions of the NLSE, we employ the asymptotic analysis method to obtain the expressions of asymptotic solitons, and analyze the interaction properties based on the soliton physical quantities(especially the soliton accelerations and interaction forces);whereas for the bounded two-soliton solution, we numerically calculate the soliton center positions and accelerations, and discuss the soliton interaction scenarios in three typical bounded cases. Via some variable transformations, we also obtain the inhomogeneous regular two-soliton and double-pole solutions for the vcNLSE with an integrable condition. Based on the expressions of asymptotic solitons, we quantitatively study the two-soliton interactions with some inhomogeneous dispersion profiles,particularly discuss the influence of the variable dispersion function f(t) on the soliton interaction dynamics.展开更多
The extraction of traveling wave solutions for nonlinear evolution equations is a challenge in various mathematics,physics,and engineering disciplines.This article intends to analyze several traveling wave solutions f...The extraction of traveling wave solutions for nonlinear evolution equations is a challenge in various mathematics,physics,and engineering disciplines.This article intends to analyze several traveling wave solutions for themodified regularized long-wave(MRLW)equation using several approaches,namely,the generalized algebraic method,the Jacobian elliptic functions technique,and the improved Q-expansion strategy.We successfully obtain analytical solutions consisting of rational,trigonometric,and hyperbolic structures.The adaptive moving mesh technique is applied to approximate the numerical solution of the proposed equation.The adaptive moving mesh method evenly distributes the points on the high error areas.This method perfectly and strongly reduces the error.We compare the constructed exact and numerical results to ensure the reliability and validity of the methods used.To better understand the considered equation’s physical meaning,we present some 2D and 3D figures.The exact and numerical approaches are efficient,powerful,and versatile for establishing novel bright,dark,bell-kink-type,and periodic traveling wave solutions for nonlinear PDEs.展开更多
We construct uniform expressions of such dark soliton solutions encompassing both single-valley and double-valley dark solitons for the defocusing coupled Hirota equation with high-order nonlinear effects utilizing th...We construct uniform expressions of such dark soliton solutions encompassing both single-valley and double-valley dark solitons for the defocusing coupled Hirota equation with high-order nonlinear effects utilizing the uniform Darboux transformation,in addition to proposing a sufficient condition for the existence of the above dark soliton solutions.Furthermore,the asymptotic analysis we perform reveals that collisions for single-valley dark solitons typically exhibit elastic behavior;however,collisions for double-valley dark solitons are generally inelastic.In light of this,we further propose a sufficient condition for the elastic collisions of double-valley dark soliton solutions.Our results offer valuable insights into the dynamics of dark soliton solutions in the defocusing coupled Hirota equation and can contribute to the advancement of studies in nonlinear optics.展开更多
In this work,we study the Riemann–Hilbert problem and the soliton solutions for a nonlocal Sasa–Satsuma equation with reverse-time type,which is deduced from a reduction of the coupled Sasa–Satsuma system.Since the...In this work,we study the Riemann–Hilbert problem and the soliton solutions for a nonlocal Sasa–Satsuma equation with reverse-time type,which is deduced from a reduction of the coupled Sasa–Satsuma system.Since the coupled Sasa–Satsuma system can describe the dynamic behaviors of two ultrashort pulse envelopes in birefringent fiber,our equation presented here has great physical applications.The classification of soliton solutions is studied in this nonlocal model by considering an inverse scattering transform to the Riemann–Hilbert problem.Simultaneously,we find that the symmetry relations of discrete data in the special nonlocal model are very complicated.Especially,the eigenvectors in the scattering data are determined by the number and location of eigenvalues.Furthermore,multi-soliton solutions are not a simple nonlinear superposition of multiple single-solitons.They exhibit some novel dynamics of solitons,including meandering and sudden position shifts.Also,they have the bound state of multi-soliton entanglement and its interaction with solitons.展开更多
The nonlinear Schrodinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particu...The nonlinear Schrodinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schrodinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schrodinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.展开更多
In the present paper,multiple exact soliton solutions for the old and the newly introduced(3+1)-dimensional modified Korteweg-de-Vries equation(mKdV)will be sought.The mKdV equations considered feature fractional deri...In the present paper,multiple exact soliton solutions for the old and the newly introduced(3+1)-dimensional modified Korteweg-de-Vries equation(mKdV)will be sought.The mKdV equations considered feature fractional derivative orders in both the space and time variables.A variety of soliton solutions ranging from hyperbolic to periodic function solutions will be constructed using simple ansatze for the equations.Finally,the algebraic equations to be obtained along the way and graphical representations will be carried out by utilizing the Mathematica software.展开更多
Nonlinear Schrödinger-type equations are important models that have emerged from a wide variety of fields,such as fluids,nonlinear optics,the theory of deep-water waves,plasma physics,and so on.In this work,we ob...Nonlinear Schrödinger-type equations are important models that have emerged from a wide variety of fields,such as fluids,nonlinear optics,the theory of deep-water waves,plasma physics,and so on.In this work,we obtain different soliton solutions to coupled nonlinear Schrödinger-type(CNLST)equations by applying three integration tools known as the(G’/G^(2))-expansion function method,the modified direct algebraic method(MDAM),and the generalized Kudryashov method.The soliton and other solutions obtained by these methods can be categorized as single(dark,singular),complex,and combined soliton solutions,as well as hyperbolic,plane wave,and trigonometric solutions with arbitrary parameters.The spectrum of the solitons is enumerated along with their existence criteria.Moreover,2D,3D,and contour profiles of the reported results are also plotted by choosing suitable values of the parameters involved,which makes it easier for researchers to comprehend the physical phenomena of the governing equation.The solutions acquired demonstrate that the proposed techniques are efficient,valuable,and straightforward when constructing new solutions for various types of nonlinear partial differential equation that have important applications in applied sciences and engineering.All the reported solutions are verified by substitution back into the original equation through the software package Mathematica.展开更多
The(3+1)-dimensional Zakharov–Kuznetsov(ZK) and the new extended quantum ZK equations are functional to decipher the dense quantum plasma, ion-acoustic waves, electron thermal energy,ion plasma, quantum acoustic wave...The(3+1)-dimensional Zakharov–Kuznetsov(ZK) and the new extended quantum ZK equations are functional to decipher the dense quantum plasma, ion-acoustic waves, electron thermal energy,ion plasma, quantum acoustic waves, and quantum Langmuir waves. The enhanced modified simple equation(EMSE) method is a substantial approach to determine competent solutions and in this article, we have constructed standard, illustrative, rich structured and further comprehensive soliton solutions via this method. The solutions are ascertained as the integration of exponential, hyperbolic,trigonometric and rational functions and formulate the bright solitons, periodic, compacton, bellshape, parabolic shape, singular periodic, plane shape and some new type of solitons. It is worth noting that the wave profile varies as the physical and subsidiary parameters change. The standard and advanced soliton solutions may be useful to assist in describing the physical phenomena previously mentioned. To open out the inward structure of the tangible incidents, we have portrayed the three-dimensional, contour plot, and two-dimensional graphs for different parametric values. The attained results demonstrate the EMSE technique for extracting soliton solutions to nonlinear evolution equations is efficient, compatible and reliable in nonlinear science and engineering.展开更多
We employ the Riemann-Hilbert(RH)method to study the Hirota equation with arbitrary order zero poles under zero boundary conditions.Through the spectral analysis,the asymptoticity,symmetry,and analysis of the Jost fun...We employ the Riemann-Hilbert(RH)method to study the Hirota equation with arbitrary order zero poles under zero boundary conditions.Through the spectral analysis,the asymptoticity,symmetry,and analysis of the Jost functions are obtained,which play a key role in constructing the RH problem.Then we successfully established the exact solution of the equation without reflection potential by solving the RH problem.Choosing some appropriate parameters of the resulting solutions,we further derive the soliton solutions with different order poles,including four cases of a fourthorder pole,two second-order poles,a third-order pole and a first-order pole,and four first-order points.Finally,the dynamical behavior of these solutions are analyzed via graphic analysis.展开更多
In this paper,a Crank-Nicolson-type finite difference method is proposed for computing the soliton solutions of a complex modifed Korteweg de Vries(MKdV)equation(which is equivalent to the Sasa-Satsuma equation)with t...In this paper,a Crank-Nicolson-type finite difference method is proposed for computing the soliton solutions of a complex modifed Korteweg de Vries(MKdV)equation(which is equivalent to the Sasa-Satsuma equation)with the vanishing boundary condition.It is proved that such a numerical scheme has the second order accuracy both in space and time,and conserves the mass in the discrete level.Meanwhile,the resuling scheme is shown to be unconditionally stable via the von Nuemann analysis.In addition,an iterative method and the Thomas algorithm are used together to enhance the computational efficiency.In numerical experiments,this method is used to simulate the single-soliton propagation and two-soliton collisions in the complex MKdV equation.The numerical accuracy,mass conservation and linear stability are tested to assess the scheme's performance.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12326305,11931017,and 12271490)the Excellent Youth Science Fund Project of Henan Province(Grant No.242300421158)+2 种基金the Natural Science Foundation of Henan Province(Grant No.232300420119)the Excellent Science and Technology Innovation Talent Support Program of ZUT(Grant No.K2023YXRC06)Funding for the Enhancement Program of Advantageous Discipline Strength of ZUT(2022)。
文摘Under investigation is an integrable generalization of the Fokas–Lenells equation, which can be derived from the negative power flow of a 2 × 2 matrix spectral problem with three potentials. Based on the gauge transformation of the matrix spectral problem, one kind of Darboux transformation with multi-parameters for the three-component coupled Fokas–Lenells system is constructed. As a reduction, the N-fold Darboux transformation for the generalized Fokas–Lenells equation is obtained, from which the N-soliton solution in a compact Vandermonde-like determinant form is given. Particularly,the explicit one-and two-soliton solutions are presented and their dynamical behaviors are shown graphically.
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Project supported by the National Natural Science Foundation of China(Grant No.11975145)the Program for Science&Technology Innovation Talents in Universities of Henan Province,China(Grant No.22HASTIT019)+2 种基金the Natural Science Foundation of Henan,China(Grant No.202300410524)the Science and Technique Project of Henan,China(Grant No.212102310397)the Academic Degrees&Graduate Education Reform Project of Henan Province,China(Grant No.2021SJGLX219Y)。
文摘We study the complex Sharma-Tasso-Olver equation using the Riemann-Hilbert approach.The associated Riemann-Hilbert problem for this integrable equation can be naturally constructed by considering the spectral problem of the Lax pair.Subsequently,in the case that the Riemann-Hilbert problem is irregular,the N-soliton solutions of the equation can be deduced.In addition,the three-dimensional graphic of the soliton solutions and wave propagation image are graphically depicted and further discussed.
基金supported by the National Natural Science Foundation of China(12175069 and 12235007)the Science and Technology Commission of Shanghai Municipality (21JC1402500 and 22DZ2229014)。
文摘We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with 3×3 matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter k into a single-valued parameter z. The analyticity of the Jost eigenfunctions and scattering coefficients of the Lax pair for the Sasa-Satsuma equation are analyzed in detail. According to the analyticity of the eigenfunctions and the scattering coefficients, the z-complex plane is divided into four analytic regions of D_(j) : j = 1, 2, 3, 4. Since the second column of Jost eigenfunctions is analytic in D_(j), but in the upper-half or lowerhalf plane, we introduce certain auxiliary eigenfunctions which are necessary for deriving the analytic eigenfunctions in Dj. We find that the eigenfunctions, the scattering coefficients and the auxiliary eigenfunctions all possess three kinds of symmetries;these characterize the distribution of the discrete spectrum. The asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering coefficients are also systematically derived. Then a matrix Riemann-Hilbert problem with four kinds of jump conditions associated with the problem of nonzero asymptotic boundary conditions is established, from this N-soliton solutions are obtained via the corresponding reconstruction formulae. The reflectionless soliton solutions are explicitly given. As an application of the N-soliton formula, we present three kinds of single-soliton solutions according to the distribution of discrete spectrum.
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.
文摘The nonlinear interactions between the monochromatic wave have been considered by K. Matsunchi, who also proposed one class of the nonlinear Schrdinger equation system with wave operator. We also obtain the same type of equations, which are satisfied by transverse velocity of higher frequency electron, as we study soliton in plasma physics. In this paper, under some condition we study the existence and nonexistence for this equations in the cases possessing different signs in nonlinear term.
基金supported in part by NSFC(11975145 and 11972291)the Natural Science Foundation for Colleges and Universities in Jiangsu Province(17 KJB 110020)。
文摘The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrodinger(NLS) hierarchies associated with higher-order matrix spectral problems.The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations.A new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix,corresponding to the reflectionless inverse scattering transforms,is proposed and applied to construction of soliton solutions to each system in the considered nonlocal reversetime NLS hierarchies.
基金Project supported by the National Natural Science Foundation of China (Grant No. 11261037)the High Education Science Research Fund of China (Grant No. 211034)the High Education Science Research Program of Inner Mongolia Autonomous Region, China (Grant No. NJ10045)
文摘In this paper, we studied N-soliton solutions of a new integrable equation studied by Qiao [J. Math. Phys. 48 082701 (2007)]. Firstly, we employed the Darboux matrix method to construct a Darboux transformation for the modified Korteweg-de Vries equation. Then we use the Darboux transformation and a transformation, introduced by Sakovich [J. Math. Phys. 52 023509 (2011)], to derive N-soliton solutions of the new integrable equation from the seed solution. In particular, the multiple soliton solutions are explicitly obtained and shown through some figures.
基金supported by the Key Project of the Chinese Ministry of Education(Grant No.2011015)the Natural Science Foundation of Hebei Province of China(Grant No.A2012202023)
文摘We report the analytical nonantonomous soliton solutions (NSSs) for two-component Bose-Einstein condensates with the presence of a time-dependent potential. These solutions show that the time-dependent potential can affect the velocity of NSS. The velocity shows the characteristic of both increasing and oscillation with time. A detailed analysis for the asymptotic behavior of NSSs demonstrates that the collision of two NSSs of each component is elastic.
基金the National Natural Science Foundation of China under Grant No.11772017the Fundamental Research Funds for the Central Universities
文摘In this paper,we investigate a(2+1)-dimensional variable-coefficient modified dispersive waterwave system in fluid mechanics.We prove the Painlevéintegrability for that system via the Painlevéanalysis.We find some auto-B?cklund transformations for that system via the truncated Painlevéexpansions.Bilinear forms and N-soliton solutions are constructed,where N is a positive integer.We discuss the inelastic interactions,elastic interactions and soliton resonances for the two solitons.We also graphically demonstrate that the velocities of the solitons are affected by the variable coefficient of that system.
基金Project supported by the Natural Science Foundation of Beijing Municipality (Grant No.1212007)the National Natural Science Foundation of China (Grant No.11705284)the Open Project Program of State Key Laboratory of Petroleum Resources and Prospecting,China University of Petroleum (Grant No.PRP/DX-2211)。
文摘We make a quantitative study on the soliton interactions in the nonlinear Schro¨dinger equation(NLSE) and its variable–coefficient(vc) counterpart. For the regular two-soliton and double-pole solutions of the NLSE, we employ the asymptotic analysis method to obtain the expressions of asymptotic solitons, and analyze the interaction properties based on the soliton physical quantities(especially the soliton accelerations and interaction forces);whereas for the bounded two-soliton solution, we numerically calculate the soliton center positions and accelerations, and discuss the soliton interaction scenarios in three typical bounded cases. Via some variable transformations, we also obtain the inhomogeneous regular two-soliton and double-pole solutions for the vcNLSE with an integrable condition. Based on the expressions of asymptotic solitons, we quantitatively study the two-soliton interactions with some inhomogeneous dispersion profiles,particularly discuss the influence of the variable dispersion function f(t) on the soliton interaction dynamics.
文摘The extraction of traveling wave solutions for nonlinear evolution equations is a challenge in various mathematics,physics,and engineering disciplines.This article intends to analyze several traveling wave solutions for themodified regularized long-wave(MRLW)equation using several approaches,namely,the generalized algebraic method,the Jacobian elliptic functions technique,and the improved Q-expansion strategy.We successfully obtain analytical solutions consisting of rational,trigonometric,and hyperbolic structures.The adaptive moving mesh technique is applied to approximate the numerical solution of the proposed equation.The adaptive moving mesh method evenly distributes the points on the high error areas.This method perfectly and strongly reduces the error.We compare the constructed exact and numerical results to ensure the reliability and validity of the methods used.To better understand the considered equation’s physical meaning,we present some 2D and 3D figures.The exact and numerical approaches are efficient,powerful,and versatile for establishing novel bright,dark,bell-kink-type,and periodic traveling wave solutions for nonlinear PDEs.
基金supported by the National Natural Science Foundation of China(No.12122105).
文摘We construct uniform expressions of such dark soliton solutions encompassing both single-valley and double-valley dark solitons for the defocusing coupled Hirota equation with high-order nonlinear effects utilizing the uniform Darboux transformation,in addition to proposing a sufficient condition for the existence of the above dark soliton solutions.Furthermore,the asymptotic analysis we perform reveals that collisions for single-valley dark solitons typically exhibit elastic behavior;however,collisions for double-valley dark solitons are generally inelastic.In light of this,we further propose a sufficient condition for the elastic collisions of double-valley dark soliton solutions.Our results offer valuable insights into the dynamics of dark soliton solutions in the defocusing coupled Hirota equation and can contribute to the advancement of studies in nonlinear optics.
基金supported by the Fundamental Research Funds for the Central Universities(Grant No.2022FRFK060015)supported by the China Postdoctoral Science Foundation(Grant No.2022M710969)the National Natural Science Foundation of China(No.12101159)。
文摘In this work,we study the Riemann–Hilbert problem and the soliton solutions for a nonlocal Sasa–Satsuma equation with reverse-time type,which is deduced from a reduction of the coupled Sasa–Satsuma system.Since the coupled Sasa–Satsuma system can describe the dynamic behaviors of two ultrashort pulse envelopes in birefringent fiber,our equation presented here has great physical applications.The classification of soliton solutions is studied in this nonlocal model by considering an inverse scattering transform to the Riemann–Hilbert problem.Simultaneously,we find that the symmetry relations of discrete data in the special nonlocal model are very complicated.Especially,the eigenvectors in the scattering data are determined by the number and location of eigenvalues.Furthermore,multi-soliton solutions are not a simple nonlinear superposition of multiple single-solitons.They exhibit some novel dynamics of solitons,including meandering and sudden position shifts.Also,they have the bound state of multi-soliton entanglement and its interaction with solitons.
基金supported by the National Natural Science Foundation of China (Grant No. 11675054)the Fund from Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213)the Project of Science and Technology Commission of Shanghai Municipality (Grant No. 18dz2271000)。
文摘The nonlinear Schrodinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schrodinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schrodinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.
文摘In the present paper,multiple exact soliton solutions for the old and the newly introduced(3+1)-dimensional modified Korteweg-de-Vries equation(mKdV)will be sought.The mKdV equations considered feature fractional derivative orders in both the space and time variables.A variety of soliton solutions ranging from hyperbolic to periodic function solutions will be constructed using simple ansatze for the equations.Finally,the algebraic equations to be obtained along the way and graphical representations will be carried out by utilizing the Mathematica software.
基金the financial support provided for this research via the National Natural Science Foundation of China(11771407-52071298)ZhongYuan Science and Technology Innovation Leadership Program(214200510010)the MOST Innovation Methodproject(2019IM050400)。
文摘Nonlinear Schrödinger-type equations are important models that have emerged from a wide variety of fields,such as fluids,nonlinear optics,the theory of deep-water waves,plasma physics,and so on.In this work,we obtain different soliton solutions to coupled nonlinear Schrödinger-type(CNLST)equations by applying three integration tools known as the(G’/G^(2))-expansion function method,the modified direct algebraic method(MDAM),and the generalized Kudryashov method.The soliton and other solutions obtained by these methods can be categorized as single(dark,singular),complex,and combined soliton solutions,as well as hyperbolic,plane wave,and trigonometric solutions with arbitrary parameters.The spectrum of the solitons is enumerated along with their existence criteria.Moreover,2D,3D,and contour profiles of the reported results are also plotted by choosing suitable values of the parameters involved,which makes it easier for researchers to comprehend the physical phenomena of the governing equation.The solutions acquired demonstrate that the proposed techniques are efficient,valuable,and straightforward when constructing new solutions for various types of nonlinear partial differential equation that have important applications in applied sciences and engineering.All the reported solutions are verified by substitution back into the original equation through the software package Mathematica.
文摘The(3+1)-dimensional Zakharov–Kuznetsov(ZK) and the new extended quantum ZK equations are functional to decipher the dense quantum plasma, ion-acoustic waves, electron thermal energy,ion plasma, quantum acoustic waves, and quantum Langmuir waves. The enhanced modified simple equation(EMSE) method is a substantial approach to determine competent solutions and in this article, we have constructed standard, illustrative, rich structured and further comprehensive soliton solutions via this method. The solutions are ascertained as the integration of exponential, hyperbolic,trigonometric and rational functions and formulate the bright solitons, periodic, compacton, bellshape, parabolic shape, singular periodic, plane shape and some new type of solitons. It is worth noting that the wave profile varies as the physical and subsidiary parameters change. The standard and advanced soliton solutions may be useful to assist in describing the physical phenomena previously mentioned. To open out the inward structure of the tangible incidents, we have portrayed the three-dimensional, contour plot, and two-dimensional graphs for different parametric values. The attained results demonstrate the EMSE technique for extracting soliton solutions to nonlinear evolution equations is efficient, compatible and reliable in nonlinear science and engineering.
基金supported by the National Natural Science Foundation of China under Grant No.11975306the Natural Science Foundation of Jiangsu Province under Grant No.BK20181351+2 种基金the Six Talent Peaks Project in Jiangsu Province under Grant No.JY-059the Fundamental Research Fund for the Central Universities under the Grant Nos.2019ZDPY07 and 2019QNA35the Postgraduate Research&Practice Innovation Program of Jiangsu Province under Grant No.KYCX212152.
文摘We employ the Riemann-Hilbert(RH)method to study the Hirota equation with arbitrary order zero poles under zero boundary conditions.Through the spectral analysis,the asymptoticity,symmetry,and analysis of the Jost functions are obtained,which play a key role in constructing the RH problem.Then we successfully established the exact solution of the equation without reflection potential by solving the RH problem.Choosing some appropriate parameters of the resulting solutions,we further derive the soliton solutions with different order poles,including four cases of a fourthorder pole,two second-order poles,a third-order pole and a first-order pole,and four first-order points.Finally,the dynamical behavior of these solutions are analyzed via graphic analysis.
基金This work was parially supported by the Natural Science Foundation of Beijing Munisipality(Grant No.1212007)by the Science Foundations of China University of Petroleum,Beijing(Grant Nos.2462020YXZZ004 and 2462020XKJS02).
文摘In this paper,a Crank-Nicolson-type finite difference method is proposed for computing the soliton solutions of a complex modifed Korteweg de Vries(MKdV)equation(which is equivalent to the Sasa-Satsuma equation)with the vanishing boundary condition.It is proved that such a numerical scheme has the second order accuracy both in space and time,and conserves the mass in the discrete level.Meanwhile,the resuling scheme is shown to be unconditionally stable via the von Nuemann analysis.In addition,an iterative method and the Thomas algorithm are used together to enhance the computational efficiency.In numerical experiments,this method is used to simulate the single-soliton propagation and two-soliton collisions in the complex MKdV equation.The numerical accuracy,mass conservation and linear stability are tested to assess the scheme's performance.