In this paper Lou's direct perturbation method is applied to the perturbed coupled nonlinear Schrodinger equations to obtain their asymptotical solutions, which include not only the zero-order solutions but also the ...In this paper Lou's direct perturbation method is applied to the perturbed coupled nonlinear Schrodinger equations to obtain their asymptotical solutions, which include not only the zero-order solutions but also the first-order modifications. Based on the asymptotical solutions, the effects of perturbations on soliton parameters and the collision between two solitons are then discussed in brief. Furthermore, we directly simulate the perturbed coupled nonlinear SchrSdinger equations by split-step Fourier method to check the validity of the direct perturbation method. It turns out that our analytical results are well supported by the numerical calculations.展开更多
We exhibit some new dark soliton phenomena on the general nonzero background for a defocusing three-component nonlinear Schrodinger equation. As the plane wave background undergoes unitary transformation SU(3), we obt...We exhibit some new dark soliton phenomena on the general nonzero background for a defocusing three-component nonlinear Schrodinger equation. As the plane wave background undergoes unitary transformation SU(3), we obtain the general nonzero background and study its modulational instability by the linear stability analysis. On the basis of this background, we study the dynamics of one-dark soliton and two-dark-soliton phenomena, which are different from the dark solitons studied before. Furthermore, we use the numerical method for checking the stability of the one-dark-soliton solution. These results further enrich the content in nonlinear Schrodinger systems, and require more in-depth studies in the future.展开更多
For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, the infinite dimensional Lie algebra of the classical symmetry group is found and the one-dimensional optimal system of an 8-dimensional subalgebra o...For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, the infinite dimensional Lie algebra of the classical symmetry group is found and the one-dimensional optimal system of an 8-dimensional subalgebra of the infinite Lie algebra is constructed. The reduced equations of the equations with respect to the optimal system are derived. Furthermore, the one-dimensional optimal systems of the Lie algebra admitted by the reduced equations are also constructed. Consequently, the classification of the twice optimal symmetry reductions of the equations with respect to the optimal systems is presented. The reductions show that the (1 + 2)-dimensional nonlinear Schrodinger equations can be reduced to a group of ordinary differential equations which is useful for solving the related problems of the equations.展开更多
This paper aims to construct and analyze the conforming and nonconforming virtual element methods for a class of fourth order nonlinear Schrodinger equations with trapped term.We mainly consider three types of virtual...This paper aims to construct and analyze the conforming and nonconforming virtual element methods for a class of fourth order nonlinear Schrodinger equations with trapped term.We mainly consider three types of virtual elements,including H^(2) conforming virtual element,C^(0) nonconforming virtual element and Morley-type nonconforming virtual element.The fully discrete schemes are constructed by virtue of virtual element methods in space and modified Crank-Nicolson method in time.We prove the mass and energy conservation,the boundedness and the unique solvability of the fully discrete schemes.After introducing a new type of the Ritz projection,the optimal and unconditional error estimates for the fully discrete schemes are presented and proved.Finally,two numerical examples are investigated to confirm our theoretical analysis.展开更多
In this article, a compact finite difference scheme for the coupled nonlinear Schrodinger equations is studied. The scheme is proved to conserve the original conservative properties. Unconditional stability and conver...In this article, a compact finite difference scheme for the coupled nonlinear Schrodinger equations is studied. The scheme is proved to conserve the original conservative properties. Unconditional stability and convergence in maximum norm with order O(τ2 + h4) are also proved by the discrete energy method. Finally, numerical results are provided to verify the theoretical analysis.展开更多
In this paper,we propose a compact scheme to numerically study the coupled stochastic nonlinear Schrodinger equations.We prove that the compact scheme preserves the discrete stochastic multi-symplectic conservation la...In this paper,we propose a compact scheme to numerically study the coupled stochastic nonlinear Schrodinger equations.We prove that the compact scheme preserves the discrete stochastic multi-symplectic conservation law,discrete charge conservation law and discrete energy evolution law almost surely.Numerical experiments confirm well the theoretical analysis results.Furthermore,we present a detailed numerical investigation of the optical phenomena based on the compact scheme.By numerical experiments for various amplitudes of noise,we find that the noise accelerates the oscillation of the soliton and leads to the decay of the solution amplitudes with respect to time.In particular,if the noise is relatively strong,the soliton will be totally destroyed.Meanwhile,we observe that the phase shift is sensibly modified by the noise.Moreover,the numerical results present inelastic interaction which is different from the deterministic case.展开更多
Twin-core optical fibers are applied in such fields as the optical sensing and optical communication,and propagation of the pulses,Gauss beams and laser beams in the non-Kerr media is reported.Studied in this paper ar...Twin-core optical fibers are applied in such fields as the optical sensing and optical communication,and propagation of the pulses,Gauss beams and laser beams in the non-Kerr media is reported.Studied in this paper are the coupled cubic-quintic nonlinear Schrodinger equations with variable coefficients,which describe the effects of quintic nonlinearity for the ultrashort optical pulse propagation in a twin-core optical fiber or non-Kerr medium.Based on the integrable conditions,bilinear forms are derived,and dark-dark soliton solutions can be constructed in terms of the Gramian via the Kadomtsev-Petviashvili hierarchy reduction.Propagation and interaction of the dark-dark solitons are presented and discussed through the graphic analysis.With different values of the delayed nonlinear response effect b(z),where z represents direction of the propagation,the linear-and parabolic-shaped one dark-dark soltions can be derived.Interactions between the parabolic-and periodic-shaped two dark-dark solitons are presented with b(z)as the linear and periodic functions,respectively.Directions of velocities of the two dark-dark solitons vary with z and the amplitudes of the solitons remain unchanged can be observed.Interactions between the two dark-dark solitons of different types are displayed,and we observe that the velocity of one soliton is zero and direction of the velocity of the other soliton vary with z.We find that those interactions are elastic.展开更多
Based on the inverse scattering transform for the coupled nonlinear Schrodinger (NLS) equations with vanishing boundary condition (VBC), the multisoliton solution has been derived by some determinant techniques of...Based on the inverse scattering transform for the coupled nonlinear Schrodinger (NLS) equations with vanishing boundary condition (VBC), the multisoliton solution has been derived by some determinant techniques of some special matrices and determinants, especially the Cauchy-Binet formula. The oneand two-soliton solutions have been given as the illustration of the general formula of the multisoliton solution. Moreover, new nonsymmetric solutions corresponding to different number of zeros of the scattering data on the upper and lower half plane are discussed.展开更多
The purpose of this paper is to study the long time asymptotic behavior for a nonlinear Schrdinger equations with magnetic effect. Under certain conditions, we prove the existence and nonexistence of the non-trivial f...The purpose of this paper is to study the long time asymptotic behavior for a nonlinear Schrdinger equations with magnetic effect. Under certain conditions, we prove the existence and nonexistence of the non-trivial free asymptotic solutions. In addition, the decay estimates of the solutions are also obtained.展开更多
In this paper, the periodic initial value problem for the following class of nonlinear Schrodinger equation of high order i partial derivative u/partial derivative t + (-1)(m) partial derivative(m)/partial derivative ...In this paper, the periodic initial value problem for the following class of nonlinear Schrodinger equation of high order i partial derivative u/partial derivative t + (-1)(m) partial derivative(m)/partial derivative x(m) (a(x)partial derivative(m)u/partial derivative x(m)) + beta(x)q(\u\(2))u + f(x, t)u = g(x, t) is considered. A leap-frog finite difference scheme is given, and convergence and stability is proved. Finally, it is shown by a numerical example that numerical result is coincident with theoretical result.展开更多
In this paper, we investigate standing waves in discrete nonlinear Schr?dinger equations with nonperiodic bounded potentials. By using the critical point theory and the spectral theory of self-adjoint operators, we pr...In this paper, we investigate standing waves in discrete nonlinear Schr?dinger equations with nonperiodic bounded potentials. By using the critical point theory and the spectral theory of self-adjoint operators, we prove the existence and infinitely many sign-changing solutions of the equation. The results on the exponential decay of standing waves are also provided.展开更多
We prove the asymptotic stability of solitary waves to 1D nonlinear Schrodinger equations in the subcritical case with symmetry and spectrum assumptions.One of the main ideas is to use the vector fields method develop...We prove the asymptotic stability of solitary waves to 1D nonlinear Schrodinger equations in the subcritical case with symmetry and spectrum assumptions.One of the main ideas is to use the vector fields method developed by S.Cuccagna,V.Georgiev,and N.Visciglia[Comm.Pure Appl.Math.,2013,6:957-980]to overcome the weak decay with respect to t of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term.Meanwhile,we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schrodinger equations obtained by G.Staffilani[Duke Math.J.,1997,86(1):109-142]to control the high moments of the solutions emerging from the vector fields method.展开更多
The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrodinger equation set in an unbounded domain.We first derive absorbing boundary conditions ...The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrodinger equation set in an unbounded domain.We first derive absorbing boundary conditions for the fractional system by using the unified approach introduced in[47,48]and a linearization procedure.Then,the initial boundary-value problem for the fractional system with ABCs is discretized,a stability analysis is developed and the error estimate O(h^(2)+τ)is stated.To accel-erate the L1-scheme in time,a sum-of-exponentials approximation is introduced to speed-up the evaluation of the Caputo fractional derivative.The resulting algorithm is highly efficient for long time simulations.Finally,we end the paper by reporting some numerical simulations to validate the properties(accuracy and efficiency)of the derived scheme.展开更多
The main purpose of this paper is to consider the initial-boundary value problem for the 1D mixed nonlinear Schrodinger equation ut=iαu_(xx)+βu^(2)u_(x)+γ|u|^(2)u_(x)+i|u|^(2)u on the half-line with inhomogeneous b...The main purpose of this paper is to consider the initial-boundary value problem for the 1D mixed nonlinear Schrodinger equation ut=iαu_(xx)+βu^(2)u_(x)+γ|u|^(2)u_(x)+i|u|^(2)u on the half-line with inhomogeneous boundary condition.We combine Laplace transform method with restricted norm method to prove the local well-posedness and continuous dependence on initial and boundary data in low regularity Sobolev spaces.Moreover,we show that the nonlinear part of the solution on the half-line is smoother than the initial data.展开更多
By using the solutions of an auxiliary Lame equation, a direct algebraic method is proposed to construct the exact solutions of N-coupled nonlinear Schrodinger equations. The abundant higher-order exact periodic solut...By using the solutions of an auxiliary Lame equation, a direct algebraic method is proposed to construct the exact solutions of N-coupled nonlinear Schrodinger equations. The abundant higher-order exact periodic solutions of a family of N-coupled nonlinear Schrodinger equations are explicitly obtained with the aid of symbolic computation and they include corresponding envelope solitary and shock wave solutions.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
In this work, we will derive numerical schemes for solving 3-coupled nonlinear Schrödinger equations using finite difference method and time splitting method combined with finite difference method. The result...In this work, we will derive numerical schemes for solving 3-coupled nonlinear Schrödinger equations using finite difference method and time splitting method combined with finite difference method. The resulting schemes are highly accurate, unconditionally stable. We use the exact single soliton solution and the conserved quantities to check the accuracy and the efficiency of the proposed schemes. Also, we use these methods to study the interaction dynamics of two solitons. It is found that both elastic and inelastic collision can take place under suitable parametric conditions. We have noticed that the inelastic collision of single solitons occurs in two different manners: enhancement or suppression of the amplitude.展开更多
In this paper,we study the Cauchy problem for the nonlinear Schrodinger equations with Coulomb potential i■_(t)u+△u+k/|x|u=λ/|u|^(p-l)u with 1<p≤5 on R^(3).Our results reveal the influence of the long range pot...In this paper,we study the Cauchy problem for the nonlinear Schrodinger equations with Coulomb potential i■_(t)u+△u+k/|x|u=λ/|u|^(p-l)u with 1<p≤5 on R^(3).Our results reveal the influence of the long range potential K|x|^(-1)on the existence and scattering theories for nonlinear Schrodinger equations.In particular,we prove the global existence when the Coulomb potential is attractive,i.e.,when K>0,and the scattering theory when the Coulomb potential is repulsive,i.e.,when K≤O.The argument is based on the newlyestablished interaction Morawetz-type inequalities and the equivalence of Sobolev norms for the Laplacian operator with the Coulomb potential.展开更多
Dynamics of three nonisospectral nonlinear Schrdinger equations(NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the sta...Dynamics of three nonisospectral nonlinear Schrdinger equations(NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the standard nonlinear Schrdinger equation(NLSE) and its first two nonisospectral counterparts, for which we derive solutions and infinitely many conserved quantities. Then, exact solutions of the three NNLSEs are derived in double Wronskian terms. Moreover,we analyze the dynamics of the solitons in the presence of the nonisospectral effects by demonstrating how the shapes,velocities, and wave energies change in time. In particular, we obtain a rogue wave type of soliton solutions to the third NNLSE.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant No 10575087) and the Natural Science Foundation of Zheiiang Province of China (Grant No 102053). 0ne of the authors (Lin) would like to thank Prof. Sen-yue Lou for many useful discussions.
文摘In this paper Lou's direct perturbation method is applied to the perturbed coupled nonlinear Schrodinger equations to obtain their asymptotical solutions, which include not only the zero-order solutions but also the first-order modifications. Based on the asymptotical solutions, the effects of perturbations on soliton parameters and the collision between two solitons are then discussed in brief. Furthermore, we directly simulate the perturbed coupled nonlinear SchrSdinger equations by split-step Fourier method to check the validity of the direct perturbation method. It turns out that our analytical results are well supported by the numerical calculations.
基金Project supported by the National Natural Science Foundation of China(Grant No.11771151)the Guangdong Natural Science Foundation of China(Grant No.2017A030313008)+1 种基金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We exhibit some new dark soliton phenomena on the general nonzero background for a defocusing three-component nonlinear Schrodinger equation. As the plane wave background undergoes unitary transformation SU(3), we obtain the general nonzero background and study its modulational instability by the linear stability analysis. On the basis of this background, we study the dynamics of one-dark soliton and two-dark-soliton phenomena, which are different from the dark solitons studied before. Furthermore, we use the numerical method for checking the stability of the one-dark-soliton solution. These results further enrich the content in nonlinear Schrodinger systems, and require more in-depth studies in the future.
基金supported by the Natural science foundation of China(NSF),under grand number 11071159.
文摘For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, the infinite dimensional Lie algebra of the classical symmetry group is found and the one-dimensional optimal system of an 8-dimensional subalgebra of the infinite Lie algebra is constructed. The reduced equations of the equations with respect to the optimal system are derived. Furthermore, the one-dimensional optimal systems of the Lie algebra admitted by the reduced equations are also constructed. Consequently, the classification of the twice optimal symmetry reductions of the equations with respect to the optimal systems is presented. The reductions show that the (1 + 2)-dimensional nonlinear Schrodinger equations can be reduced to a group of ordinary differential equations which is useful for solving the related problems of the equations.
基金supported by the NSF of China(Grant Nos.11801527,11701522,11771163,11671160,1191101330)by the China Postdoctoral Science Foundation(Grant No.2018M632791)by the Key Scientific Research Projects of Higher Eduction of Henan(Grant No.19A110034).
文摘This paper aims to construct and analyze the conforming and nonconforming virtual element methods for a class of fourth order nonlinear Schrodinger equations with trapped term.We mainly consider three types of virtual elements,including H^(2) conforming virtual element,C^(0) nonconforming virtual element and Morley-type nonconforming virtual element.The fully discrete schemes are constructed by virtue of virtual element methods in space and modified Crank-Nicolson method in time.We prove the mass and energy conservation,the boundedness and the unique solvability of the fully discrete schemes.After introducing a new type of the Ritz projection,the optimal and unconditional error estimates for the fully discrete schemes are presented and proved.Finally,two numerical examples are investigated to confirm our theoretical analysis.
基金Supported by the National Natural Science Foundation of China(No.11201041)
文摘In this article, a compact finite difference scheme for the coupled nonlinear Schrodinger equations is studied. The scheme is proved to conserve the original conservative properties. Unconditional stability and convergence in maximum norm with order O(τ2 + h4) are also proved by the discrete energy method. Finally, numerical results are provided to verify the theoretical analysis.
基金This work was supported by the National Natural Science Foundation of China(Nos.91530118,91130003,11021101,11290142,11471310,11601032,11301234,11271171)the Provincial Natural Science Foundation of Jiangxi(Nos.20142BCB23009,20161ACB20006,20151BAB201012).
文摘In this paper,we propose a compact scheme to numerically study the coupled stochastic nonlinear Schrodinger equations.We prove that the compact scheme preserves the discrete stochastic multi-symplectic conservation law,discrete charge conservation law and discrete energy evolution law almost surely.Numerical experiments confirm well the theoretical analysis results.Furthermore,we present a detailed numerical investigation of the optical phenomena based on the compact scheme.By numerical experiments for various amplitudes of noise,we find that the noise accelerates the oscillation of the soliton and leads to the decay of the solution amplitudes with respect to time.In particular,if the noise is relatively strong,the soliton will be totally destroyed.Meanwhile,we observe that the phase shift is sensibly modified by the noise.Moreover,the numerical results present inelastic interaction which is different from the deterministic case.
基金the National Natural Science Foundation of China under Grant Nos.11772017,11805020,11272023 and 11471050the Fund of State Key Laboratory of Information Photonics and Optical Communications(Beijing University of Posts and Telecommunications),China(IPOC:2017ZZ05)the Fundamental Research Funds for the Central Universities of China under Grant No.2011BUPTYB02
文摘Twin-core optical fibers are applied in such fields as the optical sensing and optical communication,and propagation of the pulses,Gauss beams and laser beams in the non-Kerr media is reported.Studied in this paper are the coupled cubic-quintic nonlinear Schrodinger equations with variable coefficients,which describe the effects of quintic nonlinearity for the ultrashort optical pulse propagation in a twin-core optical fiber or non-Kerr medium.Based on the integrable conditions,bilinear forms are derived,and dark-dark soliton solutions can be constructed in terms of the Gramian via the Kadomtsev-Petviashvili hierarchy reduction.Propagation and interaction of the dark-dark solitons are presented and discussed through the graphic analysis.With different values of the delayed nonlinear response effect b(z),where z represents direction of the propagation,the linear-and parabolic-shaped one dark-dark soltions can be derived.Interactions between the parabolic-and periodic-shaped two dark-dark solitons are presented with b(z)as the linear and periodic functions,respectively.Directions of velocities of the two dark-dark solitons vary with z and the amplitudes of the solitons remain unchanged can be observed.Interactions between the two dark-dark solitons of different types are displayed,and we observe that the velocity of one soliton is zero and direction of the velocity of the other soliton vary with z.We find that those interactions are elastic.
基金Supported by the National Natural Science Foundation of China(10705022),Joint Funds of the National Natural Science Foundation of China(U1232109)
文摘Based on the inverse scattering transform for the coupled nonlinear Schrodinger (NLS) equations with vanishing boundary condition (VBC), the multisoliton solution has been derived by some determinant techniques of some special matrices and determinants, especially the Cauchy-Binet formula. The oneand two-soliton solutions have been given as the illustration of the general formula of the multisoliton solution. Moreover, new nonsymmetric solutions corresponding to different number of zeros of the scattering data on the upper and lower half plane are discussed.
基金Supported by the National Natural Science Foundation of China.
文摘The purpose of this paper is to study the long time asymptotic behavior for a nonlinear Schrdinger equations with magnetic effect. Under certain conditions, we prove the existence and nonexistence of the non-trivial free asymptotic solutions. In addition, the decay estimates of the solutions are also obtained.
文摘In this paper, the periodic initial value problem for the following class of nonlinear Schrodinger equation of high order i partial derivative u/partial derivative t + (-1)(m) partial derivative(m)/partial derivative x(m) (a(x)partial derivative(m)u/partial derivative x(m)) + beta(x)q(\u\(2))u + f(x, t)u = g(x, t) is considered. A leap-frog finite difference scheme is given, and convergence and stability is proved. Finally, it is shown by a numerical example that numerical result is coincident with theoretical result.
基金Supported by Science and technology plan foundation of Guangzhou(No.201607010218)by Public Research&Capacity-Building Project of Guangdong(No.2015A070704059).
文摘In this paper, we investigate standing waves in discrete nonlinear Schr?dinger equations with nonperiodic bounded potentials. By using the critical point theory and the spectral theory of self-adjoint operators, we prove the existence and infinitely many sign-changing solutions of the equation. The results on the exponential decay of standing waves are also provided.
文摘We prove the asymptotic stability of solitary waves to 1D nonlinear Schrodinger equations in the subcritical case with symmetry and spectrum assumptions.One of the main ideas is to use the vector fields method developed by S.Cuccagna,V.Georgiev,and N.Visciglia[Comm.Pure Appl.Math.,2013,6:957-980]to overcome the weak decay with respect to t of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term.Meanwhile,we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schrodinger equations obtained by G.Staffilani[Duke Math.J.,1997,86(1):109-142]to control the high moments of the solutions emerging from the vector fields method.
基金supported by the NSFC under grants 11771035,91430216,U1530401supported by the NSFC under grants Nos.11571128,11771162support of the French ANR grant BOND(ANR-13-BS01-0009-01)and the LIASFMA(funding from the University of Lorraine).
文摘The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrodinger equation set in an unbounded domain.We first derive absorbing boundary conditions for the fractional system by using the unified approach introduced in[47,48]and a linearization procedure.Then,the initial boundary-value problem for the fractional system with ABCs is discretized,a stability analysis is developed and the error estimate O(h^(2)+τ)is stated.To accel-erate the L1-scheme in time,a sum-of-exponentials approximation is introduced to speed-up the evaluation of the Caputo fractional derivative.The resulting algorithm is highly efficient for long time simulations.Finally,we end the paper by reporting some numerical simulations to validate the properties(accuracy and efficiency)of the derived scheme.
文摘The main purpose of this paper is to consider the initial-boundary value problem for the 1D mixed nonlinear Schrodinger equation ut=iαu_(xx)+βu^(2)u_(x)+γ|u|^(2)u_(x)+i|u|^(2)u on the half-line with inhomogeneous boundary condition.We combine Laplace transform method with restricted norm method to prove the local well-posedness and continuous dependence on initial and boundary data in low regularity Sobolev spaces.Moreover,we show that the nonlinear part of the solution on the half-line is smoother than the initial data.
基金Project supported by the National Natural Science Foundation of China (Grant No 10461006).
文摘By using the solutions of an auxiliary Lame equation, a direct algebraic method is proposed to construct the exact solutions of N-coupled nonlinear Schrodinger equations. The abundant higher-order exact periodic solutions of a family of N-coupled nonlinear Schrodinger equations are explicitly obtained with the aid of symbolic computation and they include corresponding envelope solitary and shock wave solutions.
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In this work, we will derive numerical schemes for solving 3-coupled nonlinear Schrödinger equations using finite difference method and time splitting method combined with finite difference method. The resulting schemes are highly accurate, unconditionally stable. We use the exact single soliton solution and the conserved quantities to check the accuracy and the efficiency of the proposed schemes. Also, we use these methods to study the interaction dynamics of two solitons. It is found that both elastic and inelastic collision can take place under suitable parametric conditions. We have noticed that the inelastic collision of single solitons occurs in two different manners: enhancement or suppression of the amplitude.
基金The authors were supported by NSFC(12126409,12026407,11831004)the J.Zheng was also supported by Beijing Natural Science Foundation(1222019)。
文摘In this paper,we study the Cauchy problem for the nonlinear Schrodinger equations with Coulomb potential i■_(t)u+△u+k/|x|u=λ/|u|^(p-l)u with 1<p≤5 on R^(3).Our results reveal the influence of the long range potential K|x|^(-1)on the existence and scattering theories for nonlinear Schrodinger equations.In particular,we prove the global existence when the Coulomb potential is attractive,i.e.,when K>0,and the scattering theory when the Coulomb potential is repulsive,i.e.,when K≤O.The argument is based on the newlyestablished interaction Morawetz-type inequalities and the equivalence of Sobolev norms for the Laplacian operator with the Coulomb potential.
基金the National Natural Science Foundation of China(Grant Nos.11601312,11631007,and 11875040)
文摘Dynamics of three nonisospectral nonlinear Schrdinger equations(NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the standard nonlinear Schrdinger equation(NLSE) and its first two nonisospectral counterparts, for which we derive solutions and infinitely many conserved quantities. Then, exact solutions of the three NNLSEs are derived in double Wronskian terms. Moreover,we analyze the dynamics of the solitons in the presence of the nonisospectral effects by demonstrating how the shapes,velocities, and wave energies change in time. In particular, we obtain a rogue wave type of soliton solutions to the third NNLSE.