Conservation laws of the generalized nonlocal nonlinear Schrodinger equation

The derivations of several conservation laws of the generalized nonlocal nonlinear Schrodinger equation are presented. These invaxiants are the number of particles, the momentum, the angular momentum and the Hamiltonian in the quantum mechanical analogy. The Lagrangian is also presented.

Asymptotical solutions of coupled nonlinear Schrodinger equations with perturbations

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.

A NONLINEAR SCHRODINGER EQUATION WITH COULOMB POTENTIAL

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.

Rogue Waves in the(2+1)-Dimensional Nonlinear Schrodinger Equation with a Parity-Time-Symmetric Potential

The (2+1)-dimension nonlocal nonlinear Schrödinger (NLS) equation with the self-induced parity-time symmetric potential is introduced, which provides spatially two-dimensional analogues of the nonlocal NLS equation introduced by Ablowitz et al. [Phys. Rev. Lett. 110 (2013) 064105]. General periodic solutions are derived by the bilinear method. These periodic solutions behave as growing and decaying periodic line waves arising from the constant background and decaying back to the constant background again. By taking long wave limits of the obtained periodic solutions, rogue waves are obtained. It is also shown that these line rogue waves arise from the constant background with a line profile and disappear into the constant background again in the plane.

Dark and multi-dark solitons in the three-component nonlinear Schrodinger equations on the general nonzero background

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.

Four-soliton solution and soliton interactions of the generalized coupled nonlinear Schrodinger equation

Based on the generalized coupled nonlinear Schr¨odinger equation,we obtain the analytic four-bright–bright soliton solution by using the Hirota bilinear method.The interactions among four solitons are also studied in detail.The results show that the interaction among four solitons mainly depends on the values of solution parameters;k1 and k2 mainly affect the two inboard solitons while k3 and k4 mainly affect the two outboard solitons;the pulse velocity and width mainly depend on the imaginary part of ki(i=1,2,3,4),while the pulse amplitude mainly depends on the real part of ki(i=1,2,3,4).

Lie Symmetries,1-Dimensional Optimal System and Optimal Reductions of(1+2)-Dimensional Nonlinear Schrodinger Equation

For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, classical symmetry algebra is found and 1-dimensional optimal system, up to conjugacy, is constructed. Its symmetry reductions are performed for each class, and someexamples of exact invainvariant solutions are given.

Generalized Darboux Transformation and Rational Solutions for the Nonlocal Nonlinear Schrodinger Equation with the Self-Induced Parity-Time Symmetric Potential

In this paper, I construct a generalized Darboux transformation for the nonlocal nonlinear Schrodinger equation with the self-induced parity-time symmetric potential. The N-order rational solution is derived by the iterative rule and it can be expressed by the determinant form. In particular, I calculate first-order and second-order rational solutions and obtain their figures according to different parameters.

A nonlinear Schrodinger equation for gravity waves slowly modulated by linear shear flow

Assume that a fluid is inviscid, incompressible, and irrotational. A nonlinear Schr?dinger equation(NLSE) describing the evolution of gravity waves in finite water depth is derived using the multiple-scale analysis method. The gravity waves are influenced by a linear shear flow, which is composed of a uniform flow and a shear flow with constant vorticity. The modulational instability(MI) of the NLSE is analyzed, and the region of the MI for gravity waves(the necessary condition for existence of freak waves) is identified. In this work, the uniform background flows along or against wave propagation are referred to as down-flow and up-flow, respectively. Uniform up-flow enhances the MI, whereas uniform down-flow reduces it. Positive vorticity enhances the MI, while negative vorticity reduces it. Hence, the influence of positive(negative)vorticity on MI can be balanced out by that of uniform down(up) flow. Furthermore, the Peregrine breather solution of the NLSE is applied to freak waves. Uniform up-flow increases the steepness of the free surface elevation, while uniform down-flow decreases it. Positive vorticity increases the steepness of the free surface elevation, whereas negative vorticity decreases it.

A Family of Exact Solutions for the Nonlinear Schrodinger Equation

In this paper, the nonlinear Schrodinger (NLS) equation was analytically solved. Firstly, the stationary solutions of NLS equation were explicitly given by the elliptic functions. Then a family of exact solutions of NLS equation were obtained from these stationary solutions by a method for finding new exact solutions from the stationary solutions of integrable evolution equations.

The Interaction and Degeneracy of Mixed Solutions for Derivative Nonlinear Schrodinger Equation

The mixed solutions of the derivative nonlinear Schr&#246;dinger equation from the trivial seed (zero solution) are derived by using the determinant representation. By adjusting the interaction and degeneracy of mixed solutions, it is possible to obtain different types of solutions: phase solutions, breather solutions, phase-breather solutions and rogue waves.

Lie Symmetries,One-Dimensional Optimal System and Optimal Reduction of(2+1)-Coupled nonlinear Schrodinger Equations

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.

Soliton excitations and interaction in alpha helical protein with interspine coupling in modified nonlinear Schrodinger equation

The three-coupling modified nonlinear Schr?dinger(MNLS) equation with variable-coefficients is used to describe the dynamics of soliton in alpha helical protein. This MNLS equation with variable-coefficients is firstly transformed to the MNLS equation with constant-coefficients by similarity transformation. And then the one-soliton and two-soliton solutions of the variable-coefficient-MNLS equation are obtained by solving the constant-coefficient-MNLS equation with Hirota method. The effects of different parameter conditions on the soliton solutions are discussed in detail. The interaction between two solitons is also discussed. Our results are helpful to understand the soliton dynamics in alpha helical protein.

A local refinement purely meshless scheme for time fractional nonlinear Schrodinger equation in irregular geometry region

A local refinement hybrid scheme(LRCSPH-FDM)is proposed to solve the two-dimensional(2D)time fractional nonlinear Schrodinger equation(TF-NLSE)in regularly or irregularly shaped domains,and extends the scheme to predict the quantum mechanical properties governed by the time fractional Gross-Pitaevskii equation(TF-GPE)with the rotating Bose-Einstein condensate.It is the first application of the purely meshless method to the TF-NLSE to the author’s knowledge.The proposed LRCSPH-FDM(which is based on a local refinement corrected SPH method combined with FDM)is derived by using the finite difference scheme(FDM)to discretize the Caputo TF term,followed by using a corrected smoothed particle hydrodynamics(CSPH)scheme continuously without using the kernel derivative to approximate the spatial derivatives.Meanwhile,the local refinement technique is adopted to reduce the numerical error.In numerical simulations,the complex irregular geometry is considered to show the flexibility of the purely meshless particle method and its advantages over the grid-based method.The numerical convergence rate and merits of the proposed LRCSPH-FDM are illustrated by solving several 1D/2D(where 1D stands for one-dimensional)analytical TF-NLSEs in a rectangular region(with regular or irregular particle distribution)or in a region with irregular geometry.The proposed method is then used to predict the complex nonlinear dynamic characters of 2D TF-NLSE/TF-GPE in a complex irregular domain,and the results from the posed method are compared with those from the FDM.All the numerical results show that the present method has a good accuracy and flexible application capacity for the TF-NLSE/GPE in regions of a complex shape.

CONSERVATIVE CONFORMING AND NONCONFORMING VEMS FOR FOURTH ORDER NONLINEAR SCHRODINGER EQUATIONS WITH TRAPPED TERM

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.

Sharp Condition for Global Existence of Supercritical Nonlinear Schrodinger Equation with a Partial Confinement

We study the L^(2)-supercritical nonlinear Schrodinger equation(NLS) with a partial confinement,which is the limit case of the cigar-shaped model in Bose-Einstein condensate(BEC). By constructing a cross constrained variational problem and establishing the invariant manifolds of the evolution fow, we show a sharp condition for global existence.

On blow-up criterion for the nonlinear Schrodinger equation systems

In this paper,we study the blow-up problem of nonlinear Schrodinger equations{i■_(t)v+△u+(|u|^(2)+|v|^(2))u=0,(t,x)∈R^(1+n),i■_(t)v+△u+(|u|^(2)+|v|^(2))u=0,(t,x)∈R^(1+n),u(0,x)=u_(0)(x),v(0,x)=v0(x),and prove that the solution of negative energy(E(u,v)<O)blows up in finite or infinite time.

LONG-TIME BEHAVIOR OF FINITE DIFFERENCE SOLUTIONS OF A NONLINEAR SCHRODINGER EQUATION WITH WEAKLY DAMPED

A weakly demped Schrodinger equation possessing a global attractor are considered. The dynamical properties of a class of finite difference scheme axe analysed. The exsitence of global attractor is proved for the discrete system. The stability of the difference scheme and the error estimate of the difference solution are obtained in the autonomous system case. Finally, long-time stability and convergence of the class of finite difference scheme also are analysed in the nonautonomous system case.

Solving forward and inverse problems of the nonlinear Schrodinger equation with the generalized PT-symmetric Scarf-Ⅱpotential via PINN deep learning

In this paper,based on physics-informed neural networks(PINNs),a good deep learning neural network framework that can be used to effectively solve the nonlinear evolution partial differential equations(PDEs)and other types of nonlinear physical models,we study the nonlinear Schrodinger equation(NLSE)with the generalized PT-symmetric Scarf-Ⅱpotential,which is an important physical model in many fields of nonlinear physics.Firstly,we choose three different initial values and the same Dinchlet boundaiy conditions to solve the NLSE with the generalized PT-symmetric Scarf-Ⅱpotential via the PINN deep learning method,and the obtained results are compared with ttose denved by the toditional numencal methods.Then,we mvestigate effect of two factors(optimization steps and activation functions)on the performance of the PINN deep learning method in the NLSE with the generalized PT-symmetric Scarf-Ⅱpotential.Ultimately,the data-driven coefficient discovery of the generalized PT-symmetric Scarf-Ⅱpotential or the dispersion and nonlinear items of the NLSE with the generalized PT-symmetric Scarf-Ⅱpotential can be approximately ascertained by using the PINN deep learning method.Our results may be meaningful for further investigation of the nonlinear Schrodmger equation with the generalized PT-symmetric Scarf-Ⅱpotential in the deep learning.

A LEAP FROG FINITE DIFFERENCE SCHEME FOR A CLASS OF NONLINEAR SCHRODINGER EQUATIONS OF HIGHORDER

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.