Nonautonomous solitons in the continuous wave background of the variable-coefficient higher-order nonlinear Schrodinger equation

We reduce the variable-coefficient higher-order nonlinear Schrodinger equation （VCHNLSE） into the constantcoefficient （CC） one. Based on the reduction transformation and solutions of CCHNLSE, we obtain analytical soliton solutions embedded in the continuous wave background for the VCHNLSE. Then the excitation in advancement and sustainment of soliton arrays, and postponed disappearance and sustainment of the bright soliton embedded in the background are discussed in an exponential system.

Nondegenerate solitons of the(2+1)-dimensional coupled nonlinear Schrodinger equations with variable coefficients in nonlinear optical fibers

We derive the multi-hump nondegenerate solitons for the(2+1)-dimensional coupled nonlinear Schrodinger equations with propagation distance dependent diffraction,nonlinearity and gain(loss)using the developing Hirota bilinear method,and analyze the dynamical behaviors of these nondegenerate solitons.The results show that the shapes of the nondegenerate solitons are controllable by selecting different wave numbers,varying diffraction and nonlinearity parameters.In addition,when all the variable coefficients are chosen to be constant,the solutions obtained in this study reduce to the shape-preserving nondegenerate solitons.Finally,it is found that the nondegenerate two-soliton solutions can be bounded to form a double-hump two-soliton molecule after making the velocity of one double-hump soliton resonate with that of the other one.

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.

Deformed soliton,breather,and rogue wave solutions of an inhomogeneous nonlinear Schrodinger equation

We use the 1-fold Darboux transformation （DT） of an inhomogeneous nonlinear Schrdinger equation （INLSE） to construct the deformed-soliton, breather, and rogue wave solutions explicitly. Furthermore, the obtained first-order deformed rogue wave solution, which is derived from the deformed breather solution through the Taylor expansion, is different from the known rogue wave solution of the nonlinear Schrdinger equation （NLSE）. The effect of inhomogeneity is fully reflected in the variable height of the deformed soliton and the curved background of the deformed breather and rogue wave. By suitably adjusting the physical parameter, we show that a desired shape of the rogue wave can be generated. In particular, the newly constructed rogue wave can be reduced to the corresponding rogue wave of the nonlinear Schrdinger equation under a suitable parametric condition.

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).

Dynamics of fundamental and double-pole breathers and solitons for a nonlinear Schr¨odinger equation with sextic operator under non-zero boundary conditions

We study the dynamics of fundamental and double-pole breathers and solitons for the focusing and defocusing nonlinearSchr¨odinger equation with the sextic operator under non-zero boundary conditions. Our analysis mainly focuses onthe dynamical properties of simple- and double-pole solutions. Firstly, through verification, we find that solutions undernon-zero boundary conditions can be transformed into solutions under zero boundary conditions, whether in simple-pole ordouble-pole cases. For the focusing case, in the investigation of simple-pole solutions, temporal periodic breather and thespatial-temporal periodic breather are obtained by modulating parameters. Additionally, in the case of multi-pole solitons,we analyze parallel-state solitons, bound-state solitons, and intersecting solitons, providing a brief analysis of their interactions.In the double-pole case, we observe that the two solitons undergo two interactions, resulting in a distinctive “triangle”crest. Furthermore, for the defocusing case, we briefly consider two situations of simple-pole solutions, obtaining one andtwo dark solitons.

Exact Solutions for a Higher-Order Nonlinear Schrodinger Equation in Atmospheric Dynamics

By giving prior assumptions on the form of the solutions, we succeed to find several exact solutions for a higher-order nonlinear Schroetinger equation derived from one important model in the study of atmospheric and ocean dynamical systems. Our analytical solutions include bright and dark solitary waves, and periodical solutions, which can be used to explain atmospheric phenomena.

Binary Darboux transformation and multi-dark solitons for a higher-order nonlinear Schrodinger equation in the inhomogeneous optical fiber

Dark solitons in the inhomogeneous optical fiber are studied in this manuscript via a higher-order nonlinear Schr?dinger equation,since dark solitons can be applied in waveguide optics as dynamic switches and junctions or optical logic devices.Based on the Lax pair,the binary Darboux transformation is constructed under certain constraints,thus the multi-dark soliton solutions are presented.Soliton propagation and collision are graphically discussed with the group-velocity dispersion,third-and fourth-order dispersions,which can affect the solitons’velocities but have no effect on the shapes.Elastic collisions between the two dark solitons and among the three dark solitons are displayed,while the elasticity cannot be influenced by the above three coefficients.

Schrodinger Equation with a Cubic Nonlinearity Sech-Shaped Soliton Solutions

We first analyze the sech-shaped soliton solutions, either spatial or temporal of the 1D-Schr?dinger equation with a cubic nonlinearity. Afterwards, these solutions are generalized to the 2D-Schr?dinger equation in the same configuration and new soliton solutions are obtained. It is shown that working with dimensionless equations makes easy this generalization. The impact of solitons on modern technology is then stressed.

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.

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.

Quantitative analysis of soliton interactions based on the exact solutions of the nonlinear Schr?dinger equation

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.

Dynamics analysis of higher-order soliton solutions for the coupled mixed derivative nonlinear Schrodinger equation

In this paper,the dynamics of the higher-order soliton solutions for the coupled mixed derivative nonlinear Schrodinger equation¨are investigated via generalized Darboux transformation.Given a pair of linearly independent solutions of the Lax pair,the oneto three-soliton solutions are obtained via algebraic iteration.Furthermore,two and three solitons are respectively displayed via numerical simulation.Moreover,the dynamics of solitons are illustrated with corresponding evolution plots,such as elastic collisions,inelastic collisions,and bound states.It is found that there are some novel phenomena of interactions among solitons,which may provide a theoretical basis for studying optical solitons in experiments.

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.

Symmetric and antisymmetric vector solitons for the fractional quadric-cubic coupled nonlinear Schrodinger equation

The fractional quadric-cubic coupled nonlinear Schrodinger equation is concerned,and vector symmetric and antisymmetric soliton solutions are obtained by the square operator method.The relationship between the Lévy index and the amplitudes of vector symmetric and antisymmetric solitons is investigated.Two components of vector symmetric and antisymmetric solitons show a positive and negative trend with the Lévy index,respectively.The stability intervals of these solitons and the propagation constants corresponding to the maximum and minimum instability growth rates are studied.Results indicate that vector symmetric solitons are more stable and have better interference resistance than vector antisymmetric solitons.

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.

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.

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.

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.