The Solitary Waves for Cubic-Quintic Nonlinear Schrödinger Equation with Variable Coefficients

The cubic-quintic nonlinear Schrödinger equation (CQNLS) plays important parts in the optical fiber and the nuclear hydrodynamics. By using the homogeneous balance principle, the bell type, kink type, algebraic solitary waves, and trigonometric traveling waves for the cubic-quintic nonlinear Schrödinger equation with variable coefficients (vCQNLS) are derived with the aid of a set of subsidiary high-order ordinary differential equations (sub-equations for short). The method used in this paper might help one to derive the exact solutions for the other high-order nonlinear evolution equations, and shows the new application of the homogeneous balance principle.     

New Optical Solitons in High-Order Dispersive Cubic-Quintic Nonlinear Schrödinger Equation

By using the generalized tanh-function method, we find bright and dark solitary wave solutions to an extended nonlinear Schrödinger equation with the third-order and fourth-order dispersion and the cubic-quintic nonlinear terms, describing the propagation of extremely short pulses. At the same time, we also obtained other types of exact solutions.     

Self-Similar Solutions of Variable-Coefficient Cubic-Quintic Nonlinear Schrödinger Equation with an External Potential

An improved homogeneous balance principle and an F-expansiontechnique are used to construct exact self-similar solutions to the cubic-quintic nonlinear Schrödinger equation. Such solutions exist under certain conditions, and impose constraints on the functions describing dispersion, nonlinearity, and the external potential. Some simple self-similar waves are presented.     

New Exact Travelling Wave and Periodic Solutions of Discrete Nonlinear Schrödinger Equation

Some new exact travelling wave and period solutions of discrete nonlinear Schrödinger equation are found by using a hyperbolic tangent function approach, which was usually presented to find exact travelling wave solutions of certain nonlinear partial differential models. Now we can further extend the new algorithm to other nonlinear differential-different models.     

Solitary Wave Solutions of a Generalized Derivative Nonlinear Schrödinger Equation

With the aid of a class of nonlinear ordinary differential equation (ODE) and its various positive solutions, four types of exact solutions of the generalized derivative nonlinear Schrödinger equation (GDNLSE) have been found out, which are the bell-type solitary wave solution, the algebraic solitary wave solution, the kink-type solitary wave solution and the sinusoidal traveling wave solution, provided that the coefficients of GDNLSE satisfy certain constraint conditions. For more general GDNLSE, the similar results are also given.     

Analytical Periodic Wave and Soliton Solutions of the Generalized Nonautonomous Nonlinear Schrödinger Equation in Harmonic and Optical Lattice Potentials

We construct analytical periodic wave and soliton solutions to the generalized nonautonomous nonlinear Schrödinger equation with time- and space-dependent distributed coefficients in harmonic and optical lattice potentials. We utilize the similarity transformation technique to obtain these solutions. Constraints for the dispersion coefficient, the nonlinearity, and the gain (loss) coefficient are presented at the same time. Various shapes of periodic wave and soliton solutions are studied analytically and physically. Stability analysis of the solutions is discussed numerically.     

Higher-Order Rogue Wave Pairs in the Coupled Cubic-Quintic Nonlinear Schrödinger Equations

We study some novel patterns of rogue wave in the coupled cubic-quintic nonlinear Schrödinger equations. Utilizing the generalized Darboux transformation, the higher-order rogue wave pairs of the coupled system are generated. Especially, the first-and second-order rogue wave pairs are discussed in detail. It demonstrates that two classical fundamental rogue waves can be emerged from the first-order case and four or six classical fundamental rogue waves from the second-order case. In the second-order rogue wave solution, the distribution structures can be in triangle, quadrilateral and ring shapes by fixing appropriate values of the free parameters. In contrast to single-component systems, there are always more abundant rogue wave structures in multi-component ones. It is shown that the two higher-order nonlinear coefficients ρ₁ and ρ₂ make some skews of the rogue waves.     

Soliton and Rogue Wave Solution of the New Nonautonomous Nonlinear Schrödinger Equation

In this paper, a new type (or the second type) of transformation which is used to map the variable coefficient nonlinear Schrödinger (VCNLS) equation to the usual nonlinear Schrödinger (NLS) equation is given. As a special case, a new kind of nonautonomous NLS equation with a t-dependent potential is
introduced. Further, by using the new transformation and making full use of the known soliton and rogue wave solutions of the usual NLS equation, the corresponding kinds of solutions of a special model of the new nonautonomous NLS equation are discussed respectively. Additionally, through using the new transformation, a new expression, i.e., the non-rational formula, of the rogue wave of a special VCNLS equation is given analytically.
The main differences between the two types of transformation mentioned above are listed by three items.     

Rogue Wave Solutions for Nonlinear Schrödinger Equation with Variable Coefficients in Nonlinear Optical Systems

In this paper, the generalized Darboux transformation is constructed to variable coefficient nonlinear Schrödinger (NLS) equation. The N-th order rogue wave solution of this variable coefficient NLS equation is obtained by determinant expression form. In particular, we present rogue waves from first to third-order through some figures and analyze their dynamics.     

Exact Solutions of Five Complex Nonlinear Schrödinger Equations by Semi-Inverse Variational Principle

In this paper, we establish exact solutions for five complex nonlinear Schrödinger equations. The semi-inverse variational principle (SVP) is used to construct exact soliton solutions of five complex nonlinear Schrödinger equations. Many new families of exact soliton solutions of five complex nonlinear Schrödinger equations are successfully obtained.     

Rogue Wave with a Controllable Center of Nonlinear Schrödinger Equation

The rogue waves with a controllable center are reported for the nonlinear Schrödinger equation in terms of rational-like functions by using a direct method. The position of these solutions can be controlled by choosing different center parameters and this may describe the possible formation mechanisms for optical, oceanic, and matter rogue wave phenomenon in optical fibres, the deep ocean, Bose-Einstein condensates respectively.     

Direct Perturbation Method for Derivative Nonlinear Schrödinger Equation

We extend Lou's direct perturbation method for solving the nonlinear Schrödinger equation to the case of the derivative nonlinear Schrödinger equation (DNLSE). By applying this method, different types of perturbation solutions are obtained. Based on these approximate solutions, the analytical forms of soliton parameters, such as the velocity, the width and the initial position, are carried out and the effects of perturbation on solitons are analyzed at the same time. A numerical simulation of perturbed DNLSE finally verifies the results of the perturbation method.     

Multi-Symplectic Splitting Method for Two-Dimensional Nonlinear Schrödinger Equation

Using the idea of splitting numerical methods and the multi-symplectic methods, we propose a multi-symplectic splitting(MSS) method to solve the two-dimensional nonlinear Schrödinger equation (2D-NLSE) in this paper. It is further shown that the method constructed in this way preserve the global symplecticity exactly. Numerical experiments for the plane wave solution and singular solution of the 2D-NLSE show the accuracy and effectiveness
of the proposed method.     

Rogue Waves of the Higher-Order Dispersive Nonlinear Schrödinger Equation

In this paper,the rogue waves of the higher-order dispersive nonlinear Schrödinger (HDNLS) equation are investigated,which describes the propagation of ultrashort optical pulse in optical fibers.The rogue wave solutions of HDNLS equation are constructed by using the modified Darboux transformation method.The explicit first and second-order rogue wave solutions are presented under the plane wave seeding solution background.The nonlinear dynamics and properties of rogue waves are discussed by analyzing the obtained rational solutions.The influence of little perturbation ε on the rogue waves is discussed with the help of graphical simulation.     

Explicit N-Fold Darboux Transformations and Soliton Solutions for Nonlinear Derivative Schrödinger Equations

An explicit N-fold Darboux transformation for a coupled of derivative nonlinear Schrödinger equations is constructed with the help of a gauge transformation of spectral problems. As a reduction, the Darboux transformation for well-known Gerdjikov-Ivanov equation is further obtained, from which a general form of N-soliton solutions for Gerdjikov-Ivanov equation is given.     

Exact Solutions to Three-Dimensional Schrödinger Equation with an Exponentially Position-Dependent Mass

For an exponentially position-dependent mass, we obtain the exact solutions of the three-dimensional Schrödinger equation by using coordinate transformation method for the reference problems with Coulomb potential, Kratzer potential, and spherically square potential well of infinite depth, respectively. The explicit expressions for the energy eigenvalues and the corresponding eigenfunctions of the three systems are presented.     

Darboux Transformation and Soliton Solutions for InhomogeneousCoupled Nonlinear Schrödinger Equations with Symbolic Computation

With the aid of computation, we consider the variable-coefficient coupled nonlinear Schrödinger equations with the effects of group-velocity dispersion, self-phase modulation and cross-phase modulation, which have potential applications in the long-distance communication of two-pulse propagation in inhomogeneous optical fibers. Based on the obtained nonisospectral linear eigenvalue problems (i.e. Lax pair), we construct the Darboux transformation for such a model to derive the optical soliton solutions. In addition, through the one- and two-soliton-like solutions, we graphically discuss the features ofpicosecond solitons in inhomogeneous optical fibers.     

Exact Solutions of Schrödinger Equation for the Position-Dependent Effective Mass Harmonic Oscillator

A one-dimensional harmonic oscillator with position-dependent effective mass is studied. We quantize the oscillator to obtain a quantum Hamiltonian, which is manifestly Hermitian in configuration space, and the exact solutions to the corresponding Schrödinger equation are obtained analytically in terms of modified Hermite polynomials. It is shown that the obtained solutions reduce to those of simple harmonic oscillator as the position dependence of the mass vanishes.     

Application of Exp-Function Method to Discrete Nonlinear Schrödinger Lattice Equation with Symbolic Computation

In this paper, we present an extended Exp-function method to differential-difference equation(s). With the help of symbolic computation, we solve discrete nonlinear Schrödinger lattice as an example, and obtain a series of general solutions in forms of Exp-function.