Solitary wave solutions for nonlinear Schrödinger equation with non-polynomial nonlinearity

A new kind of non-polynomial nonlinearity is introduced in the nonlinear Schrödinger equation (NLSE) and the conditions are determined for which it admits solitary wave solutions. The study is done for two cases: one in which the nonlinear interaction is of the non-polynomial form and second in which cubic nonlinearity is also included along with the radical nonlinearity. Dark and bright solitary waves solutions are obtained in the respective cases. Further, later case is extended to conditions for which corresponding equation reduces to driven quadratic-cubic NLSE possessing cnoidal solutions with plane wave phase, which reduces to bright soliton for a certain parameter.     

The extended (G′/G)-expansion method and travelling wave solutions for the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity

In this paper, we construct the travelling wave solutions to the perturbed nonlinear Schrödinger’s equation (NLSE) with Kerr law non-linearity by the extended (G′/G)-expansion method. Based on this method, we obtain abundant exact travelling wave solutions of NLSE with Kerr law nonlinearity with arbitrary parameters. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions and rational functions.     

Stationary solution of the nonlinear Schrödinger's equation with log law nonlinearity by Lie symmetry analysis

In this paper, the nonlinear Schrödinger equation is studied in the presence of log law nonlinearity. The technique of Lie group analysis is used to carry out the integration of this equation with spatially dependent damping. The results will turn out in terms of Gaussons.     

Chaos control in the nonlinear Schrdinger equation with Kerr law nonlinearity

The nonlinear Schrdinger equation with Kerr law nonlinearity in the two-frequency interference is studied by the numerical method.Chaos occurs easily due to the absence of damping.This phenomenon will cause the distortion in the process of information transmission.We find that fiber-optic transmit signals still present chaotic phenomena if the control intensity is smaller.With the increase of intensity,the fiber-optic signal can stay in a stable state in some regions.When the strength is suppressed to a certain value,an unstable phenomenon of the fiber-optic signal occurs.Moreover we discuss the sensitivities of the parameters to be controlled.The results show that the linear term coefficient and the environment of two quite different frequences have less effects on the fiber-optic transmission.Meanwhile the phenomena of vibration,attenuation and escape occur in some regions.     

New and more general traveling wave solutions for nonlinear Schrödinger equation

An extended Fan sub-equation method is used to seek some new and more general traveling wave solutions of nonlinear Schrödinger equation (NLSE). The important fact of this method is to take the full advantage of clear relationship among general elliptic equation involving five parameters and other existing sub-equations involving three parameters. It is preferable to use this method to solve NLSE because this method gives us all the solutions obtained previously by the application of at least four methods (the method of using Riccati equation, or auxiliary ordinary differential equation method, or first kind elliptic equation or the generalized Riccati equation as mapping equation) in a unified manner. So it is shown that this method is concise and its applications are promising.     

Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients

A broad class of exact self-similar solutions to the nonlinear Schr?dinger equation (NLSE) with distributed dispersion, nonlinearity, and gain or loss has been found. Appropriate solitary wave solutions applying to propagation in optical fibers and optical fiber amplifiers with these distributed parameters have also been studied. These solutions exist for physically realistic dispersion and nonlinearity profiles in a fiber with anomalous group velocity dispersion. They correspond either to compressing or spreading solitary pulses which maintain a linear chirp or to chirped oscillatory solutions. The stability of these solutions has been confirmed by numerical simulations of the NLSE.     

Reconstruction of singularities for solutions of Schrödinger's equation

We determine the behavior in time of singularities of solutions to some Schrödinger equations onR ⁿ . We assume the Hamiltonians are of the formH ₀+V, where \(H_0 = 1/2\Delta + 1/2 \sum\limits_{k = 1}^n { \omega _k^2 x_k^2 } \) , and whereV is bounded and smooth with decaying derivatives. When all ω _k =0, the kernelk(t,x,y) of exp (?itH) is smooth inx for every fixed (t,y). When all ω₁ are equal but non-zero, the initial singularity “reconstructs” at times \(t = \frac{{m\pi }}{{\omega _1 }}\) and positionsx=(?1) ^m y, just as ifV=0;k is otherwise regular. In the general case, the singular support is shown to be contained in the union of the hyperplanes \(\{ x|x_{js} = ( - 1)^l js_{y_{js} } \} \) , when ω _j t/π=l _j forj=j ₁,...,j _r .     

Local decay of scattering solutions to Schrödinger's equation

The main theorem asserts that ifH=+gV is a Schrödinger Hamiltonian with short rangeV, L _compact ² (IR³), andR>0, then exp(iHt) _{S L} ² _(|x|<R)=O(t ^–1/2), ast where _S is projection onto the orthogonal complement of the real eigenvectors ofH. For all but a discrete set ofg,O(t ^–1/2) may be replaced byO(t ^–3/2).Research supported by the National Science Foundation under grants NSF GP 34260 and MCS 72-05055 A04     

Exact solutions of nonlinear Schrodinger’s equation with dual power-law nonlinearity by extended trial equation method

In this paper, we acquire the soliton solutions of the nonlinear Schrodinger’s equation with dual power-law nonlinearity. Primarily, we use the extended trial equation method to find exact solutions of this equation. Then, we attain some exact solutions including soliton solutions, rational and elliptic function solutions of this equation using the extended trial equation method.     

Semiclassical trajectory-coherent states of the nonlinear Schrödinger equation with unitary nonlinearity

Semiclassically concentrated states of the nonlinear Schrödinger equation (NLSE) with unitary nonlinearity, representing multidimensional localized wave packets, are constructed on the basis of the Maslov complex germ theory. A system of ordinary differential equations of Hamilton-Ehrenfest (HE) type, describing the motion of the wave packet centroid, is derived. The structure of the HE system is strongly influenced by the initial conditions of the Cauchy problem for the NLSE. Wave packets of Gaussian type are constructed in an explicit form. Possible use of the solutions constructed in the problem of optical pulse propagation in a nonlinear medium with nonstationary dispersion is discussed.     

Exact spatial soliton solution for nonlinear Schrödinger equation with nonperiodic modulation of nonlinearity and linear refractive index

In this paper, we analyze (2 + 1)-dimensional nonlinear Schrödinger equation with nonperiodic modulation of nonlinearity and linear refractive index in the transverse direction, and obtain an exact solution in explicit form using an ansatz method. Finally, the stability of the solution is discussed numerically, and the result reveals that the solution is stable to the finite initial perturbations.     

Exact chirped multi-soliton solutions of the nonlinear Schr(o|¨)dinger equation with varying coefficients

In this letter, exact chirped multi-soliton solutions of the nonlinear Schrodinger (NLS) equation with varying coefficients are found. The explicit chirped one- and two-soliton solutions are generated. As an example, an exponential distributed control system is considered, and some main features of solutions are shown. The results reveal that chirped soliton can all be nonlinearly compressed cleanly and efficiently in an optical fiber with no loss or gain, with the loss, or with the gain. Furthermore, under the same initial condition, compression of optical soliton in the optical fiber with the loss is the most dramatic. Also, under nonintegrable condition and finite initial perturbations, the evolution of chirped soliton has been demonstrated by simulating numerically.     

Rapidly decaying solutions of the nonlinear Schrödinger equation

We consider global solutions of the nonlinear Schrödinger equation

On the exact solutions of nonlinear Schrödinger equation with spin current

An integrable nonlinear Schrödinger (NLS) equation driven by spin polarized current governing the magnetization dynamics of a ferromagnetic nanowire is considered. The exact soliton solution of the NLS equation propagating along the direction of wire axis which is also the current direction along which nonuniform magnetization occurs is obtained through the application of exponential function method. The solution of the system admits a class of solitons such as kink and periodic solitons in the nanowire along the direction of the electric current.