A geometric characterization of the nonlinear Schr?dinger equation and its applications

We prove that the nonlinear Schr?dinger equation of attractive type (NLS⁺) describes just spherical surfaces (SS) and the nonlinear Schr?dinger equation of repulsive type (NLS^-) determines only pseudospherical surfaces (PSS). This implies that, though we show that given two differential PSS (resp. SS) equations there exists a local gauge transformation (despite of changing the independent variables or not) which transforms a solution of one into any solution of the other, it is impossible to have such a gauge transformation between the NLS⁺ and the NLS^-.     

A geometric characterization of the nonlinear Schrödinger equation and its applications

Science China Mathematics - We prove that the nonlinear Schrödinger equation of attractive type (NLS+) describes just spherical surfaces (SS) and the nonlinear Schrödinger equation of...     

非线性Schrodinger方程的一种数值模拟方法

该文通过对非线性Schr\"{o}dinger方程增加耗散项,提出了一种新的三层线性差分格式.证明了该格式满足连续方程所具有的两个守恒量及收敛性和稳定性.通过数值例子与已知格式进行比较,结果表明该格式计算简单且具有较高精度.     

Solitary wave interaction for a higher-order nonlinear Schrodinger equation

Solitary wave interaction for a higher-order version of thenonlinear Schrödinger (NLS) equation is examined. An asymptotictransformation is used to transform a higher-order NLS equationto a higher-order member of the NLS integrable hierarchy, ifan algebraic relationship between the higher-order coefficientsis satisfied. The transformation is used to derive the higher-orderone- and two-soliton solutions; in general, the N-soliton solutioncan be derived. It is shown that the higher-order collisionis asymptotically elastic and analytical expressions are foundfor the higher-order phase and coordinate shifts. Numericalsimulations of the interaction of two higher-order solitarywaves are also performed. Two examples are considered, one satisfiesthe algebraic relationship derived from asymptotic theory, andthe other does not. For the example which satisfies the algebraicrelationship, the numerical results confirm that the collisionis elastic. The numerical and theoretical predictions for thehigher-order phase and coordinate shifts are also in strongagreement. For the example which does not satisfy the algebraicrelationship, the numerical results show that the collisionis inelastic and radiation is shed by the solitary wave collision.As the bed of radiation shed by the waves decays very slowly(like t^–), it is computationally infeasible to calculatethe final phase and coordinate shifts for the inelastic example.An asymptotic conservation law is derived and used to test thefinite-difference scheme for the numerical solutions.     

A new approach to some nonlinear geometric equations in dimension two

We give new arguments for several Liouville type results related to the equation −Δ u = Ke^2u. The new approach is based on the holomorphic function associated with any solution, which plays a similar role as the Hopf differential for harmonic maps from a surface.     

Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances

The nonlinear Schrödinger (NLS) equation can be derived as an amplitude equation describing slow modulations in time and space of an underlying spatially and temporarily oscillating wave packet. The purpose of this paper is to prove estimates, between the formal approximation, obtained via the NLS equation, and true solutions of the original system in case of non-trivial quadratic resonances. It turns out that the approximation property (APP) holds if the approximation is stable in the system for the three-wave interaction (TWI) associated to the resonance. We construct a counterexample showing that the NLS equation can fail to approximate the original system if instability occurs for the approximation in the TWI system. In the unstable case we give some arguments why the validity of the APP can be expected for spatially localized solutions and why it cannot be expected for non-localized solutions. Although, we restrict ourselves to a nonlinear wave equation as original system we believe that the results hold in more general situations, too.     

Numerical solution of the nonlinear Schrodinger equation by feedforward neural networks

We present a method to solve boundary value problems using artificial neural networks (ANN). A trial solution of the differential equation is written as a feed-forward neural network containing adjustable parameters (the weights and biases). From the differential equation and its boundary conditions we prepare the energy function which is used in the back-propagation method with momentum term to update the network parameters. We improved energy function of ANN which is derived from Schrodinger equation and the boundary conditions. With this improvement of energy function we can use unsupervised training method in the ANN for solving the equation. Unsupervised training aims to minimize a non-negative energy function. We used the ANN method to solve Schrodinger equation for few quantum systems. Eigenfunctions and energy eigenvalues are calculated. Our numerical results are in agreement with their corresponding analytical solution and show the efficiency of ANN method for solving eigenvalue problems.     

Exact solutions of a two-dimensional nonlinear Schrödinger equation

In the present study, we converted the resulting nonlinear equation for the evolution of weakly nonlinear hydrodynamic disturbances on a static cosmological background with self-focusing in a two-dimensional nonlinear Schrödinger (NLS) equation. Applying the function transformation method, the NLS equation was transformed to an ordinary differential equation, which depended only on one function ξ and can be solved. The general solution of the latter equation in ζ leads to a general solution of NLS equation. A new set of exact solutions for the two-dimensional NLS equation is obtained.     

非稳定非线性薛定谔方程的马尔钦科方程

该文基于对非稳定非线性薛定愕方程作反散射变换得到的Ｚａｋｈａｒｏｖ－Ｓｈａｂａｔ方程,直接对积分核作变换,导出马尔钦科方程．得到的马尔钦科方程在形式上与一般非线性薛定谔方程得到的一样简单明了,且不存在逆变换的自洽困难．     

Evolution equation of the Gauss curvature under hypersurface flows and its applications

In this paper, we derive evolution equation of the integral of the Gauss curvature on an evolving hypersurface. As an application, we obtain a monotone quantity on the level surface of the potential function on a 3-dimensional steady gradient Ricci soliton with positive sectional curvature, and prove that such a soliton is rotationally symmetric outside of a compact set under a curvature decaying assumption. Along the way we will also apply our evolution equation to some other cases.     

Validity of the KdV equation for the modulation of periodic traveling waves in the NLS equation

Bethuel et al.  and  and Chiron and Rousset [3] gave very nice proofs of the fact that slow modulations in time and space of periodic wave trains of the NLS equation can approximately be described via solutions of the KdV equation associated with the wave train. Here we give a much shorter proof of a slightly weaker result avoiding the very detailed and fine analysis of ,  and . Our error estimates are based on a suitable choice of polar coordinates, a Cauchy–Kowalevskaya-like method, and energy estimates.     

Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres

A generalized method, which is called the generally projective Riccati equation method, is presented to find more exact solutions of nonlinear differential equations based upon a coupled Riccati equation. As an application of the method, we choose the higher-order nonlinear Schrodinger equation to illustrate the method. As a result more new exact travelling wave solutions are found which include bright soliton solutions, dark soliton solution, new solitary waves, periodic solutions and rational solutions. The new method can be extended to other nonlinear differential equations in mathematical physics.     

Modified method of simplest equation and its application to nonlinear PDEs

We search for traveling-wave solutions of the class of PDEswhere A_p(Q),B_r(Q),C_s(Q),D_u(Q) and F(Q) are polynomials of Q. The basis of the investigation is a modification of the method of simplest equation. The equations of Bernoulli, Riccati and the extended tanh-function equation are used as simplest equations. The obtained general results are illustrated by obtaining exact solutions of versions of the generalized Kuramoto-Sivashinsky equation, reaction-diffusion equation with density-dependent diffusion, and the reaction-telegraph equation.     

On the existence of full dimensional KAM torus for fractional nonlinear Schrodinger equation

In this paper,\ we study fractional nonlinear Schrodinger equation (FNLS) with periodic boundary condition $$ \textbf{i}u_{t}=-(-\Delta)^{s_{0}} u-V*u-\epsilon f(x)|u|^4u,\ ~~x\in \mathbb{T}, ~~t\in \mathbb{R}, ~~s_{0}\in (\frac12,1),~~~~~~~~~~~~~~~~~~~~~~~~~~~~(0.1) $$ where $(-\Delta)^{s_{0}}$ is the Riesz fractional differentiation defined in [21] and $V*$ is the Fourier multiplier defined by $\widehat{V*u}(n)=V_n\widehat{u}(n),\ V_n\in\left[-1,1\right],$ and $f(x)$ is Gevrey smooth. We prove that for $0\leq|\epsilon|\ll1$ and appropriate $V$,\ the equation (0.1) admits a full dimensional KAM torus in the Gevrey space satisfying $ \frac12e^{-rn^{\theta}}\leq \left|q_n\right|\leq 2e^{-rn^{\theta}}, \theta\in (0,1),$ which generalizes the results given by [8-10] to fractional nonlinear Schrodinger equation.     

A characterization of the Lagrangian pseudosphere

We characterize the Lagrangian pseudosphere as the only branched Lagrangian immersion of a sphere in complex Euclidean plane with constant length mean curvature vector.

     

Bifurcations and exact solutions of nonlinear Schrodinger equation with an anti-cubic nonlinearity

In this paper, we consider the nonlinear Schr\"{o}dinger equation with an anti-cubic nonlinearity. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the corresponding planar dynamical system under different parameter conditions. Corresponding to different level curves defined by the Hamiltonian, we derive all exact explicit parametric representations of the bounded solutions (including periodic peakon solutions, periodic solutions, homoclinic solutions, heteroclinic solutions and compacton solutions).     

A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations

This paper is concerned with an operator equation Ax+Bx+Cx=x on ordered Banach spaces, where A is an increasing α-concave operator, B is an increasing sub-homogeneous operator and C is a homogeneous operator. The existence and uniqueness of its positive solutions is obtained by using the properties of cones and a fixed point theorem for increasing general β-concave operators. As applications, we utilize the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for two classes nonlinear problems which include fourth-order two-point boundary value problems for elastic beam equations and elliptic value problems for Lane-Emden-Fowler equations.     

Rogon-like solutions excited in the two-dimensional nonlocal nonlinear Schrödinger equation

We report the analytical one- and two-rogon-like solutions for the two-dimensional nonlocal nonlinear Schrödinger equation by means of the similarity transformation. These obtained solutions can be used to describe the possible physical mechanisms for rogue-like wave phenomenon. Moreover, the free function of space y involved in the obtained solutions excites the abundant structures of rogue-like wave propagations. The Hermite-Gaussian function of space y (normalized function) is, in particular, chosen to depict the dynamical behaviors for rogue-like wave phenomenon.     

Comment on “New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”

In this comment we analyze the paper [Abdelhalim Ebaid, S.M. Khaled, New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity, J. Comput. Appl. Math. 235 (2011) 1984-1992]. Using the traveling wave, Ebaid and Khaled have found “new types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”. We demonstrate that the authors studied the well-known nonlinear ordinary differential equation with the well-known general solution. We illustrate that Ebaid and Khaled have looked for some exact solution for the reduction of the nonlinear Schrodinger equation taking the general solution of the same equation into account.