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$N$-soliton solutions and perturbation theory for the derivative nonlinear Scr?dinger equation with nonvanishing boundary conditions  [PDF]
V. M. Lashkin
Physics , 2007,
Abstract: We present a simple approach for finding $N$-soliton solution and the corresponding Jost solutions of the derivative nonlinear Scr\"{o}dinger equation with nonvanishing boundary conditions. Soliton perturbation theory based on the inverse scattering transform method is developed. As an application of the present theory we consider the action of the diffusive-type perturbation on a single bright/dark soliton.
Decay and scattering of solutions to three dimensional nonlinear Schr?dinger equations with potentials  [PDF]
Ze Li,Lifeng Zhao
Mathematics , 2015,
Abstract: We prove the decay and scattering of solutions to three dimensional nonlinear Schr\"odinger with a Schawtz potential. For Rollnik potentials, we obtain time decay and scattering in energy space for small initial data for NLS with pure power nonlinearity $\frac{5}{3}
Decay and scattering of solutions of nonlinear Schr?dinger equations with regular potentials  [PDF]
Xing Cheng,Ze Li
Mathematics , 2015,
Abstract: In this paper, we prove the scattering in the energy space for nonlinear Schr\"odinger equations with regular potentials in $\Bbb R^d$ namely, $i{\partial _t}u + \Delta u - V(x)u + \lambda |u|^{p - 1}u = 0$. In the first part, we prove scattering for small data for all $1+\frac{2}{d}
A remark on decay rates of solutions for a system of quadratic nonlinear Schr?dinger equations in 2D  [PDF]
Soichiro Katayama,Chunhua Li,Hideaki Sunagawa
Mathematics , 2013,
Abstract: We consider the initial value problem for a three-component system of quadratic nonlinear Schr\"odinger equations with mass resonance in two space dimensions. Under a suitable condition on the coefficients of the nonlinearity, we will show that the solution decays strictly faster than $O(t^{-1})$ as $t \to +\infty$ in $L^{\infty}$ by providing with an enhanced decay estimate of order $O((t \log t)^{-1})$. Differently from the previous works, our approach does not rely on the explicit form of the asymptotic profile of the solution at all.
Maximum Decay Rate for Finite-Energy Solutions of Nonlinear Schr?dinger Equations  [PDF]
Pascal Bégout
Mathematics , 2012,
Abstract: We give explicit time lower bounds in the Lebesgue spaces for all nontrivial solutions of nonlinear Schr\"odinger equations bounded in the energy space. The result applies for these equations set in any domain of $\R^N,$ including the whole space. This also holds for a large class of nonlinearities, thereby extending the results obtained by Hayashi and Ozawa in \cite{MR91d:35035} and by the author in \cite{beg3}.
Distributional solutions of the stationary nonlinear Schr?dinger equation: singularities, regularity and exponential decay  [PDF]
Rainer Mandel,Wolfgang Reichel
Mathematics , 2011,
Abstract: We consider the nonlinear Schr\"{o}dinger equation $-\Delta u + V(x) u = \Gamma(x) |u|^{p-1}u$ in $\R^n$ where the spectrum of $-\Delta+V(x)$ is positive. In the case $n\geq 3$ we use variational methods to prove that for all $p\in (\frac{n}{n-2},\frac{n}{n-2}+\eps)$ there exist distributional solutions with a point singularity at the origin provided $\eps>0$ is sufficiently small and $V,\Gamma$ are bounded on $\R^n\setminus B_1(0)$ and satisfy suitable H\"{o}lder-type conditions at the origin. In the case $n=1,2$ or $n\geq 3,1
A note on decay rates of solutions to a system of cubic nonlinear Schr?dinger equations in one space dimension  [PDF]
Donghyun Kim
Mathematics , 2014,
Abstract: We consider the initial value problem for a system of cubic nonlinear Schr\"odinger equations with different masses in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the small amplitude solution exists globally and decays of the rate $O(t^{-1/2}(\log t)^{-1/2})$ in $L^\infty$ as $t$ tends to infinity, if the system satisfies certain mass relations.
Self-Similar Solutions for Nonlinear Schr?dinger Equations  [PDF]
Yaojun Ye
Mathematical Problems in Engineering , 2009, DOI: 10.1155/2009/298980
Abstract: We study the self-similar solutions for nonlinear Schrödinger type equations of higher order with nonlinear term || by a scaling technique and the contractive mapping method. For some admissible value , we establish the global well-posedness of the Cauchy problem for nonlinear Schrödinger equations of higher order in some nonstandard function spaces which contain many homogeneous functions. we do this by establishing some nonlinear estimates in the Lorentz spaces or Besov spaces. These new global solutions to nonlinear Schrödinger equations with small data admit a class of self-similar solutions.
Periodic solutions of nonlinear Schr?dinger type equations
Periodic solutions of nonlinear Schro^edinger type equations

Liu Guan-Ting,Fan Tian-You,
刘官厅
,范天佑

中国物理 B , 2004,
Abstract: By using the solutions of an auxiliary elliptic equation, a direct algebraic method is proposed to construct the exact solutions of nonlinear Schr?dinger type equations. It is shown that many exact periodic solutions of some nonlinear Schr?dinger type equations are explicitly obtained with the aid of symbolic computation, including corresponding envelope solitary and shock wave solutions.
Maximum Decay Rate for the Nonlinear Schr?dinger Equation  [PDF]
Pascal Bégout
Mathematics , 2012, DOI: 10.1007/s00030-004-2003-7
Abstract: In this paper, we consider global solutions for the following nonlinear Schr\"odinger equation $iu_t+\Delta u+\lambda|u|^\alpha u=0,$ in $\R^N,$ with $\lambda\in\R$ and $0\le\alpha<\frac{4}{N-2}$ $(0\le\alpha<\infty$ if $N=1).$ We show that no nontrivial solution can decay faster than the solutions of the free Schr\"odinger equation, provided that $u(0)$ lies in the weighted Sobolev space $H^1(\R^N)\cap L^2(|x|^2;dx),$ in the energy space, namely $H^1(\R^N),$ or in $L^2(\R^N),$ according to the different cases.
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