We consider asymptotic stability of a small solitary wave to supercritical 1-dimensional nonlinear Schrödinger equations $$ iu_t+u_{xx}=Vu\pm |u|^{p-1}u \quad\text{for $(x,t)\in\mathbb{R}\times\mathbb{R}$,}$$ in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai \cite{GNT} in the 3-dimensional case using the endpoint Strichartz estimate.
To prove asymptotic stability of solitary waves, we need to show that a dispersive part $v(t,x)$ of a solution belongs to $L^2_t(0,\infty;X)$ for some space $X$. In the 1-dimensional case, this property does not follow from the Strichartz estimate alone.
In this paper, we prove that the local smoothing effect of Kato type holds global in time and combine this estimate with the Strichartz estimate to show $\|(1+x^2)^{-3/4}v\|_{L^\infty_xL^2_t}<\infty$, which implies the asymptotic stability of a solitary wave.
Multi-soliton solutions, i.e. solutions behaving as the sum of N given solitons as $t\to +\infty$, were constructed in previous works for the L2 critical and subcritical (NLS) and (gKdV) equations. In this paper, we extend the construction of multi-soliton solutions to the L2 supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.
We address stability of multi-solitons in the cubic NLS (nonlinear Schrödinger) equation on the line. By using the dressing transformation and the inverse scattering transform methods, we obtain the orbital stability of multi-solitons in the $L^2(\mathbb{R})$ space when the initial data is in a weighted $L^2(\mathbb{R})$ space.
In this article we consider the Cauchy problem for the cubic focusing nonlinear Schrö\-dinger (NLS) equation on the line with initial datum close to a particular $N$-soliton. Using inverse scattering and the $\bar{\partial}$ method we establish the decay of the $L^{\infty}$ norm of the residual term in time.
Consider a branch of unstable solitons of NLS whose linearized operators have one pair of simple real eigenvalues in addition to the zero eigenvalue. Under radial symmetry and standard assumptions, solutions to initial data from a neighbourhood of the branch either converge to a soliton, or exit a larger neighbourhood of the branch transversally. The qualitative dynamic near a branch of unstable solitons is irrespective of whether blowup eventually occurs, which has practical implications for the description of blowup of NLS with supercritical nonlinearity.
Combining the usual energy functional with a higher-order conserved quantity originating from integrability theory, we show that the black soliton is a local minimizer of a quantity that is conserved along the flow of the cubic defocusing NLS equation in one space dimension. This unconstrained variational characterization gives an elementary proof of the orbital stability of the black soliton with respect to perturbations in $H^2(\mathbb{R})$.
In this paper we prove rigidity for blowup solutions to the focusing, mass-critical nonlinear Schr{ö}dinger equation in dimensions $2 \leq d \leq 15$ with mass equal to the mass of the soliton. We prove that the only such solutions are the solitons and the pseudoconformal transformation of the solitons.
We prove the existence of global solutions to the DNLS equation with initial data in a large subset of $H^2(\mathbb R)\cap H^{1,1}(\mathbb R)$ containing a neighborhood of all solitons. We use the inverse scattering transform method, which was recently developed by D. Pelinovsky and Y. Shimabukuro, and an auto-Bäcklund transform in order to include solitons.
We study the two dimensional non-linear Schrödinger equation with two types of exponential non-linearities. It is well-known by a work of Ruf - Sani, that such models support solitary wave solutions, which are solutions of some constrained minimization problem. We show that these Ruf - Sani solitons, are spectrally unstable as well as unstable by blow-up.
We consider a Nonlinear Schrödinger Equation with a very general non linear term and with a trapping $δ$ potential on the line. We then discuss the asymptotic behavior of all its small solutions, generalizing a recent result by Masaki et al. We give also a result of dispersion in the case of defocusing equations with a non--trapping delta potential.
We investigate the validity of a soliton dynamics behavior in the semi-relativistic limit for the nonlinear Schrödinger equation in $\R^{N}, N\ge 3$, in presence of a singular external potential.
For both the cubic Nonlinear Schrödinger Equation (NLS) as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension we consider the set ${\bf M}_N$ of pure $N$-soliton states, and their associated multisoliton solutions. We prove that (i) the set ${\bf M}_N$ is a uniformly smooth manifold, and (ii) the ${\bf M}_N$ states are uniformly stable in $H^s$, for each $s>-\frac12$. One main tool in our analysis is an iterated Backlund transform, which allows us to nonlinearly add a multisoliton to an existing soliton free state (the soliton addition map) or alternatively to remove…
We study instability of a vortex soliton $e^{i(mθ+ωt)}ϕ_{ω,m}(r)$ to $$iu_t+Δu+|u|^{p-1}u=0,\quad\text{for $x\in\R^n$, $t>0$,}$$ where $n=2$, $m\in\N$ and $(r,θ)$ are polar coordinates in $\R^2$. Grillakis \cite{Gr} proved that every radially standing wave solutions are unstable if $p>1+4/n$. However, we do not have any examples of unstable standing wave solutions in the subcritical case $(p<1+n/4)$. Suppose $ϕ_{ω,m}$ is nonnegative. We investigate a limiting profile of $ϕ_{ω,m}$ as $m\to\infty$ and prove that for every $p>1$, there exists an $m_*\in \N$ such that for $m\ge m_*$, a vortex soli…
The focusing nonlinear Schrodinger equation possesses special non-dispersive solitary type solutions, solitons. Under certain spectral assumptions we show existence and asymptotic stability of solutions with the asymptoic profile (as time goes to infinity) of a linear combination of N non-colliding solitons.
We consider soliton-like solutions of a variable coefficients, subcritical nonlinear Schrodinger equation (NLS). In a previous result, we proved the existence of a pure, global-in-time, generalized soliton with prescribed asymptotic as t tends to -infinity. In addition, we described the corresponding dynamics for all time, up to a small error term. In this paper we prove that the soliton is never pure as t tends to infinity. Indeed, we give a precise sharp lower bound on the defect induced by the potential on the soliton. This result shows the existence of nontrivial dispersive effects acting …
We use the inverse scattering transform, the auto-Backlund transformation and the steepest descent method of Deift and Zhou to obtain the asymptotic stability of the solitons in the cubic NLS (nonlinear Schrodinger) equation.
We extend the global existence result for the derivative NLS equation to the case when the initial datum includes a finite number of solitons. This is achieved by an application of the Bäcklund transformation that removes a finite number of zeros of the scattering coefficient. By means of this transformation, the Riemann--Hilbert problem for meromorphic functions can be formulated as the one for analytic functions, the solvability of which was obtained recently.
The Peregrine soliton $Q(x,t)=e^{it}(1-\frac{4(1+2it)}{1+4x^2+4t^2})$ is an exact solution of the 1d focusing nonlinear schrödinger equation (NLS) $iB_t+B_{xx}=-2|B|^2B$, having the feature that it decays to $e^{it}$ at the spatial and time infinities, and with a peak and troughs in a local region. It is considered as a prototype of the rogue waves by the ocean waves community. The 1D NLS is related to the full water wave system in the sense that asymptotically it is the envelope equation for the full water waves. In this paper, working in the framework of water waves which decay non-tangentia…
We review some recent results concerning Gibbs measures for nonlinear Schroedinger equations (NLS), with implications for the theory of the NLS, including stability and typicality of solitary wave structures. In particular, we discuss the Gibbs measures of the discrete NLS in three dimensions, where there is a striking phase transition to soliton-like behavior.
We present a new method for obtaining sharp asymptotics of solutions of the defocussing nonlinear Schrödinger (NLS) equation, based on dbar methods and under essentially minimal regularity assumptions on initial data.
We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $ \I\partial_t q(x,t)+\partial_{x}^2q(x,t)+2σq^{2}(x,t)\overline{q(-x,t)}=0 $ with initial data $q(x,0)\in H^{1,1}(\mathbb{R})$. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $H^{1,1}$ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main to…
We give a unified proof of the step to find Darboux coordinates and of the ensuing Birkhoff normal forms procedure, developed in the course of the proof of asymptotic stability of solitary waves for NLS and special cases of Nonlinear Dirac equations
We consider the mass critical fractional (NLS). We show the existence of travelling waves for all mass below the ground state mass, and give a complete description of the associated profiles in the small mass limit. We therefore recover a situation similar to the one discovered in [Gerard P.; Lenzmann E.; Pocovnicu O.; Raphaël, P., A two soliton with transient turbulent regime for the one dimensional cubic half wave, submitted] for the critical case s = 1, but with a completely different asymptotic profile when the mass vanishes.
We study the one-dimensional nonlinear Schrödinger equation with the cubic-quintic combination of attractive and repulsive nonlinearities, and a trapping potential represented by a delta-function. We determine all bound states with a positive soliton profile through explicit formulas and, using bifurcation theory, we describe their behavior with respect to the propagation constant. This information is used to prove their stability by means of the rigorous theory of orbital stability of Hamiltonian systems. The presence of the trapping potential gives rise to a regime where two stable bound sta…
We consider the Cauchy problem for the defocusing Schr$\ddot{\text{o}}$dinger (NLS) equation with a nonzero background $$\begin{align} &iq_t+q_{xx}-2(|q|^2-1)q=0, \nonumber\\ &q(x,0)=q_0(x), \quad \lim_{x \to \pm \infty}q_0(x)=\pm 1. \end{align}$$ Recently, for the space-time region $|x/(2t)|<1$ which is a solitonic region without stationary phase points on the jump contour, Cuccagna and Jenkins presented the asymptotic stability of the $N$-soliton solutions for the NLS equation by using the $\bar{\partial}$ generalization of the Deift-Zhou nonlinear steepest descent method. Their large-time a…
A bi-Hamiltonian hierarchy of complex vector soliton equations is derived from geometric flows of non-stretching curves in the Lie groups $G=SO(N+1),SU(N)\subset U(N)$, generalizing previous work on integrable curve flows in Riemannian symmetric spaces $G/SO(N)$. The derivation uses a parallel frame and connection along the curves, involving the Klein geometry of the group $G$. This is shown to yield the two known $U(N-1)$-invariant vector generalizations of the nonlinear Schrodinger (NLS) equation and the complex modified Korteweg-de Vries (mKdV) equation, as well as $U(N-1)$-invariant vector…
In this paper we give the complete classification of solitons for a cubic NLS equation on the simplest network with a non-trivial topology: the tadpole graph, i.e. a ring with a half-line attached to it and free boundary conditions at the junction. The model, although simple, exhibits a surprisingly rich behavior and in particular we show that it admits: 1) a denumerable family of continuous branches of embedded solitons bifurcating from linear eigenstates and threshold resonances of the system; 2) a continuous branch of edge solitons displaying a pitchfork symmetry breaking bifurcation at the…
We transpose work by K.Yajima and by T.Mizumachi to prove dispersive and smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schrödinger equation (NLS) in 2D. As an application we extend to dimension 2D a result on asymptotic stability of ground states of NLS proved in the literature for all dimensions different from 2.
In this article we study the one-dimensional, asymptotically linear, non-linear Schrödinger equation (NLS). We show the existence of a global smooth curve of standing waves for this problem, and we prove that these standing waves are orbitally stable. As far as we know, this is the first rigorous stability result for the asymptotically linear NLS. We also discuss an application of our results to self-focusing waveguides with a saturable refractive index.
Black solitons are identical in the nonlinear Schrödinger (NLS) equation with intensity-dependent dispersion and the cubic defocusing NLS equation. We prove that the intensity-dependent dispersion introduces new properties in the stability analysis of the black soliton. First, the spectral stability problem possesses only isolated eigenvalues on the imaginary axis. Second, the energetic stability argument holds in Sobolev spaces with exponential weights. Third, the black soliton persists with respect to addition of a small decaying potential and remains spectrally stable when it is pinned to t…
We construct a local Lipschitz graph around a soliton of the cubic focusing NLS in three dimensions on which global solutions exist, and asymptotic stability as well as scattering holds.
We consider the one dimensional focusing (cubic) Nonlinear Schrödinger equation (NLS) in the semiclassical limit with exponentially decaying complex-valued initial data, whose phase is multiplied by a real parameter. We prove smooth dependence of the asymptotic solution on the parameter. Numerical results supporting our estimates of important quantities are presented.
The soliton resolution conjecture for the focusing nonlinear Schrodinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multi-soliton solution. Considered to be one of the fundamental open problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation till date. This paper proves a "statistical version" of this conjecture at mass-subcr…
