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Recovery of the Nonlinearity From the Modified Scattering Map
Abstract
We consider a class of one-dimensional nonlinear Schrödinger equations of the form $$ \begin{align*} & (i\partial_{t}+\Delta)u = [1+a]|u|^{2} u. \end{align*}$$For suitable localized functions $a$, such equations admit a small-data modified scattering theory, which incorporates the standard logarithmic phase correction. In this work, we prove that the small-data modified scattering behavior uniquely determines the inhomogeneity $a$.
... where B ⊂ H 1 is the common domain of S a and S b . 4]. Let a, b ∈ W 1,∞ , and let S a , S b denote the corresponding scattering maps for (1.1) with nonlinearities a|u| p u and b|u| p u, respectively. ...
... The work [9] proves an analogue of Theorem 1.2 for a more general class of nonlinearities in two dimensions; however, the results presented here do not suffice to establish a stability estimate in this more general setting. In the case that modified scattering holds, the recent work [4] also shows that the small-data modified scattering behavior also suffices to determine the inhomogeneity present in the nonlinearity. A stability estimate in this setting would also require some new ideas compared to what is presented here. ...
... We then rely on the fact that the free evolution of a Gaussian may be computed explicitly (and is still Gaussian), a fact that has already been exploited in the related works [4,9,13]. Using the scaling symmetry for the linear Schrödinger equation, we can therefore express the main term (1.4) in the form F σ * a(x 0 ), where c −1 σ −5 F σ forms a family of approximate identities as σ → 0 for suitable c > 0. Using the explicit form of F σ , we can estimate the difference ...
We prove stability estimates for the problem of recovering the nonlinearity from scattering data. We focus our attention on nonlinear Schr\"odinger equations of the form \[ (i\partial_t+\Delta)u = a(x)|u|^p u \] in three space dimensions, with $p\in[\tfrac43,4]$ and $a\in W^{1,\infty}$.
... We next show (3). Fix t, t 0 ∈ I. Then by Proposition 2.1, we obtain ...
The inverse scattering problem for the two-dimensional nonlinear Klein-Gordon equation $u_{tt}-\Delta u + u = \mathcal{N}(u)$ is studied. We assume that the unknown nonlinearity $\mathcal{N}$ of the equation satisfies $\mathcal{N}\in C^\infty(\mathbb{R};\mathbb{R})$, $\mathcal{N}^{(k)}(y)=O(|y|^{\max\{ 3-k,0 \}})$ ($y \to 0$) and $\mathcal{N}^{(k)}(y)=O(e^{c y^2})$ ($|y| \to \infty$) for any $k=0,1,2,\cdots$. Here, $c$ is a positive constant. We establish a reconstraction formula of $\mathcal{N}^{(k)}(0)$ ($k=3,4,5,\cdots$) by the knowledge of the scattering operator for the equation. As an application, we also give an expression for higher order G\^{a}teaux differentials of the scattering operator at 0.
We consider the 1d cubic nonlinear Schrödinger equation with an external potential V that is non-generic. Without making any parity assumption on the data, but assuming that the zero energy resonance of the associated Schrödinger operator is either odd or even, we prove global-in-time quantitative bounds and asymptotics for small solutions. First, we use a simple modification of the basis for the distorted Fourier transform (dFT) to resolve the (possible) discontinuity at zero energy due to the presence of a resonance and the absence of symmetry of the solution. We then use a refined analysis of the low frequency structure of the (modified) nonlinear spectral distribution, and employ smoothing estimates in the setting of non-generic potentials.
We study the inverse problem of recovery a compactly supported non-linearity in the semilinear wave equation utt-Δu+α(x)|u|2u=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{tt}-\Delta u+ \alpha (x) |u|^2u=0$$\end{document}, in two and three dimensions. We probe the medium with complex-valued harmonic waves of wavelength h and amplitude h-1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^{-1/2}$$\end{document}, then they propagate in the weakly non-linear regime; and measure the transmitted wave when it exits suppα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{supp}\,}}\alpha $$\end{document}. We show that one can extract the Radon transform of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} from the phase shift of such waves, and then one can recover α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}. We also show that one can probe the medium with real-valued harmonic waves and obtain uniqueness for the linearized problem.
This article is concerned with the small data problem for the cubic nonlinear
Schr\"odinger equation (NLS) in one space dimension, and short range
modifications of it. We provide a new, simpler approach in order to prove that
global solutions exist for data which is small in $H^{0,1}$. In the same
setting we also discuss the related problems of obtaining a modified scattering
expansion for the solution, as well as asymptotic completeness.
We prove Asymptotic Completeness of one dimensional NLS with long range nonlinearities. We also prove existence and expansion of asymptotic solutions with large data at infinity.
We present a general algorithm to show that a scattering operator associated to a semilinear dispersive equation is real analytic, and to compute the coefficients of its Taylor series at any point. We illustrate this method in the case of the Schrodinger equation with power-like nonlinearity or with Hartree type nonlinearity, and in the case of the wave and Klein-Gordon equations with power nonlinearity. Finally, we discuss the link of this approach with inverse scattering, and with complete integrability.
Using the two-dimensional nonlinear Schrödinger equation as a model example, we present a general method for recovering the nonlinearity of a nonlinear dispersive equation from its small-data scattering behavior. We prove that under very mild assumptions on the nonlinearity, the wave operator uniquely determines the nonlinearity, as does the scattering map. Evaluating the scattering map on well-chosen initial data, we reduce the problem to an inverse convolution problem, which we solve by means of an application of the Beurling–Lax Theorem.
We consider the initial-value problem for the one-dimensional cubic nonlinear Schrödinger equation with a repulsive delta potential. We prove that small initial data in a weighted Sobolev space lead to global solutions that decay in $L^{\infty }$ and exhibit modified scattering.
We consider the cubic nonlinear Schrödinger equation with an exceptional potential. We obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions. We prove the existence of modified scattering for this model, that is, linear scattering modulated by a phase. Our approach is based on the spectral theorem for the perturbed linear Schrödinger operator and a factorization technique, that allows us to control the resonant nonlinear term. We make some parity assumptions in order to control the small-energy behavior of the scattering coefficients and of the wave functions.
We consider an inverse scattering problem for a non-linear Schrödinger equation with a non-linearity of power-type. Applying the time-dependent technique, developed by Enss and Weder, we prove that the non-linearity is uniquely determined from a scattering data defined via the scattering operator. We also obtain a reconstruction formula which is valid for large scattering data.
We consider the cubic nonlinear Schrödinger equation with a potential. We obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions to the Cauchy problem.
We present a new and simpler proof that the nonlinear scattering operator S is analytic on energy space. We apply it in particular to a fourthorder nonlinear wave equation in Rn. In addition, we prove that S determines the scatterer uniquely and that for small powers there is no scattering.
We shall consider the inverse scattering problem for time dependent version of Hartree equation and nonlinear Klein–Gordon equation. The uniqueness theorem on identifying the cubic convolution nonlinearity from the knowledge of the scattering operator will be shown.
In this paper it will be shown that a potential $V(x)$ and a constant $\lambda$ are uniquely determined from the scattering operator $S$ associated with the nonlinear Schrödinger equation
\[
i\frac{\partial u}{\partial t}+(-\Delta+V)u+\lambda(|x|^{-\sigma}*|u|^{2})u=0 ,
\]
and the corresponding unperturbed equation
\[
i\frac{\partial u}{\partial t}-\Delta u=0 .
\]
We study the asymptotic behavior in time of solutions to the Cauchy problems for the nonlinear Schrodinger equation with a critical power nonlinearity and the Hartree equation. We prove the existence of modified scattering states and the sharp time decay estimate in the uniform norm of solutions to the Cauchy problem with small initial data. This estimate is very important for the proof of the existence of modified scattering states to the nonlinear Schrodinger equations with a critical nonlinearity and the Hartree equation. In order to derive the desired estimates we introduce a certain phase function since the previous methods, based solely on a priori estimates of the operator x + it acting on the solution without specifying any phase function, do not work for the critical case under consideration. The well-known nonexistence of the usual L 2 scattering states shows that our result is sharp.
Scattering theory compares the behavior in the distant future and past of a system evolving in time. It is called nonlinear if the system evolves in a nonlinear fashion. Consider a one-parameter group of operators on some linear space X: $$
U(t)U(s) = U(t + s);U(o) = I
$$ -∞ <t, s <+∞. We think of U(t)f as representing the state at time t beginning with a state f ε X at time zero. We are interested in the behavior of U(t)f as t → ±∞ and in the relationship between the behavior at +∞ and at −∞.
In this paper I present a method to uniquely reconstruct a potential V(x) from thescattering operator associated to the nonlinear Schrodinger equation and thecorresponding unperturbed equation where . I uniquely reconstruct the potential V by considering scattering states that have small amplitude and high velocity. In the small amplitude limit the main contribution to scattering comes from the potential V andsince moreover, the scattering state has high velocity the classical translation dominated the solution and the quantum spreading is a lower order term. These two effects lead to asimplification of the scattering process that allows me to uniquely reconstruct the potentialV.
We study the inverse scattering problem for the three dimensional nonlinear Schrödinger equation with the Yukawa potential. The nonlinearity of the equation is nonlocal. We reconstruct the potential and the nonlinearity by the knowledge of the scattering states. Our result is applicable to reconstructing the nonlinearity of the semi-relativistic Hartree equation.
We study the inverse scattering problem for the nonlinear Schrödinger equa-tion and for the nonlinear Klein-Gordon equation with the generalized Hartree type non-linearity. We reconstruct the nonlinearity from knowledge of the scattering operator, which improves the known results.
The authors compute the long-time asymptotics for solutions of the NLS equation just under the assumption that the initial data lies in a weighted Sobolev space. In earlier work (see e.g. [DZ1],[DIZ]) high orders of decay and smoothness are required for the initial data. The method here is a further development of the steepest descent method of [DZ1], and replaces certain absolute type estimates in [DZ1] with cancellation from oscillations.
In this paper we consider the inverse scattering problem for the non‐linear Schrödinger equation on the line \def\dr{{\rm d}}$$i{\partial\over\partial t}u(t,x)=‐{\dr^2\over\dr x^2}u(t,x)+V_0(x)u(t,x)+\sum_{j=1}^{\infty}V_j(x)|u|^{2(j_0+j)}u(t,x)$$\nopagenumbers\end We prove, under appropriate conditions, that the small‐amplitude limit of the scattering operator determines uniquely V j , j =0,1,… . Our proof gives also a method for the reconstruction of the V j , j =0,1,… . Copyright © 2001 John Wiley & Sons, Ltd.
We prove that in multidimensional short--range potential scattering the high velocity limit of the scattering operator of an N--body system determines uniquely the potential. For a given long--range potential the short--range potential of the N--body system is uniquely determined by the high velocity limit of the modified Dollard scattering operator. Moreover, we prove that any one of the Dollard scattering operators determines uniquely the total potential. We obtain as well a reconstruction formula with error term. Our simple proof uses a geometrical time--dependent method. I Introduction Let us consider first an N--body quantum mechanical system in n 2 space dimensions with interactions given by local short--range pair potentials (multiplication operators). By ams Classification 35P, 35Q, 81U; PACS number: 03.65.Nk y Fellow Sistema Nacional de Investigadores. Research Partially supported by Deutscher Akademischer Austauschdienst 1 S we denote the scattering operator betwee...
In this paper we solve the multidimensional inverse scattering problem for the nonlinear Klein-Gordon equation on R n ; n 2: @ 2 @t 2 u(x; t) Gamma Deltau(x; t) + u(x; t) + V 0 (x)u(x; t) + 1 X j=1 V j (x)juj 2(j0 +j) u(x; t) = 0: We prove that the small-amplitude limit of the scattering operator determines uniquely all the V j ; j = 0; 1; Delta Delta Delta. Our proof gives, as well, a method for the reconstruction of the V j ; j = 0; 1; Delta Delta Delta. ams classification 35P, 35Q, 35R and 81U. y Research partially supported by Proyecto PAPIIT-DGAPA IN 105799. z Fellow Sistema Nacional de Investigadores. 1 1 Introduction Let us consider the following nonlinear Klein-Gordon equation with a potential @ 2 @t 2 u(x; t) Gamma Deltau(x; t) + u(x; t) + V 0 u(x; t) + F (x; u) = 0; u(x; 0) = f 1 ; @ @t u(x; 0) = f 2 ; (1.1) where u(x; t) is a complex-valued funcion of x 2 R n ; t 2 R, and F (x; u) is a complex-valued function. Moreover, Delta := P n j=1...
In this paper I prove a L^p-L^p' estimate for the solutions of the
one-dimensional Schroedinger equation with a potential in L^1_gamma where in
the generic case gamma > 3/2 and in the exceptional case (i.e. when there is a
half-bound state of zero energy) gamma > 5/2. I use this estimate to construct
the scattering operator for the nonlinear Schroedinger equation with a
potential. I prove moreover, that the low-energy limit of the scattering
operator uniquely determines the potential and the nonlinearity using a method
that allows as well for the reconstruction of the potential and of the
nonlinearity.
A review of modified scattering for the 1d cubic NLS
- Murphy
A new proof of long-range scattering for critical nonlinear Schrödinger equations
- Kato
Dispersive equations
- Visan