Noise and solitons: nonlinear partial differential equations with random initial conditions

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Rapidly oscillating random perturbations

We consider in this section the case of initial conditions perturbed with a rapidly oscillating random process. We prove that for both the NLS and KdV equations the behavior of the soliton components of the solution actually shows very little dependence on the kind of perturbation, the limit behavior being controlled by a canonical system of SDEs where the only parameter of the original perturbation playing a role is its integrated covariance. Explicit computations are presented in the case of a square (box-like) initial condition
perturbed with a zero mean, stationary, rapidly oscillating process ν!(x) := ν(x/ε2): U0(x) = ￿ q + σ ε ! ￿ [0,R](x) , (2.1.1) but they can be extended to the case of perturbation of a more general initial condition defined by a bounded, compactly supported function q(x). The rapidly oscillating fluctua- tions of the initial condition can model the high-frequency additive noise of the light source generating the pulse in nonlinear fiber optics, for instance. We will show in subsection 2.1.2 that for rapidly oscillating processes (small values of ε) the limit system governing the stability of the soliton components reads as a set of SDEs and it is formally equivalent to the system where the initial condition contains a white noise ( ˙W ) perturbation U0(x) = ￿ q + √2ασ ˙Wx ￿ [0,R](x) , (2.1.2) where α is the integrated covariance of the process ν. This shows that to study the soliton components in the limit of rapid oscillations, the only required parameter of the statis- tics of ν is its integrated covariance. Notice however that we cannot directly use a white noise to perturb the initial condition, as the IST requires some integrability conditions on the initial condition (for example, U0 ∈ L1 for NLS), which are not satisfied by a white noise.

The Inverse Scattering Transform method

The essential technical instrument we will use to deal with the two nonlinear evolution equations we analyze is the inverse scattering transform. This method is a non-linear analogue of the Fourier transform, which can be used to solve many linear partial differential equations. The basic principle is the same, and with both methods the calculation of the solution of the initial value problem proceeds in three steps, as follows:
1. the forward problem: the initial conditions in the original “physical” variables are transformed into some spectral coefficients. For the Fourier transform they are simply the Fourier coefficients, while for the IST they are called scattering data. These scattering data characterize a spectral problem in which the initial conditions play the role of a potential. The Jost coefficients (a, b) together with the set of the discrete eigenvalues (ζn) of the spectral problem and the associated norming constants ρn compose the set of scattering data.
2. time dependence: the Fourier coefficients or the scattering data evolve according to simple, explicitly solvable ordinary differential equations. It is easy to obtain their value at any future time.
3. the inverse problem: the evolved solution in the original variables is recovered fromthe evolved solution in the transformed variables. This is done using the inverse Fourier transform or solving the Gelfand-Levitan-Marchenko equation (for the IST).

Limit of rapidly oscillating processes

This subsection contains a rigorous justification for the use of the IST. We have already remarked that to be able to apply the IST the initial condition U0 needs to satisfy some integrability condition, L1 for NLS and (2.1.8) for KdV. These hypotheses are satisfied by initial conditions of the form (2.1.1) for any ε > 0 if ν is bounded. Our objective is to show that the IST applied to these random initial conditions gives a problem that reads as a canonical system of SDEs in the limit ε → 0. Thanks to the convergence result of Theorem 2.2 below, this limit system can be used to study the behavior of rapidly oscillating initial conditions (0 < ε ￿ 1), as we shall do in the following sections. We stress that our interest is in the study of rapidly oscillating initial conditions, which are physically more relevant than the limit case of infinitely rapid oscillations and for which the IST can be applied in a rigorous way. We make the following assumptions (standard in the diffusion approximation theory, [FGPS]) on the process ν: Hypothesis 2.1. Let ν(x) be a real, homogeneous, ergodic, centered, bounded, Markov stochastic process, with finite integrated covariance ￿ ∞ 0 E[ν(0)ν(x)] dx = α < ∞ and with generator L& satisfying the Freedholm alternative.

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NLS solitons – small-intensity real noise

In this subsection we consider the example of an initial condition composed of a square function perturbed with a small, real white noise ( ˙W ). First,we use a perturbative approach to study the effects of the perturbation on solitons (proposition 2.9). Then, in the last part of this subsection, we study the effect of this perturbation on quiescent solitons (corollary 2.10). We have here U0(x) = (q+σ ˙Wx) [0,R](x). The initial condition is (2.2.1) and the system (2.1.3) for x ∈ [0,R] reads.

Table of contents 

1 Introduction 
1.1 Noise and solitons: nonlinear partial differential equations with random initial conditions
1.2 Imaging with noise: a linear partial differential equation with random initial conditions
1.3 The stochastic linear transport equation: regularization by noise.
2 Noise andsolitons 
2.1 Rapidly oscillating random perturbations
2.1.1 The Inverse Scattering Transform method
2.1.2 Limit of rapidly oscillating processes
2.2 Stability of solitons
2.2.1 NLS solitons – deterministic background solution
2.2.2 NLS solitons – small-intensity real noise
2.2.3 NLS solitons – small-intensity complex noise
2.2.4 KdV solitons – deterministic background solution
2.2.5 KdV solitons – small-amplitude random perturbation with q > 0
2.2.6 KdV solitons – small-amplitude random perturbations with q=0 .
2.2.7 Perturbation of NLS and KdV solitons: similarities and differences .
2.3 Examples
2.3.1 Creation of solitons from a rapidly oscillating real noise
2.3.2 Complex white noise: no soliton creation
2.3.3 Complex noise of constant intensity: no soliton creation for fixed ξ
2.3.4 Complex telegraph noise: no soliton creation for fixed ξ
2.3.5 Density of states – Ornstein-Uhlenbeck process
2.3.6 Density of states – general case
2.4 Appendix
3 Imaging with noise 
3.1 Mathematical formulation
3.1.1 The wave equation with random sources
3.1.2 The background Green’s function
3.1.3 The scattering operator and the Born approximation
3.1.4 The imaging problem
3.1.5 The classical setting: multi-offset sources
3.2 Stationary random sources
3.3 Incoherence by blending
3.4 High frequency analysis
3.4.1 The imaging functional in a high frequency regime
3.4.2 Expected contrast of the estimated perturbation
3.4.3 Fluctuations and typical contrast
3.4.4 Comments
3.5 Applications and simulations
3.5.1 Simultaneous source exploration
3.5.2 Seismic forward simulations
3.5.3 Numerical illustrations
3.6 Appendices
3.6.1 Appendix A: Statistical stability for a continuum of point sources
3.6.2 Appendix B: Resolution analysis
4 The stochastic transport equation 
4.1 Introduction
4.2 Convergence results
4.3 Existence of weakly differentiable solutions
4.4 Uniqueness of weakly differentiable solutions
4.5 Appendix A: technical results
4.6 Appendix B: Sobolev regularity of random fields

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