On nonlinear Schrödinger equations with random initial data

<abstract><p>This note is concerned with the global well-posedness of nonlinear Schrödinger equations in the continuum with spatially homogeneous random initial data.</p></abstract>

On space-time properties of solutions for nonlinear evolutionary equations with random initial data

We consider space-time properties of periodic solutions of nonlinear wave equations, nonlinear Schrödinger equations and KdV-type equations with initial data from the support of the Gibbs’ measure. For the wave and Schrödinger equations we establish the best Hölder exponents. We also discuss KdV-type equations which are more difficult due to a presence of the derivative in the nonlinearity.

SELECTION OF THE OPTIMUM ROUTE IN AN EXTENDED TRANSPORTATION NETWORK UNDER UNCERTAINTY

Relevance. For a given values set of extensive transport network sections lengths an exact method has been developed for finding optimal routes. The method provides an approximate solution when the initial data - are random variables with known distribution laws, as well as if these data are not clearly specified. Fora special case with a normal distribution of the numerical characteristics of the network, solution is brought to the final results. Method. An exact method of deterministic routing is proposed, which gives an approximate solution in case of random initial data. The method is extended to the case when the initial data are described in theory of fuzzy sets terms. The problem of stability assessing of solutions to problems of control the theory under conditions of uncertainty of initial data is considered. Results. A method of optimal routes finding is proposed when the initial data are deterministic or random variables with known distribution densities. A particular case of a probabilistic - theoretical description of the initial data is considered when can be obtained a simple solution of problem. Proposed method for obtaining an approximate solution in the general case for arbitrary distribution densities of random initial data. The situation is common when the initial data are not clearly defined. A simple computational procedure proposed for obtaining a solution. A method for stability assessing of solutions to control problems adopted under conditions of uncertainty in the initial data, is considered.

Shell-crossing in a ΛCDM Universe

ABSTRACT Perturbation theory is an indispensable tool for studying the cosmic large-scale structure, and establishing its limits is therefore of utmost importance. One crucial limitation of perturbation theory is shell-crossing, which is the instance when cold-dark-matter trajectories intersect for the first time. We investigate Lagrangian perturbation theory (LPT) at very high orders in the vicinity of the first shell-crossing for random initial data in a realistic three-dimensional Universe. For this, we have numerically implemented the all-order recursion relations for the matter trajectories, from which the convergence of the LPT series at shell-crossing is established. Convergence studies performed at large orders reveal the nature of the convergence-limiting singularities. These singularities are not the well-known density singularities at shell-crossing but occur at later times when LPT already ceased to provide physically meaningful results.

Solving the 4NLS with white noise initial data

We construct global-in-time singular dynamics for the (renormalized) cubic fourth-order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the ‘random-resonant / nonlinear decomposition’, which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche type argument, this singular component is nonlinear, consisting of arbitrarily high powers of the random initial data. We also employ a random gauge transform, leading to random Fourier restriction norm spaces. For this problem, a contraction argument does not work, and we instead establish the convergence of smooth approximating solutions by studying the partially iterated Duhamel formulation under the random gauge transform. We reduce the crucial nonlinear estimates to boundedness properties of certain random multilinear functionals of the white noise.

The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data

We consider the focusing energy-critical quintic nonlinear wave equation in 3D Euclidean space. It is known that this equation admits a one-parameter family of radial stationary solutions, called solitons, which can be viewed as a curve in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s-1}_x({{\mathbb{R}}}^3)$, for any $s&gt; 1/2$. By randomizing radial initial data in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s-1}_x({{\mathbb{R}}}^3)$ for $s&gt; 5/6$, which also satisfy a certain weighted Sobolev condition, we produce with high probability a family of radial perturbations of the soliton that give rise to global forward-in-time solutions of the focusing nonlinear wave equation that scatter after subtracting a dynamically modulated soliton. Our proof relies on a new randomization procedure using distorted Fourier projections associated to the linearized operator around a fixed soliton. To our knowledge, this is the 1st long-time random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime.