Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are also surveyed, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation. The solitons are considered in one, two, and three dimensions. Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of one-dimensional solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for theoretical and experimental studies alike. In both the one-dimensional and two-dimensional cases, the mechanism that creates solitons in NLs in principle is different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.

Solitons in nonlinear lattices

Wave collapse in physics: principles and applications to light and plasma waves

Chapter 5 - Optical vortices and vortex solitons

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The Discrete Nonlinear Schrödinger Equation

We study the propagation of intense optical beams in layered Kerr media. With appropriate shapes, beams with a power close to the self-focusing threshold are shown to propagate over long distances as quasi-stationary waveguides in cubic media supporting a periodic nonlinear refractive index.

Self-guiding light in layered nonlinear media

The scattering properties of regularizing finite-range potentials constructed in the form of squeezed rectangles, which approximate the first and second derivatives of the Dirac delta function δ(x), are studied in the zero-range limit. Particularly, for a countable set of interaction strength values, a non-zero transmission through the point potential δ'(x), defined as the weak limit (in the standard sense of distributions) of a special dipole-like sequence of rectangles, is shown to exist when the rectangles are squeezed to zero width. A tripole sequence of rectangles, which gives in the weak limit the distribution δ''(x), is demonstrated to exhibit the total transmission for a countable sequence of the rectangle's width that tends to zero. However, this tripole sequence does not admit a well-defined point interaction in the zero-range limit, making sense only for a finite range of the regularizing rectangular-like potentials.

https://iopscience.iop.org/article/10.1088/0305-4470/36/27/311/pdf

On the existence of resonances in the transmission probability for interactions arising from derivatives of Dirac's delta function

A nonlinear dynamical model of molecular monolayers arranged in Scheibe aggregates is derived from a proper Hamiltonian. Thermal fluctuations of the phonons are included. The resulting equation for the excitons is the two dimensional nonlinear Schr\"odinger equation with noise. Two limits of the complicated spectrum of the noise are considered: time independent, spatially white noise, simply corresponding to disorder in the arrangement of the molecules, and pure white noise. Parameter values are found by comparison with experiments by M\"obius and Kuhn [Isr. J. Chem. 18, 375 (1979)] and order of magnitude estimates given where experiments are not available. The temperature dependent coherence time is found from numerical simulations. Experiments show that the excitons stay coherent during their lifetime. This is in correspondence with the model at temperatures lower than 3 K. To increase this limiting temperature it is found that the dipole-dipole coupling and the exciton-phonon coupling must be decreased significantly.

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Temperature effects in a nonlinear model of monolayer Scheibe aggregates

Multiplicative Gaussian white noise is included in the two-dimensional nonlinear Schrodinger equation. The collective coordinate method is used to derive an effective stochastic equation for the width of the wave function. From this equation the coherence time of the ground state solitary wave solution, tcoh , is found to be proportional to D-1/3 (D is the variance of the noise). Direct numerical simulations indicate tcohαD-0.4

White noise in the two-dimensional nonlinear schrödinger equation

Azobenzene-containing polymers exhibit strong surface-relief features when irradiated with polarized light. Currently proposed theories do not explain all the observed features. Here we propose a theory based on elastic deformation of the polymer due to interaction between dipoles ordered through polarized light irradiation. The effects are due to the presence of a boundary layer. The observation of both wells and humps dependent on the architecture of the polymer can be explained with the present theory.

Yu. B. Gaididei

Papers

Self-guiding light in layered nonlinear media

On the existence of resonances in the transmission probability for interactions arising from derivatives of Dirac's delta function

Temperature effects in a nonlinear model of monolayer Scheibe aggregates

White noise in the two-dimensional nonlinear schrödinger equation

Theory of photoinduced deformation of molecular films