Nonlinear Random Waves

The following sections are included:

• ABBREVIATIONS USED IN THIS BOOK

• Contents

Nonlinear science is magnetic and surprising, with a list of peculiarities and features that are still incomplete and not well understood. Thus one of the latest and most impressive developments of mathematical physics is the soliton theory. It was found that the complete set of elementary excitations of certain nonlinear evolution equations (they are called exactly integrable) consists, roughly speaking, of linear waves and solitons. The latest ones are the result of the balance between nonlinearity and dispersion, and hence are essentially nonlinear objects which cannot be obtained by any modification of the linear theory. They are stable in the collisions between them in the sense that they preserve their identity after the interaction. However, on first approaching them the solitons appear as an idealized description of physical processes. In order to consider a more exact model of the real world, we have to take into account many other factors such as the randomness and fluctuations and to study how the non-linearity works together with them. The unusual stability of the solitons leads to its important role in the global behaviour of the system. It means that in order to describe the nonlinear stochastic dynamics, we have to examine the statistics of both the linear modes and the solitons. These two types of excitations are the main differences of the nonlinear stochastic systems compared with linear ones, where the problem is reduced only to the study of the linear wave statistics. Moreover, the nonlinearity being the cause of the mode interaction, it changes essentially the time statistics, and this is the reason why most of the conventional methods of the linear stochastic theory fail in the nonlinear theory…

The following sections are included:

• Mathematical Preliminaries

• Mean Field Method

• Localization

• Parabolic Equation

The following sections are included:

• Inverse Scattering Technique

• Exactly Integrable Stochastic Equations

The following sections are included:

• Born Approximation in the Nonlinear Wave Theory

• Mean Field Method in Nonlinear Theory

The following sections are included:

• The Perturbed Korteweg-de Vries Equation. One Soliton Dynamics

• Perturbation Theory for Nonlinear Schrödinger Solitons

• Perturbation Theory for Kinks of the Sine-Gordon Equation

• General Remarks

The following sections are included:

• Stochastic Dynamics of the Korteweg-de Vries Soliton

• Solitons on Inhomogeneous Lattices

• Perturbed Dynamics of the Nonlinear Schrödinger Soliton

The following sections are included:

• The Forced Dynamics of a Kink

• Multisoliton Effects in Randomly Perturbed Systems

The following sections are included:

• Direct Analytical Methods

• Intensity of a Nonsoliton Random Wave Packet in the Asymptotic Region

• Weakly Nonlinear Incoherent Waves

The following sections are included:

• Randomly Modulated Bright Solitons

• Modulated Dark Solitons

• Modulated Multisoliton Pulses

The following sections are included:

• Scattering Data of Nonlinear Disordered Layer

• On Nonstationary Transmission Through a Disordered Layer

The following sections are included:

• One-Dimensional Motion of a Newtonian Particle

• One-Dimensional Nonconservative Motion

• A Charge in an Electromagnetic Field

• General Remarks

The following sections are included:

• Introduction

• The Computer Simulation of the Simple Langevin Equation

• Stochastic Ordinary Differential Equations

• General Remarks

The following sections are included:

• Numerical Schemes

• The Nonlinear Klein-Gordon Model

• A Stochastic Nonlinear Klein-Gordon Model

The following sections are included:

• Stochastic Korteweg-de Vries Models

• Korteweg-de Vries Equation

• Nonlinear Schrödinger Equation

• Future Stochastic Simulations

The following sections are included:

• Conclusion

• Bibliography