Nonlinear Sciences / nlin.SI

Stochastic Process Associated with Traveling Wave Solutions of the Sine-Gordon Equation

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📝 Original Info

  • Title: Stochastic Process Associated with Traveling Wave Solutions of the Sine-Gordon Equation
  • ArXiv ID: 0809.3484
  • Date: 2009-01-06
  • Authors: Researchers from original ArXiv paper

📝 Abstract

Stochastic processes associated with traveling wave solutions of the sine-Gordon equation are presented. The structure of the forward Kolmogorov equation as a conservation law is essential in the construction and so is the traveling wave structure. The derived stochastic processes are analyzed numerically. An interpretation of the behaviors of the stochastic processes is given in terms of the equation of motion.

💡 Deep Analysis

Deep Dive into Stochastic Process Associated with Traveling Wave Solutions of the Sine-Gordon Equation.

Stochastic processes associated with traveling wave solutions of the sine-Gordon equation are presented. The structure of the forward Kolmogorov equation as a conservation law is essential in the construction and so is the traveling wave structure. The derived stochastic processes are analyzed numerically. An interpretation of the behaviors of the stochastic processes is given in terms of the equation of motion.

📄 Full Content

arXiv:0809.3484v2 [nlin.SI] 5 Oct 2008 Typeset with jpsj2.cls Full Paper Stochastic Process Associated with Traveling Wave Solutions of the Sine-Gordon Equation Tetsu Yajima∗and Hideaki Ujino1 † Department of Information Science, Faculty of Engineering, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi 321-8585 1Gunma National College of Technology, 580 Toriba, Maebashi, Gunma 371-8530 Stochastic processes associated with traveling wave solutions of the sine-Gordon equation are presented. The structure of the forward Kolmogorov equation as a conservation law is essential in the construction of the stochastic process as well as the traveling wave structure. The derived stochastic processes are analyzed numerically. An interpretation of the behaviors of the stochastic processes is given in terms of the equation of motion. KEYWORDS: stochastic process, sine-Gordon equation, forward Kolmogorov equation, conser- vation law, traveling wave solutions 1. Introduction Research on stochastic processes has received much attention in mathematical physics recently. For example, the stochastic L¨owner equation1 attracts much interest regarding its connection to the conformal field theory,2 as well as its original meaning as a stochastic version of the L¨owner equation.3 The stochastic cellular automaton and asymmetric simple exclusion process (ASEP), among others, are also attracting considerable interest. The ASEP describes a non-equilibrium process of interacting particles, whose steady state has been exactly ob- tained.4 Even the current at equilibrium and the corresponding phase diagram were exactly presented for the ASEP.5 Besides the ASEP, several stochastic cellular automata are invented for the successful analysis of traffic flows.6 The stochastic process associated with the Burgers equation is called Burgers’ process,7 which is applied to studies of turbulence.8 This process is a typical example of nonlinear stochastic processes, in other words, stochastic processes associated with nonlinear equations. The time evolution of the density function of Burgers’ process, for instance, is governed by the Burgers equation uu −2uux + uxx = 0, which is indeed nonlinear. Such nonlinear stochas- tic processes are expected to be an effective approach to analyzing phenomena, where both nonlinear and stochastic effects are simultaneously predominant. Among many kinds of nonlinear equations that describe various physical systems, we have a series of integrable equations. Expecting future applications to the analysis of phenomena ∗yajimat@is.utsunomiya-u.ac.jp †ujino@nat.gunma-ct.ac.jp 1/12 J. Phys. Soc. Jpn. Full Paper related to integrable systems, we shall present new nonlinear stochastic processes associated with the sine-Gordon (SG) equation φxt = m2 sin φ, m: constant, (1.1) along the same line as the Burgers process. As is well known, the SG equation is an integrable nonlinear equation that is suitable for describing the dynamics of connected pendulums under gravity,9 the motions of the dislocation of one-dimensional materials,10 the propagation of optical waves in one-dimensional space,11 and so forth. The SG equation possesses a kink solution that has localized structure, aside from periodic solutions. In this paper, we shall consider periodic and one-kink solutions having traveling wave structures, and we shall present stochastic processes associated with these solutions. We shall also perform numerical analyses of the derived evolution equations under suitable initial conditions in order to study their behaviors. The results and their explanation given in the later sections are expected to describe typical features of stochastic motion associated with integrable equations. This paper is organized as follows. In the next section, the stochastic differential equation is presented. A detailed description of the derived stochastic processes is given in §3. In §4, numerical calculations of the stochastic equations for some kinds of traveling waves are pre- sented, and interpretations of the results of the numerical analyses are shown in §5. The final section is devoted to conclusions, including discussions on possible experimental realization. 2. Forward Kolmogorov Equation as a Conservation Law of the Sine-Gordon Equation We shall consider a one-dimensional Ito diffusion,12 which is a scalar stochastic variable Xt obeying the stochastic equation dXt = b(Xt, t) dt + σ(Xt, t) dBt, (2.1) where the real-valued functions b and σ are called the drift and diffusion coefficients, respec- tively. The one-dimensional Brownian motion is denoted Bt. The density function p(x, t) obeys the forward Kolmogorov equation ∂p ∂t = A∗p, (2.2a) where A∗is defined by A∗= −∂ ∂x [b(x, t) · ] + 1 2 ∂2 ∂x2  σ2(x, t) ·  . (2.2b) The operator A∗is called the conjugate operator of the generator A of eq. (2.1). Equation (2.2a) can be interpreted as a conservation law of p(x, t), with the flux −bp + (σ2p)x/2. Among the conservation laws of the SG e

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