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

T yp eset with jpsj2.cls < v er.1 .2 > Full P aper Sto c hastic Pro cess Asso ciated with T ra v eling W a v e Solutions of the Sine-Gordon Equation T etsu Y aj ima ∗ and Hideaki Uj ino 1 † Dep artment of Information Scienc e, F aculty of Engine ering, Utsunomiya University, 7-1-2 Y oto, Utsunomiya, T o chigi 321-858 5 1 Gunma National Col le ge of T e chnolo gy, 580 T orib a, Maeb ashi, Gunma 371-8530 Sto c hastic process es asso ciated with tra veling wa ve solutions of the sine- Gordon equatio n are presented. The str uc tur e of the for w ar d Kolmogor o v equa tion as a conser v ation law is essential in the construction of the sto c ha s tic pro cess a s well as the traveling w av e structure. The deriv ed stochastic pro cesses are analyzed numerically . An int e rpretation of the b ehaviors of the stochastic pro cesses is given in terms o f the equation of motion. KEYWORDS: stochastic process, sine-Gordon e quation, forwa rd Kolmog o ro v equation, co nser- vation law, traveling wave so lutions 1. In tro duction Researc h on sto c h astic p ro cesses has receiv ed m uch atten tion in mathematic al physic s recen tly . F or example, the sto chastic L¨ owner equation 1 attracts m uch interest regarding its connection to the conformal field theory , 2 as well as its original meaning as a sto c hastic ve r sion of the L¨ owner equation. 3 The sto c hastic cellular automaton an d asymmetric simple exclusion pro cess (ASEP), among others, are also attracting considerable in terest. The ASEP describ es a non-equilibrium pro cess of int eracting p articles, whose steady state h as b een exactly ob- tained. 4 Ev en the curr en t at equilibrium and the corresp ondin g ph ase d iagram were exactly present ed for th e ASEP. 5 Besides the ASEP , sev eral sto c hastic cellular automata are in ven ted for the successful analysis of traffic flo w s. 6 The sto c hastic p r ocess asso ciated with the Burgers equation is called Bur gers’ pro cess, 7 whic h is applied to studies of tur b ulence. 8 This pro cess is a t ypical example of nonlinear sto c hastic pro cesses, in other w ords , sto c h astic p ro cesses asso ciated with nonlinear equations. The time ev olution of the densit y function of Burgers’ pr ocess, for instance, is go verned by the Burgers equation u u − 2 uu x + u xx = 0, wh ic h is ind eed n onlinear. Su c h nonlinear sto c has- tic pro cesses are exp ected to b e an effectiv e app roac h to analyzing phenomena, where b oth nonlinear and sto c h astic effects are simultaneously predominant. Among man y kinds of nonlinear equations that describ e v arious physical systems, we h a v e a series of integrable equations. Exp ecting futur e applications to the analysis of phenomena ∗ ya jimat@is .utsunomiya-u.ac.j p † ujino@nat.gunma-ct.ac.jp 1/12 J. Phys. Soc . Jpn. Full P aper related to int egrable systems, we sh all present new nonlinear sto c hastic pr ocesses asso ciated with the sine-Gordon (SG) equation φ xt = m 2 sin φ, m : constan t , (1.1) along the same line as the Bu rgers pro cess. As is w ell kn o wn, the SG equation is an inte grab le nonlinear equation th at is su itable for d escribing the dyn amics of connected p en dulums u nder gra vit y, 9 the m otions of the d islocation of one-dimensional materials, 10 the propagation of optical w av es in on e-dimen sional space, 11 and so f orth. The SG equati on p ossesses a kink solution that h as lo calized str u cture, aside fr om p erio dic solutions. In this pap er, we shall consider p erio dic and one-kink solutions ha ving tra v eling wa v e structures, and w e sh all presen t sto c hastic pro cesses associated w ith these solutions. W e shall also p erf orm numerica l analyses of the deriv ed evo lu tion equations und er suitable initial cond itions in order to study their b eha viors. The results and their explanation given in the later sections are exp ected to describ e t ypical features of s to c h astic motion asso ciated w ith in tegrable equations. This pap er is organized as follo w s. In th e next section, the s to c h astic different ial equation is p resen ted. A detailed d escription of th e d eriv ed sto c hastic pro cesses is giv en in § 3. In § 4, n u merical calculations of the sto c hastic equations for some kinds of tra veling wa v es are p re- sen ted, and in terpretations of th e results of the n u merical analyses are shown in § 5. The final section is devo ted to conclusions, includ ing d iscussions on p ossible exp erimental r ealiz ation. 2. F orw ard Kolmogoro v Equation as a Conserv ation La w of the Sine-Gordon Equation W e shall consider a one-dimensional Ito diffu s ion, 12 whic h is a scalar sto c hastic v ariable X t ob eying the sto c hastic equation d X t = b ( X t , t ) d t + σ ( X t , t ) d B t , (2.1) where the real-v alued fun ctions b and σ are called the dr ift an d diffusion co efficien ts, resp ec- tiv ely . T he on e-dimen sional Bro wn ian m otio n is d enoted B t . The dens ity function p ( x, t ) ob eys the forw ard Kolmogoro v equation ∂ p ∂ t = A ∗ p, (2.2a) where A ∗ is defi n ed by A ∗ = − ∂ ∂ x [ b ( x, t ) · ] + 1 2 ∂ 2 ∂ x 2  σ 2 ( x, t ) ·  . (2.2b) The op erator A ∗ is called the conjugate op erator of the generator A of eq. (2.1). E quation (2.2a) can b e interpreted as a conserv ation law of p ( x, t ), with the fl ux − bp + ( σ 2 p ) x / 2. Among the conserv ation la ws of the SG equation, w e shall consid er ( φ 2 x ) t = ( − 2 m 2 cos φ ) x . (2.3) 2/12 J. Phys. Soc . Jpn. Full P aper Equations (2.2a) and (2 .3) ha v e the same form when we define p = N φ 2 x , where N is the normalization factor. Thus, w e can read − bφ 2 x + 1 2 ( σ 2 φ 2 x ) x = − 2 m 2 cos φ + A ( t ) , (2.4) where A ( t ) is an arbitrary fu n ction. When φ is a tra v eling w av e solution, φ = φ ( x − v t ) , v : constan t, (2.5) the dr ift an d d iffusion co efficien ts are separated into simple forms. S ince the deriv ativ es of th e v ariable are giv en as φ x = φ ′ , φ t = − v φ ′ , (2.6) and φ xt = − v φ ′′ , we find that − vφ ′′ = m 2 sin φ from eq. (1.1) . Multiplying b oth sid es by φ ′ and in tegrating by x , w e ha ve φ ′ 2 = φ 2 x = 2 m 2 v (cos φ − C ) , C : constan t . (2.7) Th us, w e can eliminate cos φ from (2.4) and derive bφ 2 x − 1 2 ( σ 2 φ 2 x ) x + A ( t ) = 2 m 2 C + v φ 2 x . (2.8) In tr o du cing a n ew function F = σ 2 φ 2 x , we can rewrite eq. (2.8) as bφ 2 x = 2 m 2 C + v φ 2 x + F x − A 2 . Because b and σ should b e b ounded for all x v alues ev en at the zeros of φ x , w e assume that F and A are give n by F ( x, t ) = α n + 1 φ n +1 x , A = 4 m 2 C. Then the co efficien ts of eq. (2.2a) are determin ed b y b = v + α 2 φ xx φ n − 2 x , σ 2 = α n + 1 φ n − 1 x , n ≥ 2 . (2.9) The ab ov e restriction to n prev ents b and σ from div erging. Hence, the sto c hastic d ifferen tial equation eq. (2.1) b ecomes d X t =  v + α 2 φ xx φ n − 2 x  d t + r α n + 1 φ ( n − 1) / 2 x d B t , (2.10) where v is the ve lo cit y of the tr av eling w av e, n ≥ 2, and α are constants. Equation (2.10) can b e considered to describ e the sto c hastic motion of a particle moving und er the influ ence of a tra veling wa v e solution of th e SG equation. When th e function φ x approac hes zero ex- p onen tially , suc h as the kink solution, eq. (2.10) is also w ell-defined for 1 ≤ n < 2, since φ xx φ − 1 x = (ln φ x ) x . 3/12 J. Phys. Soc . Jpn. Full P aper 3. T ra v eling W a ve Solutions of the Sine-Gordon Equation a nd t he Co efficien t of Asso ciated Sto c hastic Differen tial Equation W e shall consider th e co efficien ts in eq. (2.10) for sev eral t yp es of solutions φ of the SG equation (1 .1). I n eq. (2.7), the v elo cit y v is allo w ed only n ega tiv e v alues for C ≥ 1 and only p ositiv e v alues for C ≤ − 1, w hereas b oth p ositiv e and negativ e v v alues are allo w ed for | C | < 1. Among the tra ve lin g wa v e solutions, w e c ho ose those that hav e negativ e vel o cities: φ ( z ) = 2 arctan " r C − 1 C + 1 tn √ C + 1 mz √ − 2 v , r 2 C + 1 !# , C > 1 (3.1a) = 2 arctan    r 1 − C 1 + C " cn mz √ − v , r 1 + C 2 !# − 1    , | C | < 1 (3.1b) = 4 arctan h e mz / √ − v i , C = 1 . (3.1c) The functions cn( z , k ) and tn( z , k ) are Jacobian elliptic functions with a m o du lus k . W e note that the solution is p erio dic for C > 1 and | C | < 1, whereas the solution forms a solitary wa v e for C = 1. In the case asso ciate d with p erio dic solutions, their su pp orts need to b e compact for these cases or a p erio dic b oundary condition for x is required, since the normalization factor of the density f unction should b e finite. The solution eq. (3.1c) is the kink solution, and we can consider the sto c hastic equation on th e whole s pace. Hereafter, the parameter n is c hosen to b e either n = 1 or 5. In the former case, its sto c hastic effect is spatially uniform . In the latter case, on the other hand , its sto c hastic effect dep ends linearly on the densit y function of X t . In Figs. 1 and 2, w e sho w tw o examples of the drift co efficien ts σ in th e case of n = 1, and we displa y n o diffus ion co efficien ts b ecause they are constan ts. Figure 1 sh o ws a t ypical configuration of b at t = 0 for the p erio dic solution eq. (3.1a) with C > 1. The parameter C cannot hav e a v alue at | C | < 1, b ecause p erio dical zeros of φ x in the denominator of b yield singularities in b , and the sto c hastic pr ocess is not well-defined. Since b is defin ed by eq. (2.9), it is p erio dic o wing to the p erio dicit y of φ . In the case asso ciated with the kin k s olution, C = 1, the fi gure of b is giv en in Fig. 2. Th e coefficient b h as a lo calized s tructure around the cen ter of the kink. In Figs. 3 and 4, we sh all sho w tw o t ypical examples of the d rift and diffusion co efficien ts in the case of n = 5. Figure 3 shows the co efficien ts for p erio dic solutions C > 1. Both b and σ ha ve the same p erio d, whic h is half of the p erio d of the solution φ . On the other hand, in the case of | C | < 1, the app earances of b and σ are similar to those shown in the case of C > 1, but their p erio d is the same as that of φ . In Fig. 4, we show the co efficien ts of eq. (2.10) for the kink solution (3.1c). Both of the co efficien ts h a v e lo calized str uctures around the cent er of the kink. Sto c hastic b ehavio r of the v ariable X t app ears only in the vicinit y of the k in k, and the drift co efficien t c hanges in the 4/12 J. Phys. Soc . Jpn. Full P aper -4 -2 2 4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0.2 b x Fig. 1. D r ift co efficients b for the sto chastic differential e q uation eq. (2.10 ) with n = 1 asso ciated with the p erio dic solution (3.1 a) for C > 1. -2 -1 1 2 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0.2 b x Fig. 2. D r ift co efficients b for the sto chastic differential e q uation eq. (2.10 ) with n = 1 asso ciated with the kink solution (3.1c), derived by s etting C = 1. same neighborh oo d. The p erio dicities or lo caliz ed structures affect the b eha vior of the sample paths of X t . 4. Numerical Results of the Sto c hastic Equations In this section, w e shall analyze eq. (2.10) numerical ly and study the time evo lu tion of X t , asso ciate d w ith the solution eq. (3.1). F or all the simulatio n s, w e shall tak e m = 1. The order n , wh ic h d etermines the figu r e of the diffusion co efficien t, is selected to b e 1 and 5. W e n ote that all the s ample paths f or eac h case start fr om a fixed p oin t, although the initial p oin ts should b e distributed in accordance with the initial distribution p ( x, t = 0). Ho we ver, as we will see shortly , the effect of this omission remains only in the early stage an d disapp ears after the sto c hastic v ariable X t is captured by the region where the sto c h astic effect is sufficien tly strong. 5/12 J. Phys. Soc . Jpn. Full P aper -4 -2 2 4 -2 -1.5 -1 -0.5 0.5 1 b x (a) x σ -4 -2 2 4 0.2 0.4 0.6 0.8 1 (b) Fig. 3. Examples of th e co efficients of the stochastic differen tial equation (2.10) with n = 5, asso ciated with the s o lution under C > 1. (a) The drift co efficient b . (b) The diffusion co efficient σ . 4.1 Time evolution of the sto cha stic variable asso ciate d with the p erio dic solutions for n = 1 In Fig. 5, fiv e sample p aths for the solution eq. (3.1a) are sho w n together. The parameters in b , σ and φ are selected as C = 1 . 2 , α = 1 , v = 0 . 5 , (4.1) and the initial condition is set as X 0 = 0. The paths fl uctuate around their initial p oints in the early stage. Although th ey split into some groups of p aths, the v ariable X t c hanges along sev eral lines parallel to eac h other. The a ve r age ve lo cit y of the motion of X t is observ ed to b e appro ximately the same as that of th e p erio dic solution. 4.2 Time evolution of the sto chas tic variable asso ciate d with the kink solution for n = 1 Another example is a sto c hastic p ro cess asso ciated with the 1-kink solution eq. (3.1c). The co efficients are give n by b ( x, t ) = v − αm 2 p | v | tanh m ( x − v t ) p | v | . (4.2) 6/12 J. Phys. Soc . Jpn. Full P aper x b -3 -2 -1 1 2 3 -1.5 -1 -0.5 0.5 (a) x σ -3 -2 -1 1 2 3 0.2 0.4 0.6 0.8 1 (b) Fig. 4. Examples of the co efficien ts of the s to chastic different ial equa tion eq. (2.10) with n = 5, asso ciated with the kink solution ( C = 1 ). (a) Drift co efficient b . (b) Diffusion c o efficient σ . W e start with a set of p arameters, C = 1 , α = 1 , v = − 0 . 5 , (4.3) and the initial condition X 0 = − 3. The results of the five paths are sho w n in Fig. 6. As we can see f r om th e fi gu r e, all the paths show similar features. T he paths start from x = X 0 , linger around the initial p oint o wing to the effect of Bro wnian motion, and fi nally tra vel along a line on the X - t plane. Although some of the paths disp la y large d eviations, or sta y around simple v alues o ccasionally , the av erage incline of eac h path coincides with the v elo cit y of the p eaks of the p er io dic solution of φ . 4.3 Sample p aths in the c ase asso ciate d with the kink solution for n = 5 A t the end of this section, we shall sh o w the b eha vior of X t for n = 5. Because the explicit forms of the drif t co efficien t for p erio dic s olutions in Fig. 3(a) are similar to th e iteration of b asso ciate d with the kink solution in Fig. 4(a), w e chose the same parameters listed in eq. (4.3), with the initial condition X 0 = − 1 . 5. The b eha v iors of X t are sho w n in Fig. 7. The paths 7/12 J. Phys. Soc . Jpn. Full P aper t X t 2 4 6 8 10 12 -8 -6 -4 -2 0 2 Fig. 5. Sample paths for X t asso ciated with the p erio dic s o lution of the SG equa tion with n = 1 a nd C > 1. t X t 2 4 6 8 10 12 -7 -6 -5 -4 -3 -2 -1 0 Fig. 6. Sample paths for X t asso ciated with the kink solution of the SG e q uation with n = 1. initially draw curve s near a simple lin e with small fluctuations, and some of the paths un dergo nearly the same amoun t of p osition sh ifts and b egin oscillating with rather large amp litud es. In b oth of th e initial and final states, the a ve rage incline of the paths are ob s erv ed to b e the same. Although eac h path of X t fluctuates intensiv ely , and sh ifts from the initial line, its mean path is parallel to a set of lines in th e initial state. T h e mean p aths in the final state trace the line dra wn by th e cent er of the kink that passes through the origin, wh ile the paths in the initial state ha ve a common in tercept corresp ond ing to X 0 . T he time when a path falls into a state with a s tr ong oscillati on differs from time to time. 5. Asymptotic Beha viors of Sample P a ths In the n umerical results shown in the pr evious section, the b eha viors of the sample paths for X t in sufficien tly large time regio n s im p ly the existence of some deterministic pr ocess that driv es the mean mo v ements of the sto c hastic v ariable, and we shall roughly d escrib e this effect in terms of the equation of motion generated from eq. (2.10). When we omit the second 8/12 J. Phys. Soc . Jpn. Full P aper t X t 1 2 3 4 5 6 7 8 -5 -4 -3 -2 -1 0 Fig. 7. Sample paths for X t asso ciated with the kink solution of the SG e q uation with n = 5. term in eq. (2.1), we ha ve a non s tochasti c time ev olution la w of X t , and we can fin d that the relation d X t = b ( X t , t ) d t holds. This means that the v ariable X t represent s the p osition of a particle whose v elo cit y is b ( X t , t ). Since w e consider tra ve ling wa v e solutions of eq. (1.1) and the drift co efficien t given b y eq. (2.9), the dep end ences of b ( x, t ) on space and time v ariables ha ve the same structur e as those of φ , which means b = b ( x − v t ). Different iating the relation ˙ x = b ( x − v t ), we ha ve the relation ¨ x = d d t b ( x, t ) = b ′ · ( ˙ x − v ) = b ′ ˙ x − v b ′ . (5.1) When we consider a small region around a giv en p oint of x , and approxima te the graph of b b y its tangen t whose incline is β , eq. (5.1) redu ces to ¨ x = β ˙ x − β v . (5.2) This equation is the same as that for a particle with resistivit y prop ortional to its velocit y under a constan t force. F or n ega tive β , the motion results in a state of constan t v elo cit y v , whereas the v alue of x dive r ges for p ositiv e β . Th u s, there app ear attractors for x at p oin ts where b oth b ( x − v t ) = v and b ′ ( x − v t ) < 0 are satisfied. As an example of this explanation, let u s first roughly d escrib e the case asso ciated with the kink solution f or n = 1. The exact figure of th e dr ift co efficien t is giv en in Fig. 2. The co efficien t b is almost constan t in r egions far fr om the cente r of the kink , whereas it conspicuously decreases with x arou nd the cen ter. In far regions f rom the cen ter of the kink, the v ariable x mo ve s almost at a constan t v elo cit y to w ard the attractiv e region. After x is caugh t in the area where the time ev olution of the v ariable x is approximat ed by eq. (5.2), the path app roac hes the tra jectory of the cen ter of the kink, since th e attractiv e r egio n also tra v els at a velocit y v . Next, we shall explain the b eh a viors of th e paths for eac h case of th e sim ulation in d eta il. In the case of n = 1 and C > 1, there app ear p erio dic in terv als, whose cente r s are th e attractor of x . The initial p oint, X 0 = 0, is one of the repu lsing p oints, an d X t tra ve ls a wa y from the initial p oin t at an a v erage ve lo cit y of b ( X t , t ). The motion of X t sho ws a small fluctuation 9/12 J. Phys. Soc . Jpn. Full P aper due to the term σ d B t . Finally , eac h path X t is tr app ed in one of the attractors and mov es at an a v erage velocit y v . T he three p aths in Fig. 5 are lo cate d around the tra jectories of the attracto rs. Since the initial velocit y of X t has a negativ e v alue, b ( X 0 , 0) = v < 0, the num b er of paths in th e region of large X t tends to b e s maller than that for smaller X t . The other case of n = 1, the r esult asso ciated with the kink solution, can b e explained in the same wa y . Only a single attractor app ears in th is case, and all the p aths fall in to th e line x = v t in Fig. 6. At the initial p osition X 0 = − 3, b is app ro ximately b ≃ 0 . 2, and X t increases gradu ally with s to c h astic flu ctuatio n s. After the p oin t t ≃ 4, th e paths are caught b y the attractor n ear th e cen ter of the kink, and X t is dr iv en by the attractor. In the case n = 5 and C = 1, we should consider the prop ert y of the diffusion co efficien t as w ell as the drift co efficien t, b ecause σ is not a constan t. As in the case of n = 1, the p oin t b ( x − v t ) = 0 with b ′ < 0 pla ys the role as an attractor, bu t the intensit y of the sto c h astic effect is v alid only around the cent er of the kink. When the X 0 is sufficien tly far fr om the cen ter, X t sho ws n o sto c h astic effect and trav els at a velocit y v , and X t v aries w ith little sto c hastic oscillat ion. In the case sh o wn in Fig . 7, b is approximat ely b ≃ v = − 0 . 5 and σ is r ather small but cannot b e negligible near X 0 = − 1 . 5. T h us, X t initially mo ves at a v elo cit y v app ro ximately , with little fluctuation. After X t is caugh t b y the attractor, it shifts its tra jectory n ear the cen ter of the kink and b egins to oscillate at a large amplitude b ecause of large v alues of σ . The p oint wh en X t b egins to oscillate is determined p urely in a sto c hastic w ay , and this causes some of the paths to fluctuate extremely (Fig. 7), b ut others trav el in a rather deterministic wa y . 6. Discussions and Conclusions In this p ap er, w e ha ve presente d new sto c hastic equations, asso ciat ed with the tra ve lin g w av e solution of the SG equation. By virtue of the structure of φ , th e un kno wn function of the SG equation, as a trav eling wa v e, and the assumption that th e d ensit y fu nction p ( x, t ) is prop ortional to φ x , we hav e foun d that the co efficien ts of the sto c hastic equation eq. (2.1) can b e expressed in terms of the deriv ativ es of φ , b ecause the co efficien ts in the equation for X t should not h a v e sin gu larities. W e hav e also p erform ed n um erical analyses of the time ev olution of X t under some initial conditions. The paths dra wn b y X t sho w c haracteristic app earances, namely , the a verag e ˙ X t asymptotically coincides with the velocit y of the tra veling w av e. Th ese b eha viors of the paths can b e explained b y considering attractiv e p oin ts generated from the drift co efficien t, w hic h m ov es at the same v elo cit y as th e solution itself. Thus, the paths X t are observ ed to b e d riv en by the solution of th e SG equation. The results derived in th is p ap er are d istinctiv e in th at we can consider sto c hastic motions of a particle, wh ere the densit y function is expressed in terms of the tra veling wa v e solution of a solv able equation. Since the trav eling wa v e solutions of the S G equ atio n are well kno wn, this f eature enables us to deriv e v arious physical quanti ties related to sto chastic evolutio n s 10/12 J. Phys. Soc . Jpn. Full P aper accurately . In addition, we can exp ect sev eral app licat ions of the resu lt presen ted in this pap er. The b eha viors of solitons und er external effects suc h as impur ities 13, 14 or flu ctuatio n s in metamate r ials 15 ha ve attracted muc h in terest and ha ve b een studied in v arious systems. Although man y results ha v e b een derive d from these stud ies, these studies fo cus on effects that act on solitons; the sto c hastic effect caused by solitons has not b een studied so far. Applying the results in this pap er to the analysis of p henomena in real m ateria ls, we can find h o w solitons affect original systems as feedbac ks, an d ho w p articles excited in media tra vel while in teracting with solitons. Since the SG equation is related to man y p h ysical systems, the derive d mo del can p ro vide go o d descriptions of s u c h systems includ ing sto c hastic effects. F or example, the mechanica l mo del of the motion of connected rotators, whic h are applied forces d u e to twisted connection prop ortional to t wist angle under gra v ity, 9 or a dislo cation in sinusoidal p erio dic p oten tials 10 with heat flu ctuatio n , and the optical w av es in t wo-le vel atoms 11 with heat excitat ions, are suc h candid ates. Th erefore, we can exp ect th at the sto c h astic pro cesses derived in this p ap er can b e obs er ved exp erimenta lly . Su c h exp eriments in actual media or d er iv ations of sto c hastic equations from physical mo dels are c hallenging p roblems. A t the end of the p ap er, we w ou ld like to discuss extensions of the d eriv ed results. W e used the sp ecia l structure of tra v eling w av es to derive the stoc h astic d ifferen tial equati on, namely , their co efficien ts b and σ in explicit forms. Because of this r estrictio n , th e result is not applicable to the case asso ciated with a solution that has a time-dep endent p rofile suc h as a t wo-solit on solution. When we construct sto c h astic pro cesses f or general solutions, w e can consider sto c hastic p henomena asso ciate d with the SG equation u nder v arious conditions. Another extension is to consider s toc hastic equations for other in tegrable equations. F or exam- ple, the nonlinear S chr¨ odinger (NLS) equation is on e of the well- k n o wn mo dels in integ r able systems. The metho d of s tochasti c quan tization is known to b e effectiv e w ay for connecting the Schr¨ odinger equation with a sto c hastic differential equation, and the sto c hastic equation asso ciate d with the NLS equation can b e used to describ e the motion of particles in v arious physic al mo dels s u c h as plasma p h ysics or n onlinear optics. There are m an y applications to b e considered, and these are futur e p roblems to b e examined. Ac kno w le dgment This wo r k is partially su pp orted by a Grant-in-Aid for S cien tific Research (C) 13640 395 from the Japan So ciet y for the Promotion of Science (JSP S). 11/12 J. Phys. Soc . Jpn. Full P aper References 1) O . Schramm: Isra e l J. Math. 118 (2 000) 2 21. 2) M. Bauer a nd D. Bernard: Phys. Rep. 432 (200 6) 11 5 . 3) K . L¨ owner: Ma th. Ann. 89 (19 2 3) 103. 4) B . Derrida, M. R. Ev ans, V. Hakim, and V. Pasquier: J. Phys. A 26 (1 9 93) 1 4 93. 5) T . Sasamoto : J. Phys. A 32 (19 99) 7 1 09. 6) K . Nishina ri, M. F ukui, and A. Schadschneider: J. P h ys. A 3 7 (2004 ) 3101 . 7) H. Osada and S. Kotani: J. Math. So c. Jpn. 37 (1985) 275. 8) W. A. W oyczy ´ nsk i: Bur gers-KPZ T urbulenc e (Springe r -V erlag, Berlin Heidelb erg, 1 998). 9) A. C. Scott: Amer. J. P h ys. 37 (19 69) 52. 10) J. F renkel and T. K on trov a: J. Phys. USSR 1 (1939) 137. 11) G. L. Lamb, J r.: Rev. Mo d. Phys. 43 (1971) 99. 12) B. Øksendal: Sto chastic Differ ent ial Equations (Spr ing er, Berlin Heidelb erg, 2 003) 6th ed. 13) T. Iizuk a and M. W adati: J. Phys. So c. Jpn. 61 (1992) 434 4. 14) T. Iizuk a and M. W adati: J. Phys. So c. Jpn. 61 (1992) 307 7. 15) T. Tsurumi: J. Ph ys. Soc . Jpn. 77 (2008) 07400 6. 12/12