Some Nonlinear Equations with Double Solutions: Soliton and Chaos

Transcription

1 Some Noliear Equatios with Double Solutios: Solito a Chaos Yi-Fag Chag Departmet of Physics, Yua Uiversity, Kumig, 659, Chia ( yifagchag@hotmail.com) Abstract The fuametal characteristics of solito a chaos i oliear equatio are completely ifferet. But all oliear equatios with a solito solutio may erive chaos. While oly some equatios with a chaos solutio have a solito. The coitios of the two solutios are ifferet. Whe some parameters are certai costats, the solito is erive; while these parameters vary i a certai regio, the bifurcatio-chaos appears. It coects a chaotic cotrol probably. The ouble solutios correspo possibly to the wave-particle uality i quatum theory, a coect the ouble solutio theory of the oliear wave mechaics. Some oliear equatios possess solito a chaos, whose ew meaigs are iscusse briefly i mathematics, physics a particle theory. Key wors: oliear equatio; solito; chaos; uality; physical meaig. MSC: 5Q5; 65P; A; 7D5. Itrouctio It is well ow that some oliear equatios have the solito solutios [], while all oliear equatios have the chaos solutios. The solito a chaos possess may ifferet characteristic: a solito has the same shapes a velocities i a travellig process a eve if through a collisio, it has a efiite trace which is aalogous to a classical particle; the chaos is a uiversal pheomeo for various oliear systems, it escribes a orer a itrisic stochastic motio which appears to be irregular a cofuse. Therefore, they form usually two remarable aspects, respectively. But the both relatios are beig otice icreasigly. Abullaev summarize the yamical chaos of solitos a breathers for the sie-goro equatio, the oliear Schroiger equatio a the Toa chai, etc[]. Recetly, some iscusse the relatios amog chaos a the KV equatio [], the perturbe sie-goro equatio [], the complex Gizburg-Laau equatio [5], which have the solito solutio. Warbos has eve suggeste a iea: chaotic solitos (chaoitos) i the coservative systems [6]. We prove that some equatios have solito solutios a chaos solutios, a their coitios are ifferet [7]. The possible meaigs of the ouble solutios are iscusse here.. From Solito to Chaos..The oliear Schroiger equatio xx i t, () has a solito solutio [] e s h x uet i u sec [ ( )]exp[ ( )( x uct)], ()

2 where the variable x u t e. Let exp[ i u e ( x uct)] v, () the equatio () may become v [ C av v ] /, () where a ( u / ) ( u u / ). Whe C=, the solito solutio is v e e c / a sech. (5) From this let v a / si x, the equatio is x' a si x, (6) which has the chaos solutio. For a stable state whose eergy is H, if =-b<, the equatio will be '' H b, (7) whose itegral is ' ( C H b ) /. (8) Let C H / b, so ' b H ( ). b (9) Whe H / b, s H b th b ( C ). () It is the simplest solito with a bell shape. Usig a substitutio Hx / b for Eq. (9), a it become a ifferece equatio H. () It is a ow equatio, which has the chaos solutio, a its parameter etermie the bifurcatio-chaos is H /. Moreover, this equatio may iclue the Higgs equatio a the Gizburg-Laau equatio...the Dirac equatio has show the existece of a oegeerate, isolate, zero-eergy, c-umber solutio. Its solutios may be moopoles, yos a solitos [8,9,]. The oliear Dirac equatio is m l ( ). () It is the Heiseberg uifie equatio [] whe m=. The probability esity, x x x a, so [ l ( ) m ] [ l ( ) m ] x l ( )( ) l ( ) l ( ). () Let ( ) x u t, the equatio is

3 whose solutio is l ( ), () exp( l c ). (5) It is aalogous to a solito sice ( e c ) ( ) a / ( ). Usig a substitutio ( l x ) / 8 for Eq. (), the the correspoig ifferece equatio is l, (6) which has the chaos solutio a the parameter l /...For the Korteweg-e Vries equatio t x xxx, (7) let x ut, usig two orer itegrals, the ' ( ) / u C C. (8) For the solito solutio, the itegral costats shoul be C C, so Eq. (8) is ' ( u ) /, (9) whose solito solutio is u u sec h ( ). () Usig a substitutio u[ ( u / )( x) ] /, the ifferece equatio is u. () I a u 8 regio, the values of bifurcatio-chaos are u=,5,..., For the cubic Klei-Goro equatio m a, () let ( x ut) / u, so ( a m C) /, () If C=, a>, a m sec h ( m C ). () It is the simplest solito with a i shape. Moreover, a m ( ) / m is the same with Eq. (), so it has the chaos solutio. Further, all oliear equatios have chaos. (5). From Chaos to Solito.. The simplest ifferece equatio with the chaos solutio is. (6) It may correspo to a ifferetial equatio of first orer

4 x' x, (7) a a partial ifferetial equatio of seco orer xx tt a b. (8) It becomes to a oriary ifferetial equatio by a way o solito solutio, i.e., Eq. (7). Whe x / for Eq. (6), x th( C) (9) is amely a solito solutio. A bifurcatio-chaos regio, x [, ] / correspos to / For sigle stable solutio.75,i.e.,.5, so the coitio o x / is satisfie ecessarily i the regio, the solito ca exist. / While for two-brach regio,.5.75, i.e.,.5 four-braches to chaos,.5.5, i.e.,.89 ecessary coitio i which the solito appears is solutio a a part of two-brach regio. For the rest x / /.89; for a regio from.85. Sice x, the, it correspos to the regio of sigle / oes ot hol geerally...the logistic equatio F F( E F) () t correspos to a ifferece equatio E, () whose parameter is ( E) /. I the regio, two braches appear for E, four braches appear for E 5, etc., there is the chaos for E ( ) /.7. The equatio () has the solutio E F C exp( Et). () Whe t, Eq. () is aalogous to a solito sice F E / ( C) for t= a F E for t. It shows that the state will reach to stable at last as time icreases cotiuously...the ifferece equatio with a chaos solutio si( ) () correspos to a partial ifferetial equatio of seco orer xx tt si( ), () amely, the sie-goro equatio. It has the solito solutio x ut tg [exp( )]. (5) u Oly some chaos equatios have the solito solutios..discussio Further, we iscuss some possible meaigs of the ouble solutios possesse by these equatios briefly. I the mathematical aspect, whe some parameters are a certai costat, the solito is erive; while these parameters vary i a certai regio, the bifurcatio-chaos appears. Therefore,

5 the former correspos to a stable state, a the latter is a chageable process. I the physical aspect, Szebehely a McKezie iscusse that the three-boy problem i gravitatioal fiel possesses chaotic behaviors []. We prove that the gravitatioal wave is a type of oliear wave, a shoul be ifferet to electromagetic wave a have ew characteristics, for example, as solitos []. Perhaps, the ouble solutios are two ifferet states. These parameters are the orer parameters. These states ofte epe o the itegral costats, the bouary coitios a the iitial coitios. It explais agai that the solutios of the oliear equatios epe o the iitial values sesitively. It coects the chaos cotrol by a metho of parameter-cotrol. Whe we cotrol the orer parameter i the oliear system, chaos appear, isappear, sychroize [], eve a etermiatioal solito is prouce, for ifferet parameteric values. For example, the solito ca be erive i a propagatio of shallow water waves, but if the flow rate reaches a certai value, there will form the turbulece. Moreover, the solito solutio correspos to particle eve it may be a egeerate oublet with Fermi umber ( / ) [7,8]. The chaos solutio seems to correspo to the fiel, icluig the stochastic fiel. It will probably coect the ouble solutio theory of the e Broglie-Bohm oliear wave mechaics. I this case the wave-particle uality is a wave-particle sythesis, where the particle is escribe by the mobile sigularity of solito of the wave equatio [5]. The ouble solutios show a simultaeous existece o etermiism a probabilism quatitatively from a aspect i some oliear systems. The solito equatios a the chaos equatios have the wiely applie omais, i which above ouble solutios will show may meaig results or a goo eal of elightemet. Refereces.A.C.Scott,et al., Proc.of IEEE.6,(97)..F.Kh.Abullaev, Phys.Rep.79,(989)..Yu.N.Zaio, Sov.Tech.Phys.Lett.8,787(99)..G.Filatrella,et al., Phys.Lett.A78,8(99). 5.S.Popp,et al., Phys.Rev.Lett.7,88(99). 6.P.J.Werbos, Chaos, Solito, Fractals,,(99). 7.Yi-Fag Chag, Joural of Yua Uiversity. 6,8(). 8.R.Jaciw a C.Rebbi, Phys.Rev.D,98(976). 9.R.Jaciw a J.R.Schrieffer, Nucl.Phys.B9,5(98)..H.Grosse, Phys.Rep.,97(986)..W.Heiseberg, Rev.Mo.Phys.9,69(957)..V.Szebehely & R.McKezie, Celestial Mech.,(98)..Yi-Fag Chag, Apeiro,,(996)..C.K.Dua, S.S.Yag, Wali Mi, et al., Chaos,Solito,Fractals. 9,9(998). 5.L.e Broglie, No-liear Wave Mechaics. Elsevier, 96. 5