Exact solutions of the generalized Kuramoto Sivashinsky equation

Transcription

1 Exact solutions of the generalized Kuramoto Sivashinsky equation N.A. Kudryashov Department ofapplied Mathematics, Moscow Engineering Physics Institute, 31 Kashirskoe Avenue, Moscow , USSR Received 13 March 1990; revised manuscript received 1 May 1990; accepted for publication 3 May 1990 Communicated by A.R. Bishop Transformations for the solutions obtained by the Weiss Tabor Carnevale method are used for investigation of several classes of analytical solutions of the generalised Kuramoto Sivashinsky equation which is nonintegrableby means of the usual inverse scattering transform method. In 1983 Weiss, Tabor and Carnevale [1] offered some classes of analytical solutions of nonintegrable the Painlevé property expansion for partial differ- equations. In this connection the method was apential equations and demonstrated its efficiency for plied to get analytical solutions of the Burgers Korobtaining Bäcklund transformations and Lax pairs teweg de Vries equation and the Kawahara equafor some types of integrable equations (Korteweg tion [31.Studying the system of equations after de Vries, Boussinesq, Kadomtsev Petviashvili, etc.) substitution of eq. (2) into eq. (1) and setting the using the method of inverse scattering transform, expressions at the same orders of F(x, 1) equal to The essence ofthe Weiss Tabor Carnevale method zero provided some analytical solutions of the gencan be presented in several ways. For instance, we eralized Ginzburg Landau [4,5] and Kuramoto Sihave the nonlinear partial differential equation vashinsky (KS) equations [6 8]. ~ (1) The generalized Kuramoto Sivashinsky equation.. occupiesa prominent position in describing physical If the solution of the equation has the form processes in unstable systems [ This equation has the form u= ~ ~ (2) 1=0 u, +uu,. ~ (3) where p is an integer, then (1) is supposed to have where u (x, I) is a function characterising some physthe Painlevé property. In (2) u 1 are coefficients, and ical processes; a, /3 and y are constant coefficients; x F(x, t) is the new function. It is useful to note that is the coordinate, t is the time. the solutions u (x, 1) of the form (2) are single-val- Multiplying (3) by u (x, t) and integrating the obued about the movable, singularity manifolds tamed expression over x from ~ to ~ when the F(x, t)=o. condition lim~,+~u,,,~=ois fulfilled (m=o, 1,2, 3; At first the possibility of a solution of the initial u,,,,,, is the mth order derivative), we find equation of the form (2) was associated with the integrability of eq. (1) by the inverse scattering trans- ~ = a < y< ui.>, form,which corresponds to the idea ofablowitz, Ra- ôt mani and Segur [2] about the Painlevé property. Later it turned out that a certain class of differ- / 2 ~ f ~ 4 ential equations nonintegrable by the inverse scat- \ U mx/,j U mx tering transform can have the form (2). Applying the Weiss Tabor--Carnevale method one can find If a> O, y> O, eq. (4) shows that the term with the /90/s Elsevier Science Publishers By. (North-Holland) 287

2 second-order derivative in eq. (3) corresponds to (9) addition of energy to the system, the fourth-order derivative term characterises the dissipation of energy. and eq. (6) is transformed to the expression Substituting eq. (2) into eq. (3) at p=3 and setô 3lnF +l5a ö2lnf 2 ting the expressions at the same orders of F(x, t) u=60 equal to zero one can find the resonances of the solution u(x, 1) of the generalised KS equation from 8 ln F ox3 Ox +4~(l6 a2) +u 3. (10) the equation [13] 2 l3j+60)=0. (5) The primes of the variables in eqs. (9) and (10) are (j+ 1 )(J 6)(j omitted. After taking u(x, t)=v(~),~=x c On account of eq. (5) the KS equation is nonintegrable by the inverse scatteringtransform method, the speed ofthe wave), eq. (9) in the coordinate sys- 0t (c0 is tem of the travelling wave has the form Taking i~0 for j?~4and substituting eq. (2) into 2 +v~+av~+v~= 0, (11) eq. sembling (3) we theget Bäcklund the transformation of for solutions integrable re- q c0v+ ~v where q is the constant of integration. equations by means of the inverse scattering trans- Ifa=4, u form [3]: 3_ c1 =const, the transformation (10) can be written as 82 2lnF (12) ~ x v=c d 1+R+R~, R=60 nf +4~(16a fl 2/y)81 ~U 3, (6) Substitutingeq. (12) into eq. (11) we arrive at x The remaining set of equations containing u3(x, 1) 1R)Z+~JR~Zd~=0, (13) Z~+5Z ~+(5 A 5 and derivatives F~,F~,..., ~ in the general case where happens to be overdetermined, but joint for several 2 +AR ~(1 A2), classes stitution of[14], functions. In particular the following sub- A=~(c Z R ~ +~R 1 c0 + 1). (14) U3 =C1 =const, F(x, t)=c2+c3 exp(kx+wt), (7) over ~, we get Multiplying (14) by R~and integrating the result in this overdetermined set of functions suggests ~ 2 J ~ lutions of the generalised KS equation in the form of 2 )R ~D. (15) solitary binations waves of theand parameters kinks under of eq. the(3): following corn- The 5(1 constant A D is connected with the constants c 0, = = ± 12 ± 16 ±4. (8) D=c1(c0 ~c1)+~(a 5)(l A and q by the relation 2) q. Let us demonstrate that the transformation (6) can be directly used to obtain exact solutions of the generalised Kuramoto Sivashinsky equation. Forthis purpose we normalise eqs. (3) and (6) by setting It follows from eq. (13) that each solution of the equation R~+~R3+AR2 5(l A2)R ~D=0, (16) after substitution in eq. (12), leads to a solution of u=ct ~ u, x=\/~7 ~, the generalised Kuramoto Sivashinsky equation in the case u=4. = (y/a2 ) t. a= pi~j~. Denoting the real values of the relation Thuseq. (3)hastheform R3+15AR2 75(l A2)R 50D=0 (17) 288

3 as R 1, R2 and R3 with R1 ~R2~R3, the solution of Since eq. (11) is invariant under the transforeq. (16) is expressed by means of the elliptic Jacobi mations function R( ~)= R2 + (R1 R2) cn 2 ( ~,,,/~(R 1 R3), S), v + v, c0 + c0, ~ a ~ o~ (i) the solutions of the KS equation at a= 4 can be R1 R2 R1 R3 (18) obtained (18). taking into account expressions (12) and At a= l2/~/~7the transformation (10) for the so- The solution (18) is a periodic wave. At c0 = 6 lutions of the KS equation in the coordinate system (c1=d=q=0, A= 1) and c0= 9 (c1=d=0, of the travelling wave can be written in the form A=2, q=~) the solution (18) is transformed into a solitary wave: v=60 exp( 2~/~/~) ~[exp(~/v~) R(~)=l5ch 2(~). (19) d I dlnf \1 In the general case, the solution of eq. (16) and X exp(~/\/~) d~)] +v 3. (22) the solution of the KS equation can be expressed by means of elliptic Weierstrass functions. The solulions of the generalized KS equation canbe obtained from the formula Let us define the new variable Then the transformation (22) can be presented in V=ci+R+~_~~ [50D+75(l_A2)R the form 21nF (23) v c 2 R3] 2. (20) 0~ dr 60d 1 47~/~d0, R 15AR It is easily seen using eq. (18) that these solutions Substituting eq. (23) into eq. (11) at a= l2/~/~7 are also periodic (cnoidal) waves and their ampli- and using the new variables, tudes can be found as the real roots of the cubic 0 dv 12 / dv d 2v \ equation (17). q c 0v+~v 2 ~~ ~ do d02) To investigate whether the solitary wave obtained after substituting (19) into (20), d3v +3O2~ +o~)=0, (24) v=l5ch2(~)[l th(~)], (21) we find, as in the case a=4, that each solution R(0) is a soliton, one can arrange the numerical simula- of the equation tion of its interaction with the other solutions of the KS equation by means ofa finite difference method. R ~+~~(R B)3=0, (25) The calculations indicate that the solution (21) (where B is an arbitrary constant) under the does not change in time and has elastic interaction condition with other solutions. Therefore the solitary wave (21) satisfying the KS equation from a classical point of q = Ci (c 0 ~ c~) view is a soliton [15]. A simulation of the evolution ofan initial perturbation, described by eq. (30) with leads after its substitution in eq. (23) to a solution periodic boundary conditions, demonstrated that the of the KS equation in the form of a kink, initial perturbation irrespective of its form trans- 1 2oo~ forms peculiarity in time of this in an autostructure. at a=4 The is that characteristic it formed V C1 47~J~(c3+O)3 of a soliton gas consisting of solitary waves de- where c 3 is an arbitrary constant. (26) scribed by eq. (21) [161. Because eq. (11) is invariant under the transfor- 289

4 mation (i) these solutions of the KS equation can be obtained in an evident way at i= l2/\/ ~7. v=c 1 + At a= l6/~i the transformation of the solutions l9if~~ 0~R, ofthe KS equation (10) can be presented in the form d 2 ln F (32) 0=exp( i~/,,,/i~),r=50 do2 v=60 exp( 3~/~)~[exP(2~/~) After substitution of eq. (32) into eq. (11), using d / xp(~/~/5~) d ln F\1 the new variables, and taking account of X ~e d~)j +v 3. (27) = 30i q=c1(c0 ~c1), c0 c1 Setting the new variable 19~/i~ we find that each solution of the equation O=exp( _~/~Ji~) we find that the transformation (27) for R ~ +~R 3 = 0 v 3=c~can be written as gives a solution of the KS equation at ty=0 by formula (32), 1 / 3dR _02R), 60i ~ v_ c ~73\ do u=ct l9~((c3+o)2 (C +0)3) R= d 2 do2 ln F (28) Setting in (33) Substituting this expression into eq. (11) using the O=exp( i~/,~/t~), c 3 = 1, (33) new variables we obtain we arrive at the solution of eq. (3) given in refs. qc + 2) 2~ [6,8]. Exact solutions of the generalised Kuramoto 0v+2v 2.,Ji~dO 0 dv 73~/~i 16 (o~+o2d do Sivashinsky equation can also be obtained if we re- 1 ~ +302~_~+O~~ ~ 0, (29) write the transformation (6) taking into account that 73..,/7i~\ do do3) u 4(x, t) is a function of the derivatives of F(x, t) we find that all the solutions of the equation with respect to x and 1 [141, 2 82 F2 R~+~R30 (30) ~ 8~ F under the condition +~(l6 a) 8 ln F2 F~ 90 q=c~(co ~c 1), c0=c1 73~ + ~a{f;x}+15~_{f;x}+,i~a(7_~a2). (34) satisfy eq. (29) and consequently formula (28) gives the solution ofthe KS equation for a= 1 6/~/5i.They Substituting eq. (34) into eq. (6) and setting the have the form of kinks, expressions at the same powers of F(x, 1) equal to 60 / 92 20~ \ zero, we obtain that F(x, 1) satisfies the obtained set v=c1 73.J~I~ (c3 + 9)2 + ( + 9)3). (31) of equations if Sh={F;x}=m, F~/F~= c0, (35) The exact solution of the KS equation at a= 0 can be found analogously. For this case the transfor- where c0 is an arbitrary constant, Sh is the Schwarz mation of the solution (10) can be presented in the derivative. The constants m can be defined with reform spect to a, 290

5 The author thanks V.A. Arutyunov and A.L. Gay- I a=4: m 1 = ~ _~, rilov for their help in this investigation. 0=0: m1=3~, m = 2 ~ a= l2/,,~/~:m References 1 = ~94, a= 16/~J~: m1 = 146~ (36) [1]J. Weiss, M. Tabor and G. Carnevale, J. Math. Phys. 24 The general solution of eq. (35) can be written in (1983) 522. [2] M.T. Ablowitz, A. Ramani and H. Segur, J. Math. Phys. 21 the form [8] (1980) 715. [3] N.A. Kudryashov, Doki. Akad. Nauk SSSR 300 (1988) 342 F(x,t)= ay, +by2 ae~bd, (37) [Soy. Phys.Dokl. 33(1988) 336]. dy~+ey2 [4]F.CarielloandM.Tabor,PhysicaD39 (1989) 77. [5] N.A. Kudryashov, Soy. Math. Sim. 1 (1989) 151. where a, b, dand e are arbitrary constants and Yi and [6] N.A. Kudryashov, Priki. Mat. Mekh. 52 (1988) 465 [J. Y2 are two independent solutions of the linear AppI. Math. Mech. 52 (1988) 361]. equation [7] N.A. Kudryashov, Soy. Math. Sim. 1 (1989)57. [8] R. Conte and M. Musette, J. Phys. A 22 (1989)169. y + ~my= 0. [9] B.J. Cohen, J.A. Kromes, W.M. Tang and M.N. Rosenbluth, Nucl. Fusion 16 (1976) 971. The exact solutions of the generalised Kuramoto [10] Y. Kuramoto and T. Tsuzuki, Prog. Theor. Phys. 55(1976) Sivashinsky equation can be obtained after substi tuting eq. (37) into eq. (34). [11]G.I. Sivashinsky, Physica D 4 (1982) 227. [12]I. Topper and T. Karvahara, J. Phys. Soc. Japan 44 (1977) In this paper analytical solutions of the KS equa tion have been obtained with the help of the trans- [13] N.A. Kudryashov, Preprint Moscow Phys. Eng. Inst., N007- formations (10) allowing one to obtain the Pain- 87(1987). levé-type equations (16), (25), (30). It is important [14] R. Hirota, J. Phys. Soc. Japan 73 (1974)1498. to note that all these equations have resonances at [l5]n.j.zabusk~ andm.d.kruskal,phys.rev.lett. 15(1965) and j=6 [17],corresponding to resonances [16]T.Kawahara, Phys. Rev. Lett. 51(1983) 381. (5) of the initial KS equation (3). [17] A.C. Newell, M. Tabor and Y.B. Zeng, Physica D 29 (1987)1. 291