TWO-SOLITON SOLUTIONS OF THE KADOMTSEV- PETVIASHVILI EQUATION

TWO-SOLITON SOLUTIONS OF THE KADOMTSEV-PETVIASHVILI EQUATION 23 Jurnal Teknolog, 44(C) Jun 26: 23 32 Unverst Teknolog Malaysa TWO-SOLITON SOLUTIONS OF THE KADOMTSEV- PETVIASHVILI EQUATION TIONG WEI KING 1, ONG CHEE TIONG 2 & MUKHETA ISA 3 Abstract. Several fndngs on solton solutons generated by the Kadomtsev-Petvashvl (KP) equaton were dscussed n ths paper. Ths equaton s a two dmensonal of the Korteweg-de Vres (KdV) equaton. Tradtonal group-theoretcal approach can generate analytc soluton of soltons because KP equaton has nfntely many conservaton laws. By usng Hrota Blnear method, we show va computer smulaton how two soltons soluton of KP equaton produces trad, quadruplet and a non-resonance structures n solton nteractons. Keywords: equatons Solton, Hrota Blnear method, Korteweg-de Vres and Kadomtsev-Petvashvl Abstrak. Beberapa keputusan tentang penjanaan penyelesaan solton oleh persamaan Kadomtsev-Petvashvl akan dbncangkan dalam kertas n. Kaedah teor kumpulan mampu memberkan penyelesaan secara analtk kerana persamaan KP mempunya ketakterhnggaan banyaknya hukum keabadan. Dengan kaedah Blnear Hrota, dtunjukkan melalu smulas berkomputer bagamana penyelesaan dua solton persamaan KP mampu menghaslkan strukturstruktur trad, kuadruplet dan struktur tak beresonan dalam nteraks solton. Kata kunc: Petvashvl Solton, kaedah Blnear Hrota, persamaan Kortewegde Vres dan Kadomtsev- 1. INTRODUCTION There are many examples of resonance n physcs. However, resonance n solton nteracton s an nterestng phenomena. In ths paper, we wll use the Kadomtsev- Petvashvl (KP) equaton to model the two-solton nteractons as n Ong [1] and also n Anker and Freeman [2]. In partcular, the KP equaton s the two-dmensonal form of the Korteweg-de Vres (KdV) equaton. Mles [3, 4] dscovered that n the nteracton of two soltons, the nteracton regon between the ncdent soltons and the centered-shfted soltons after nteracton s essentally tself a sngle solton whch leads to a very smple conceptual pcture of the nteracton process. Ths nteractng solton s the resonant solton assocated wth the resultng soltary waves form by two ncdent soltons. 1,2&3 Department of Mathematcs, Unverst Teknolog Malaysa, 813 Skuda, Malaysa JTJUN44C[3].pmd 23

24 TIONG WEI KING, ONG CHEE TIONG & MUKHETA ISA 2. TWO-SOLITON SOLUTIONS- KP EQUATION Generally, the KP equaton can be wrtten as: 3 2 u u u u + 6u + + 3 = 3 2 x t x y y and can be smply wrtten as: ( ) u + 6uu + u + 3u = (2) t x xxx x yy We wll consder the lnearzed form of Equaton (2), then we have plane-wave solutons whose phase varable kx + my wt satsfed the dsperson relaton 3m 3 w = k (3) k N solton solutons for Equaton (2) had been solved by Satsuma [] usng Hrota Blnear method [6] as: 2 u( x, y, t) = 2 ln f 2 x 2 (1) (4) wth the functon f gven by f where and wth 2 f xx f fx = 2 2 f () a = δj + exp( η ) + l n (6) j δ j 1 = j, = j, η = kx + my wt (7) k = l + n, (8) m = n 2 l 2 (9) w 3 2 m = k k 3 () JTJUN44C[3].pmd 24

TWO-SOLITON SOLUTIONS OF THE KADOMTSEV-PETVIASHVILI EQUATION 2 and Thus, where a ε = for, j = 1 and 2. l + n ( ) ( ) ( ) (11) f = 1 + ε exp η + ε exp η + A ε ε exp η + η (12) 1 1 2 2 12 1 2 1 2 A 12 = ( n1 n2)( l1 l2) ( l + n )( l + n ) 1 2 2 1 (13) By usng Mcrosoft Vsual C++ Professonal Edton, we wll generate a computer programmng to portray the 2-solton solutons of Equaton (2) and later produce varous Encapsulated Postscrpt (eps) fles. These results were demonstrated n computer smulaton secton. 3. RESONANCES IN THE KP EQUATION There are three types of resonances n KP soltons nteracton whch are full resonance, partal resonance and non-resonance as n Ong [1]. Resonance wll only occurs when the value of A 12 s very close to zero. Therefore, the values of n 1, n 2, l 1 and l 2 wll determne the resonance structure. If we fx the values of l 1 and l 2, then n 1 and n 2 wll determne the value of A 12. We wll show ths effect n the followng secton. 3.1 Full Resonance We need to have a stuaton where n 1 = n 2 or l 1 = l 21 so that ln A 12 tends to or smply A 12 wll be zero. Therefore, we shall fx the values of l 1 and l 2 but l 1 l 2 and consder the case where n 1 = n 2. Therefore, Equaton (12) wll become: ( ) () 1 ( 2) ( 3) ( ) f = 1 + ε1 exp η1 + ε2 exp η2 Any combnatons between (1), (2) and (3) wll form a new solton. We can observe that the frst solton, S 1, second solton, S 2 and the resonant solton, S 23 can be represented by the functon f as below. Solton 12 : = 1+ ε 1 exp( η 1 ) Solton 13 : = 1+ ε 2 exp( η 2 ) Solton 23 : f ε exp( η ) ε exp( η ) (14) f () f (16) = + (17) 1 1 2 2 JTJUN44C[3].pmd 2

26 TIONG WEI KING, ONG CHEE TIONG & MUKHETA ISA When the term ln A 12 tends to, the length of resonant solton wll ncrease and ths wll form a trad as shown n Fgure 1. S 2 S 23 S 1 Fgure 1 A trad 3.2 Partal Resonance In ths case, we wll consder the case of n 1 n 2 so that the values of A 12 wll be so close to zero. Thus we have Equaton (12) agan wth A 12. The value of A 12 wll determne the length of resonant solton S 23. In ths case, we wll have an nteracton pattern called quadruplet as shown n Fgure 2, whch represents the partal resonant structure. S 1 * S 2 S 23 S 2 * S 1 Fgure 2 A quadruplet From Fgure 2, when S 1 and S 2 come nto nteracton, t wll produce a resonant solton S 23 and later break up agan to form S* 1 and S* 2 whch are actually solton S 1 and S 2 respectvely, but of course wth some phase shft. 3.3 Non-Resonance For the non-resonance case, we wll take n 1 n 2 that A 12 1. Ths means that the lne S 23 no longer exsts. Ths phenomena s called non-resonance nteractons. In ths cases, S 1 wll be centered along the lne k 1 x + m 1 y = whereas S 2 wll be centered along the lne k 2 x + m 2 y = wth JTJUN44C[3].pmd 26

TWO-SOLITON SOLUTIONS OF THE KADOMTSEV-PETVIASHVILI EQUATION 27 β β m m tan α =, where β =, β =. 1 + 1 2 1 2 1 2 ββ 1 2 k1 k2 In ths case, α s the angle of nteracton between S 1 and S 2 and β 1, β 2 are the tangents of the lnes respectvely. In ths case, we wll have an nteracton pattern called cross as shown n Fgure 3, whch represents the non-resonance structure. (18) S 2 S 1 * α S 1 S 2 * Fgure 3 A cross 4. COMPUTER SIMULATIONS By usng Mcrosoft Vsual C++ Professonal Edton, we wll generate a computer programmng to portray the 2-solton solutons of Equaton (2) and later produce varous graphcal outputs by usng GnuPlot to produce varous Encapsulated Postscrpt (eps) fles. A smple computer program wrtten to portray the 2-solton solutons of the KP equaton s gven n the followng subsecton. 4.1 Computer Program for 2-KP Soltons #nclude < ostream.h > #nclude < fstream.h > #nclude < math.h > #defne N vod man( ) { double k1, k2, A12, t, f, f1, f2, al, a2; double x[n + 1], u[n + 1], y; double n1, n2, l1, l2, ml, m2, wl, w2, Eta1, Eta2, Epslon1, Epslon2; nt ; ofstream ofp; ofp.open( 1.out,os::out); n1 = 3; n2 = 3 + exp( ); l1 = 2; l2 = 3; t = 1; k1 = l1 + n1; k2 = l2 + n2; a1 = 1; JTJUN44C[3].pmd 27

28 TIONG WEI KING, ONG CHEE TIONG & MUKHETA ISA a2 = 1.3; m1 = n1*n1 l1*l1; m2 = n2*n2 l2*l2; w1 = ((3*ml*ml/k1) + (k1*k1*k1)); w2 = ((3*m2*m2/k2) + (k2*k2*k2)); Epslon1 = a1/k1; Epslon2 = a2/k2; A12 = ((l1 l2)*(n1 n2))/((l2 + n1)*(l1 + n2)); y = 4; whle (y< = 4) { x[] = ; for( = ; < = N; ++) { Eta1 = k1*x[] + m1*y w1*t; Eta2 = k2*x[] + m2*y w2*t; f = 1 + Epslon1*exp(Etal) + Epslon2*exp(Eta2) + Epslon1*Epslon2*A12*exp(Eta1 + Eta2); f1 = k1*epslon1*exp(eta1) + k2*epslon2*exp(eta2) + (k1 + k2)*epslon1*epslon2*a12*exp(eta1 + Eta2); f2 = k1*k1*epslon1*exp(eta1) + k2*k2*epslon2*exp(eta2) + (k1 + k2)*(k1 + k2)*a12*epslon1*epslon2*exp(eta1 + Eta2); u[] = 2*((f*f2 f1*f1*)/(f*f)); f(u[].> u[] +.<) ofp << x[] << << y << << u[] << << endl; x[ + 1] = x[] +.16; } y+ = 1; } ofp.close(); } 4.2 Computer Smulatons Full Resonance For the full resonance, the values of n 1, n 2, l 1 and l 2 were chosen as below: n 1 = 3, n 2 = 3, l 1 = 2, l 2 = 3, (A 12 = ) (19) By usng these values, the computer smulaton produces trad as shown n Fgure 4. JTJUN44C[3].pmd 28

TWO-SOLITON SOLUTIONS OF THE KADOMTSEV-PETVIASHVILI EQUATION 29-4 -3-2 - Fgure 4 2 3 3D plot of a trad 4 3 2 Space, y - -2-3 -4 4 4.3 Computer Smulatons Partal Resonance On the other hand, f we choose n 1 = 3, n 2 = 3 + exp( p), l 1 = 2, l 2 = 3, (A 12 ) (2) wth p = 2 for the partal resonance, we wll get a quadruplet representng the partal resonant solton. The bgger the value of p n n 2, the longer wll be the resonant solton S 23. - - (a) p = - - Space, y - - (b) p = 2 - - Space, y JTJUN44C[3].pmd 29

3 TIONG WEI KING, ONG CHEE TIONG & MUKHETA ISA - - (c) p = 3 - - Space, y Fgure A set of quadruplet wth dfferent values of p 4.4 Computer Smulatons Non-Resonance Whle for the non-resonance, we have n 1 = 1, n 2 = 3, l 1 = 2, l 2 = 3, (A 12 = 1) (21) The values of l 1 and l 2 are specfcally chosen to ensure that the ampltude of soltons are postve. The ampltude of frst solton S 1 and second solton S 2 were 1 2 determned by k 2 wth = 1, 2 and k as defned n Equaton (8). The value of k must be postve to produce postve ampltude of the soltons. Wth ths condtons, we wll have a cross as shown n Fgure 6. 2-2 -2 - - 2 Space, y - - -2 2-2 2 Fgure 6 3D plot of a cross JTJUN44C[3].pmd 3

TWO-SOLITON SOLUTIONS OF THE KADOMTSEV-PETVIASHVILI EQUATION 31. CONCLUSION From the computer smulatons, we observed that there were three dfferent resonance structures n two-solton solutons of the KP equaton whch are full resonance, partal resonance and non-resonance structures. In partal resonance, the value of A 12 played an mportant role n determnng the length of the resonant solton S 23. In future, we ntend to study the nteractons patterns of three-solton solutons n the KP equaton and try to model some of the physcal phenomena by usng the KP equaton n marne ocean basn. REFERENCES [1] Ong, C. T. 1993. Varous Aspects of Soltons Interactons. M.Sc. Thess (Appled Mathematcs). Unverst Teknolog Malaysa. [2] Anker, D., and N. C. Freeman. 1978. Interpretaton of Three-Solton Interactons n Terms of Resonant Trads. Journal of Flud Mech. 87(1): 17-31. [3] Mles, J. W. 1977. Oblquely Interactng Soltary Waves. Journal of Flud Mech. 79(1): 7-169. [4] Mles, J. W. 1977. Resonantly Interactng Soltary Waves. Journal of Flud Mech. 79(1): 171-179. [] Satsuma, J. 1976. N-Solton Soluton of the Two-Dmensonal Korteweg-de Vres Equaton. Journal of Physcal Socety Japan. 4(1): 286-29. [6] Hrota, R. 1971. Exact Soluton of the Korteweg-de Vres Equaton for Multple Collsons of Soltons. Phys. Rev. Lett. 27(18): 1192-148. JTJUN44C[3].pmd 31