Exact Periodic Solitary Wave and Double Periodic Wave Solutions for the (2+1)-Dimensional Korteweg-de Vries Equation*

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1 Exact Periodic Solitary Wave Double Periodic Wave Solutions for the (+)-Dimensional Korteweg-de Vries Equation* Changfu Liu a Zhengde Dai b a Department of Mathematics Physics Wenshan University Wenshan China b School of Mathematics Statistics Yunnan University Kunming China Reprint requests to C. L.; chfuliu@yahoo.com.cn Z. Naturforsch. 64a (009); received October / revised December A new technique the extended ansatz function method is proposed to seek periodic solitary wave solutions of integrable systems. Exact periodic solitary wave solutions for the (+)-dimensional Korteweg-de Vries (KdV) equation are obtained by using this technique. By using the trial function method Jacobi elliptic function double periodic solutions are also constructed for this equation. This result shows that there exist periodic solitary waves in the different directions for the (+)- dimensional KdV equation. Key words: Two-Dimensional KdV Equation; Extended Ansatz Function Method; Periodic Solitary Wave Solution; Double Periodic Solution.. Introduction Boiti et al. derived the (+)-dimensional KdV equation [] u t + (uv) x + u xxx = 0 u x v y = 0 () by using the idea of the weak Lax pair. () is also called Boiti-Leon-Manna-Pempinelli equation []. For v = u y = x () reads as the ubiquitous KdV equation in dimensionless variables u t + 6uu x + u xxx = 0. () Thus () is a generalisation of the KdV equation. The rich dromion structures localized structures are revealed by Lou et al. for () [ 4] Wazwaz [5]. In the recent years many authors have applied the Auto-Bäcklund transformation method [6] the F- expansion method [7] found abundant exact solutions for many nonlinear partial differential equations. But these obtained structures do not include some periodic solitary wave solutions which is a new type of soliton solutions. Recently Dai et al. introduced a new technique the extended homoclinic test technique which is used Project supported by Chinese Natural Science Foundation (No ) Yunnan Natural Science Foundation (No. 004A000M) Yunnan Educational Science Foundation(No. 06Y04A). for seeking periodic solitary wave solutions of integrable equations obtained periodic solitary wave solutions for integrable equations [8 ]. In this paper we extend the technique mentioned above introduce a new ansatz function f (xyt)=b e (Px+Qy+Wt) + b cos(kx + Ly + Vt)+b e (Px+Qy+Wt) which is different from the function f (xyt) = + a p(xt)+ a q(yt)+ap(xt)q(yt) in [4] consider (). Then we use the trial function method [] to look for the double periodic wave solutions of (). Exact periodic solitary wave double periodic wave solutions are obtained for (). This result shows that there exists periodic solitary waves in the different directions for (). To our knowledge some periodic solitary wave double periodic wave solutions of the (+)-dimensional KdV equation are new solutions up to now. The rest of this paper is organized as follows: in Section a procedure for solving the (+)-dimensional KdV equation is given. Finally some conclusions follow.. Exact Periodic Solitary Wave Double Periodic Wave Solutions for the (+)-dimensional KdV Equation Method. We transform () into the bilinear form D y (D t + D x) f f = 0 () / 09 / $ c 009 Verlag der Zeitschrift für Naturforschung Tübingen

2 60 C. Liu Z. Dai Periodic Wave Solutions for (+)-Dimensional KdV Equation by the transformation u = ln( fxyt)) xy v = ln( f (xyt)) xx (4) where the bilinear operator D m x Dn t is defined as ( D m x Dn t a b = x ) m ( x t ) n t (5) a(xt)b(x t ) x=x t=t. Then () can be rewritten as ( f yt f f y f t + f xxxy f + f xx f xy f xxy f x f xxx f y ) = 0. (6) We express the function f in the form f (xyt)=b e (Px+Qy+Wt) + b cos(kx+ Ly +Vt) (7) + b e (Px+QY +Wt). Substituting (7) into (6) we obtain an algebraic equation for e j(px+qy +Wt) sin(kx+ Ly + vt)cos(kx+ Ly + vt). Equating all therein coefficients to zero ( j = 0) yields a set of algebraic equations for b b b PQWKL V: 6b Qb P + 4b Qb W b LV + 4b LK = 0 b Lb P + b Qb K b Qb V b Lb W + b Pb LK b Kb P Q = 0 b b QW b b VL+ b P Qb + b LK b b K b PQ b P b LK = 0 b Qb K + b Lb P + b Lb W + b Qb V b Pb LK + b Kb P Q = 0 b P Qb + b LK b + b b QW b b VL b K b PQ b P b LK = 0. (8) Solving this system of algebraic equations with the aid of Maple we can obtain the following results. Case. P = KLb W = b LK (48b Q b L b 4 ) 4 b Qb 64 b Q b V = 6 K = K L = L Q = Q b = b b = b b = b K (6b Q b L b 4 ) b Q b (9) where K L Q b b b are free parameters. Setting b = ±b wehave f (xyt)=b cosh(px + Qy +Wt)+b cos(kx+ Ly +Vt) f (xyt)=b sinh(px + Qy +Wt)+b cos(kx+ Ly +Vt). (0) Substituting (0) into (4) yields periodic soliton solutions of () as follows: { 4b u (xyt)= PQ b LK + b b (PQ KL)cosh(Px + Qy +Wt)cos(Kx+ Ly +Vt) (b cosh(px + Qy +Wt)+b cos(kx+ Ly +Vt)) + b b (PL + KQ)sinh(Px + Qy +Wt)sin(Kx+ Ly +Vt) (b cosh(px + Qy +Wt)+b cos(kx+ Ly +Vt)) { 4b v (xyt)= P b K + b b (P K )cosh(px + Qy +Wt)cos(Kx+ Ly +Vt) (b cosh(px + Qy +Wt)+b cos(kx+ Ly +Vt)) + 4b b PK sinh(px + Qy +Wt)sin(Kx+ Ly +Vt) (b cosh(px + Qy +Wt)+b cos(kx+ Ly +Vt)) () { 4b u (xyt)= PQ b LK + b b (PQ KL)sinh(Px + Qy +Wt)cos(Kx+ Ly +Vt) (b sinh(px + Qy +Wt)+b cos(kx+ Ly +Vt)) + b b (PL + KQ)cosh(Px + Qy +Wt)sin(Kx+ Ly +Vt) (b sinh(px + Qy +Wt)+b cos(kx+ Ly +Vt))

3 C. Liu Z. Dai Periodic Wave Solutions for (+)-Dimensional KdV Equation 6 { 4b v (xyt)= P b K + b b (P K )sinh(px + Qy +Wt)cos(Kx+ Ly +Vt) (b sinh(px + Qy +Wt)+b cos(kx+ Ly +Vt)) + 4b b PK cosh(px + Qy +Wt)sin(Kx+ Ly +Vt) (b sinh(px + Qy +Wt)+b cos(kx+ Ly +Vt)) () where all parameters are defined by (9). Especially when L = 0 () () become u (xyt)= 4b b KQsin(Kx+ K t)sinh(qy) (b cosh(qy)+b cos(kx+ K t)) v (xyt)= () b K (b cos(kx+ K t)cosh(qy)+b ) (b cosh(qy)+b cos(kx+ K t)) u (xyt)= 4b b KQsin(Kx+ K t)cosh(qy) (b sinh(qy)+b cos(kx+ K t)) v (xyt)= (4) b K (b cos(kx+ K t)sinh(qy)+b ) (b sinh(qy)+b cos(kx+ K t)) where b b K Q are free parameters. Case. L = L P = P W = P V = 0 Q = 0 K = 0 b = b b = b b = b. (5) Similar as previous case we set b = ±b obtain the following periodic soliton solutions of (): u (xyt)= 4b b PLsinh(Px P t)sin(ly) (b cosh(px P t)+b cos(ly)) (6) v (xyt)= 4b P (b + b cosh(px P t)cos(ly)) (b cosh(px P t)+b cos(ly)) u (xyt)= 4b b PLcosh(Px P t)sin(ly) (b sinh(px P t)+b cos(ly)) (7) v (xyt)= 4b P (b b cosh(px P t)cos(ly)) (b sinh(px P t)+b cos(ly)) where b b P L are free parameters. Fig.. Periodic soliton solution of () with b = b = K = Q = ; (a): t = ; (b): y = t = ; (c): x = 0 t =. Case. K = P w = 8P L = L P = P L Q = b 4b b v = 0 b = b b = b b = b. (8)

4 C. Liu Z. Dai Periodic Wave Solutions for (+)-Dimensional KdV Equation 6 Fig.. Periodic soliton solution of (7) with b = b = P = L = t = 0. Fig.. Periodic soliton solution of () with b = b = K = Q = ; (a): t = ; (b): y = t = ; (c): x = 0 t =. Setting b = ±b = b we get f (x yt) = b cosh(px L y + 8Pt) 4 + cos( Px + Ly) (9) f (x yt) = b sinh(px + L y + 8P t) 4 + cos( Px + Ly). Fig. 4. Periodic soliton solution of (0) with P = L = t = 0.

5 C. Liu Z. Dai Periodic Wave Solutions for (+)-Dimensional KdV Equation 6 Substituting (9) into (4) yields periodic soliton solutions of () as follows: { cosh(px 4 u (xyt)= PL L y + 8P t)cos( Px + Ly) (cosh(px 4 L y + 8P t)+cos( Px + Ly)) sinh(px 4 L y + 8P t)sin( Px + Ly) (cosh(px 4 L y + 8P t)+cos( Px + Ly)) v (xyt)=p { 4cosh(Px 4 L y + 8P t)cos( Px + Ly) (cosh(px 4 L y + 8P t)+cos( Px + Ly)) + 4 sinh(px 4 L y + 8P t)sin( Px + Ly) (cosh(px 4 L y + 8P t)+cos( Px + Ly)) (0) { 4 + sinh(px + 4 u (xyt)= PL L y + 8P t)cos( Px + Ly) (sinh(px + 4 L y + 8P t)+cos( Px + Ly)) 7cosh(Px + 4 L y + 8P t)sin( Px + Ly) (sinh(px + 4 L y + 8P t)+cos( Px + Ly)) v (xyt)=p { 7 4sinh(Px + 4 L y + 8P t)cos( Px + Ly) (sinh(px + 4 L y + 8P t)+cos( Px + Ly)) + 4 cosh(px + 4 L y + 8P t)sin( Px + Ly) (sinh(px +. 4 y + 8P t)+cos( Px + Ly)) () As shown in Figure 4 the solutions u (xyt) v (xyt) in (0) (similarly other periodic soliton solutions) are obviously two periodic solitary waves which are periodic waves along the X-direction: X = Px + Ly with period π are solitary waves along Y-direction: Y = Px 4 L y + 8P t. We note that here waves are two left-propagation waves along Y-direction: Y = Px 4 L y + 8P t. This is a interesting phenomenon to the evolution of shallow water waves. One problem which we will study is whether there exist a similar phenomenon to other equations of shallow water waves or not. Method. Suppose that () has the Jacobi elliptic function double periodic solution as follows: u = A + Bcn (ξ m) C + cn (ξ m) v = E + Fcn (ξ m) G + cn (ξ m) () where ξ = Px + Qy + Wt ABCEFGPQW are constants to be determined m is the modulus of the Jacobi elliptic function (0 m ). With the aid of Maple we substitute () into () balance the coefficients of cn(ξ m) j ( j = 0468) to yield a set of algebraic equations. By solving this set of equations we can determine ABCEFGPQ W. Then from () () we obtain the double periodic function solutions of () which read u = Q (PF 4P +W + 8m P )cn (Px + Qy +Wtm) EP P cn (Px + Qy +Wtm) v = E + Fcn (Px + Qy +Wtm) cn () (Px + Qy +Wtm)

6 64 C. Liu Z. Dai Periodic Wave Solutions for (+)-Dimensional KdV Equation u = Bcn (Px + Qy +Wtm) B + QP + cn (Px + Qy +Wtm) v = [ (4P Q WQ P B + 4P m Q)cn (Px + Qy +Wtm) QP( + cn (Px + Qy +Wtm)) + P Q 4P m Q +WQ+ P ] B QP( + cn (Px + Qy +Wtm)) (4) u = m Bcn (Px + Qy +Wtm)+B Bm Pm Q + PQ m + m cn (Px + Qy +Wtm) v = [ (4m 4 P Q m WQ m P B 8P m Q)cn (Px + Qy +Wtm) QP( m + m cn (Px + Qy +Wtm)) + WQm + 6P m Q P Q P B + P Bm 4P m 4 Q WQ QP( m + m cn (Px + Qy +Wtm)) ] (5) where B E F P Q W are free parameters.. Conclusion In this paper the (+)-dimensional KdV equation is investigated by the extended ansatz function method trial function method. Some periodic solitary wave solutions double periodic solutions are obtained for this equation. The result shows that there exists periodic solitary waves in the different directions for (+)-dimensional KdV equation. These methods are effective for seeking periodic solitary wave solutions double periodic solutions of some integrable equations. Acknowledgement The authors would like to thank the referees for valuable suggestion help. [] M. Boiti J. Leon F. Pempinelli Inverse Probl. 7 (985). [] D. S. Wang H. Li J. Math. Anal. Appl. 4 7 (008). [] S. Y. Lou J. Phys. A: Math. Gen (995). [4] S. Y. Lou H. Y. Ruan J. Phys. A: Math. Gen (00). [5] A. M. Wazwaz Appl. Math. Comput (008). [6] D. S. Wang H. Q. Zhang Intern. J. Modern Phys. C 6 9 (005). [7] D. S. Wang H. Q. Zhang Chaos Solitons Fractals 5 60 (005). [8] Z.D.DaiZ.T.LiZ.J.LiuD.L.LiPhys.Lett.A 7 00 (008). [9] Z. D. Dai Z. J. Liu D. L. Li Chin. Phys. Lett. A 5 5 (008). [0] Z. D. Dai S. L. Li Q. Y. Dai J. Huang Chaos Solitons Fractals 4 48 (007). [] Z. D. Dai M. R. Jiang Q. Y. Dai S. L. Li Chin. Phys. Lett. 065 (006). [] Z. D. Dai J. Huang M. R. Jiang S. H. Wang Chaos Solitons Fractals 6 89 (005). [] F. D. Xie Z. Y. Yan Chaos Solitons Fractals 6 08 (008).