VARIATIONAL APPROACH TO SOLITARY SOLUTIONS USING JACOBI-ELLIPTIC FUNCTIONS. Yue Wu

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1 Mathematical and Comptational Applications, Vol., No., pp. 9-93,. Association for Scientific Research VARIATIONAL APPROACH TO SOLITARY SOLUTIONS USING JACOBI-ELLIPTIC FUNCTIONS Ye W Economical Mathematics Office, Department of Economics and Management Shanghai Uniersity of Political Science and Law, Shanghai 7, China noodlew@mail.dh.ed.cn Abstract- Partial differential eqations are transformed into ordinary differential eqations, and a ariational formlation is then established. The trial fnction is chosen sing Jacobi-elliptic fnction with some nknown parameters similar to the exp-fnction method. Varios approximate solitary soltions are obtained when making the obtained ariational principle stationary with respect to each nknown parameter in the trial-fnction. The copled Zakharo-Kznetso eqations are sed as an example to elcidate the soltion procedre. Keywords- Variational theory, semi-inerse method, solitary soltion, exp-fnction method, Jacobi-elliptic fnction. INTRODUCTION Solitary soltion is a hot topics in nonlinear science. Recently many new analytical methods were appeared to find solitary soltions of arios nonlinear eqations, sch as the exp-fnction method[-], the mlti-wae method[6], the G /G -expansion method[7], the tanh-method[8], the ariational iteration method[9-], ariational methods[-]and others[-]. Most analytical methods aim to searching for exact soltions withot considering the bondary/initial conditions, sch soltions might hae no physical nderstandings as pointed ot by Ji-Han He in his reiew article[9]. In this paper we trn or attention to approximate solitary soltions sing a ariational method. Generally speaking, there exist two basic ways to describe a physical problem[3] : ( by differential eqations (DE with bondary or initial conditions; ( by ariational principles (VP. The VP model has many adantages oer its DE partner: simple and compact in form while comprehensie in content, encompassing implicitly almost all information characterizing the problem nder consideration. In this paper, we will apply the semi-inerse method[3] to the search for a ariational principle for the discssed problem, then the trial-solitary soltions are expressed sing Jacobi-elliptic Fnctions.. VARIATIONAL FORMULATION In this paper, we will consider the copled Zakharo-Kznetso(ZK eqations []: t xxx yyx 6x x ( δ λ η 6μ α ( t xxx yyx x x x

2 Y. W 9 to illstrate the soltion procedre To seek the traeling solitary soltion of eqations(-(, we se the following transformation ( x, y, t, ( x, y, t, ξ kx ly wt Eqs.(-( become: k l w6k k ( ( ( ( 6 3 ''' ' ' δk λkl w kη kμ αk ( Integrating the aboe eqations, we obtain k l '' 3k wk d ( ( ( 3 '' δk λkl w kη 3kμ αk d (6 where dand d are constants. We apply the semi-inerse method to search for its ariational principle. According to the semi-inerse method[3], we write a Lagrangian in the form: ( ( ' 3 L k l k w ( k d F( (7 where F is an nknown fnction of. The Eler-Lagrange eqation with respect to is Eq.(. The Eler-Lagrange eqation with respect to reads: δ F k (8 δ δ F where is called He s ariational deriatie with respect to [3], defined as: δ δ F F F F (9 δ t t x x In iew of Eqation (6, we set: δ F k ( δ k 3 λ kl '' ( w k η μ d ( δ α From (, the nknown F can be identified as follows 3 ' 3 d F( ( δk λkl ( ( w kη kμ ( α α α α Finally we obtain the needed Lagrangian ( ( ' 3 L k l k w ( k d ( 3 ' 3 d ( δk λkl ( ( w kη kμ α α α α and the ariational formlation (3 (

3 9 Variational Approach to Solitary Soltions sing Jacobi-elliptic Fnctions (, J Ldξ ( ( ' 3 ( k l k w k d 3 ' 3 d ( δ k λkl ( ( w kη kμ dξ α α α α (3 3. TRIAL FUNCTIONS USING JACOBI ELLIPTIC FUNCTIONS In this section, we set d d a soltion in a general form d m for simplicity. The exp-fnction method[] admits nc q p a exp( nξ b n m exp( mξ Similarly in this paper we assme the soltion can be expressed in Jacobi elliptic fnctions: d nc q m p D NC Q MP asn n n bsn m m Asn n N B sn In order to make the soltion procedre mch simpler, we consider a simple case ξ A Aisn( ξ A i sn( ξ (7 where ( ( M M ( ( (6 ξ B B isn( ξ B i sn( ξ (8 A, A, A, B, B, B are nknown constants to be frther determined. sn is sn( ξ sn ξ / m, m (<m<, is the modls of the the Jacobi elliptic fnction, ( Jacobi elliptic fnction. Sbstitting (7 and (8 into Eq.(, we obtain J A, A, A, B, B, B L( A, A, A, B, B, B dξ (9 ( J J Setting and Ai Bi solitary soltions. 3. Stationary with respect to A Making J in Eq.(9 stationary with respect to A, J A respectiely, we can obtain arios approximate ( i i and setting each coefficients of cn( ξ dn( ξ sn, (, ce am ξ sn i to zero, we obtain a set of eqations for A, A, A, B, B, B : sn ξ ( (

4 Y. W αkmA α m ( k l A 3 αk (8 3 m m A αm ( ( k l ( 3 m m w( m A 3 αkm ( m A A 3 αkm ( m B α k m A 9α A A k m α B k m 9α A A k m 3 3 α A k m α B k m 3 αm ( ( k l ( m w A α A m w 8 α A k αa A km αa km αb km 6αA k m 6αA l m 9αA km αaakm 8αAkm 8αAAkm 3αBkm αbkm 6αAkm 6αAlm αAkm 6αAkm αakm 6αAlm αalm αamw 3α A m w α Am w 9 α A k m α A k m α A l m 8 α A k m α A A k m 8 α A k m α B k m 6 α A k m 6 α A l m α A k m 6 α A A k m α A k m 8 α A A k m 3 α B k m α B k m 3 3 α A k m α A l m 8 α A A 9 α 3 α 8 α α A k m 8 α A l m α A l m α A k m α A l m α A m w α A m w α A m w 3 α A m w km A km B km A k m α A k m 6 α A l m 9 α A k m 8 α A A k m 3 α B k m 6 α A k m 3 6 α A l m 9 α A k m 6 α A k m α A k m 6 α A l m α A l m α A k m 6 α A A k m α A k m α B k m 8 α A k m 8 α A l m 8 α A k m 8 α A l m α A m w 8 α A k m α A A k m α A k m α B k m α A A k m 8 α A k m α B k m α A m w3 α A m w α A m w α A k m 6 α A A k m α A k m α B k m α A k m 3 3 α A l m 8 α A k m 8 α A l m α A m w α A k m α A k m α A l m α A k m α A l m 3 8 α A k m α A k m α A l m Soling the aboe eqations simltaneosly with the help of some a mathematical software, we obtain the following soltion A, A, (k l A, ( m 6 m(k l ( k l w( k l B,, B B ( Sbstitting ( into (7 and (8, the traeling wae soltion is obtained which

5 9 Variational Approach to Solitary Soltions sing Jacobi-elliptic Fnctions reads ( ξ (k l isn ξ ( ( 6 m(k l ( m( k l w( k l i sn (3 sn tanh ξ, we hae the following special solitary wae soltion When m, ( ( ξ (k l tanh( i ξ ( 6 (k l 8 ( k l w( k l ξ i coth ( ( 3. Stationary condition with respect to A Similarly setting J A ( i i and setting each coefficients of cn( ξ dn( ξ sn, (, ce am ξ sn i to zero, we obtain a set of eqations for A, A, A, B, B, B : sn ξ ( A k m A k m A l m A A k m A k m B k m A k m A l m 8 α α α 8 α A k m α A k m α A l m 6 α α α 8 α 8 α 3 α A k m 8 α A k m 8 α A l m α A m w 6 α A A km α A km α B km 8 α A k m 8 α A l m 3 3 α A k m 8 α A k m 8 α A l m α A m w α A m w α A km α A A km 8 α A A km 8 α A km 3 α B km α B k m α A k m 6 α A k m α A l m 6 α A l m α A A km α A km α B km 6 α A k m 6 α A l m α A k m 6 α A k m 6 α A l m α A m w α A m w α A m w α A m w α A km 9 α A km 8 α A A km α A A km 8 α A A km α A km 3 α B km 3 α B km α B km α A k m α A k m α A l m 6 α A l m 6 α A A k m 8 α A km α B k m α A k m α A k m α A l m α A l m α A k m 8 α A k m 8 α A l m 3 α A m w3 α A m w α A m w α A m w (6

6 Y. W α A km 9 α A km α A A km 8 α A A km α A k m 3 α B k m α B k m α A k m 6 α A k m 3 3 α A l m 6 α A l m α A A km α A km α B km 6 α A k 6 α 8 α 6 α 6 α m A l m A k m A k m A l m 3 3 A m w A m w A m w 3 α α α 9 α A k m α A A k m 8 α A A k m 8 α A k m 3 α B k m α B k m α A k m 6 α A k m α A l m 6 α A l m α A A km α A km α B km 6 α A k m 6 α A l α A k m 6 α A k m 6 α A l m α A m w α A m w α A m w 3 9 α A km 9 α A km 8 α A A km α A A km 8 α A A km 8 α A km 3 α B km 3 α B km α B km α A k m 6 α A k m α A l m 6 α A l m 6 α A A k m 8 α A α B k m α A k m α A k m α A l m α A l m α A km 8 α A k m 8 α A l m 3 α A m w 3 α A m w α A m w α A m w 3 9 α A km 9 α A km α A A km 8 α A A km 8 α A km m km 3 α B k m α B k m α A k m 6 α A k m α A l m 6 α A l m α A A k m α A k m α B k m 6 α A k α A l m 8 α A k m 6 α A k m 6 α A l m 3 α A m w α α A m w A m w Soling the aboe ten eqations, we obtain the following soltions Case 8 ( m ( k l A,, A ( k l A 6 6 l ( k l m m k l B 6 km( k l k 8 ( k l w 8 ( k l m w ( ( B, m ( k l 3 w k l B ( ( m (

7 96 Variational Approach to Solitary Soltions sing Jacobi-elliptic Fnctions Case 8 ( m ( k l m(k l (k l A, A, A 8 8( 7mm ( k l 3w( m( k l B 7 k 8m( m( k l 3mw( k l B ( m( k l 3w( k l B ( Case 3 A, m ( k l A, A m( m 6m ( k l ( k l mw( k l B, B, ( B The aboe cases lead to the following soltions 8 ( m ( k l ( k l 3 ( ξ isn ( ξ (3 ( 6 l ( k l m 6 m ( k l 3 6 km( k l k ( m k l w( k l 8 ( k l w 8 ( k l m w ( 3 ( ξ ( m k l 8 ( m(k l (k l 88( 7mm ( k l 3w( m( k l ξ 7 k 8m( m( k l 3mw( k l isn ( m( k l 3w( k l isn ( isn( ξ ( isn( ξ isn ( (6

8 Y. W 97 m ( k l i sn (7 6m ( k l m( m( k l mw( k l sn tanh ξ, the aboe soltions trn ot to be familiar solitary i sn (8 When m, ( soltions, which read ( k l ( k l coth( ξ i (9 ( ξ ( ( ( k l k l wk l k ( 8( k l 3 w k l icoth (3 6( k l ( k l ( tanh( k i ξ l i coth( ξ (3 ( k l w( k l ( k l w( k l k 7 8 ( k l w( k l itanh 8 3 icoth (3 ( k l tanh( ξ i ξ (33 ( 8 ( k l w( k l 6 ( k l 8 itanh ( Stationary condition with respect to B J By setting, and doing the same as illstrated before, we obtain a set of B eqations for A, A, A, B, B, B : δ μ B k m 9B k m B k l m λ

9 98 Variational Approach to Solitary Soltions sing Jacobi-elliptic Fnctions aa km 6 B ek m 3 aa km aa km B ek m 3 aa km δ δ μ μ 8 B e k m B k m B k m 8 B k B B k m 8 B k m μ μ μ μ 9 B k m 6 B B k m B k m 8 B B 3 μ 8 B B km B km μ 6 B kl mλ B kl m λ8 B kl m λ B kl m λ 3 λ B k l m B m w3 B m w B m w3 B m w B k m η 3 3 B k m η B k m η 3 B k m η km μ α A k m 6 B δ k m3 α A k m α A k m 6 B δ k m 6 B δ k m 3 3 δ μ μ μ μ B k m 8 B k B B k m B k m 9 B k m B B k m 3 8 B k m μ8 B B k m μ 9 B k m μ 6B k l m λ 6 B k l m λ B k l m λ B k l m λ B m w 3 B m w B m w B k m η η B k m 3 B k m η μ 3 3 B δ k m 9 B k m μ B k l m λ α A km 6 B δ k m 3 α A km α A km B δ k m 3 α A km B δ k m B δ k m B δ k m 8 B kmμ B B km μ B km μ9 B km μ6 B B km μ B km μ8b B k m μ 3 8 B B km μ 9 B km μ 6 B kl m λ B kl m λ 8 B kl m λ 3 3 B k l m λ B k l m λ B m w 3 B m w B m w B m w B km η 3B km η B km η 3 B km η B δk m 3 α A k m 6 B δ k m 6 B δ k m B δ k m B k m μ 8 B B k m μ 9 B k m μ 6 B k l m λ 6 B k l m λ 6 B k l m λ B k l m λ α A km 8 B δ k m 8 B δ k m B kmμ 6 B B km μ B km μ 3 8 B k l m λ8 B k l m λ B m w B k m η 3 α A km α A km 8 B kmμ B B km μ B km μ B B k m μ 8 B k m μ B m w3 B m w B m w B k m η B k m η B k m η α A km B δ k m 8 B δ k m B km μ6 B B km μ B km μ B kl m λ 8 B kl m λ B m w B km η B δ k m B δ k m 8 B km μ B kl m λ B kl mλ 3 3 B δ k m 8 B k m μ B k l m λ The aboe system admits the following soltion

10 Y. W 99 A A 6 m( δ k l λ 3 αμ, A, k m k l k l w k 3 α k μ ( ( δ λ ( δ λ ( η B, B, ( δ k l λ B (3 This leads to the following solitary soltion 6 m( δk l λ 9 3 αμ k( m( δk λl ( δk λl ( w kη (36 isn 3 α k μ 9 ( ξ ( δ k l λ isn (37 When m, sn tanh soltions:, Eqations(36- (37 admit the following solitary wae ( δ λ ( δ λ ( w η 6( 8k k l k l k δk l λ 3 αμ 3 αk μ i coth (38 ( δ k l λ coth( i ξ (39 3. Stationary condition with respect to B Similarly by setting J / B, we obtain a set of eqations for A, A, A, B, B, B : α A k m 8 B δ k m 8 B δ k m 6 B B k m μ B k m B δ k m 8 B k m μ B k l m λ B δ k m 8 B k m μ B k l m λ μ λ λ η 3 3 B km 8 B kl m 8 B kl m B m w B km α A k m 8 B δ k m 8 B δ k m 6 B B k m μ B k m μ B k m λ λ η η A km A km B k m B k m A km B k m 8 B k l m 8 B k l m B m w B m w B k m B k m 3 α α δ 6 δ α 6 δ B δ k m 9 B km μ B B km μ8 B B km μ8 B km μ 3 3 B B km μ B km μ 8 B km μ B kl m λ 6 B k l m λ B k l m λ 6 B k l m λ B m w B m w B m w B m w 3 3 B k m η B k m η B k m η B k m η μ μ

11 9 Variational Approach to Solitary Soltions sing Jacobi-elliptic Fnctions α A km 3 α A km α A km Bδ k m 6 B δ k m α A km δ δ δ μ μ B k m B k m 8 B k m 9 B k m 9 B k m 8 B B k m B B km μ 8 B B km μ 8 B km μ 6B B k m μ 8 B k m μ 3 3 B km μ B kl m λ 6 B kl m λ B kl m λ B kl m λ λ 8 B k l m 3 B m w3 B m w B m w B m w3 B k m η 3 3 η η 3 B k m B k m B k m η μ α A km α A km B δ k m 6 B δ k m α A km 6 B δ k m B δ k m 9 B k m μ9 B k m μ B B k m μ8 B B k m μ 3 μ μ μ 8 B k m B B k m B k m 8 B k m λ λ λ μ B k l m λ 6 B k l m 6 B k l m 6 B k l m 3 B m w B m w B m w 3 B km η B km η B km η α A km α A km B δ k m 6 B δ k m α A km 6 B δ k m 3 6 δ μ μ μ 6 B k m 9 B km B B km 8 B B km 8 B km μ 6 μ μ μ B B km B km 8 B km B kl m λ 6 B k l m 6 λ λ 6 B k l m 6 B k l m B m w B m w B m w B k m η B k m B k m η λ η δ δ μ μ μ μ μ 6B B k m 8B k m 6 μ λ λ λ 6 λ η η η k m η 3 3 α A k m α A k m B δ k m B δ k m α A k m B δ k m B δ k m B km μ B km μ B B km μ 6 B B km μ B km μ B B km μ B km μ B km μ 6 B kl m λ B kl m λ B kl m λ B kl m λ B m w B m w B m w B k m η B k m η B k m η 3a A k m 3a A k m a A k m B e k m 6B e k m a A k m B e k m B k m 8B k m 9B k m 9B k m 8B B k m B B k m 8B B k m 8B k m μ μ μ B k m B k l m 6B k l m B k l m B k l m λ 8B k l m 3B m w3b m wb m wb m w3b k m 3B k m B k m B Soling the aboe ten eqations reslts the following cases Case

12 Y. W 9 A ( δ λ ( η 6km( δk l λ 6 k ( m ( δk l λ 8m ( k l wk μ 9 μ 3 α k k m δk λl 3 δk λl w kη A 9 α k μ A, ( ( ( ( B 8 ( m ( δk λl, 9 μ B, ( δ k λ l B ( Case 6 ( δ k m δ k l m λ l m λ A, 3 αμ A km( m( δ k λ l m( δ k λ l ( w kη 3 α kμ, A B, m( δ k λ l B, B ( The aboe two cases yield the following solitary soltions ( δ λ ( η 6km( δk l λ 6 k ( m ( δk l λ 8m ( k l wk μ 9 μ 3 α k k( m( δk λl 3( δk λl ( w kη isn 9 α k μ ( ξ ( 8 ( m ( δk λl ( δ k λ l isn ( ξ (3 9 μ 6 ( δ k m δ k l m λ l m λ 3 αμ km m k l m k l w k 3 α kμ ( ( δ λ ( δ λ ( η isn, ( m( δ k λ l i sn ( When m, sn tanh wae soltions respectiely:, Eqations(-( reslt in the following solitary

13 9 Variational Approach to Solitary Soltions sing Jacobi-elliptic Fnctions 3 3 ( ξ ( 6k( δk l λ 6( δk l λ 6( δk l λ wk η μ 9 μ 3 α k 8k( δk λl 3( δk λl ( w kη icoth 9 α k μ 6 ( δk λl ( δ k λ l coth( ξ (6 9 μ i (7 ( δ λ ( δ λ ( η 6( δk λl 8 k k l k l w k 3 αμ 3 αkμ i tanh (8 ( δ k λ l tanh( i ξ (9. CONCLUSION This paper actally coples seeral methods, i.e., the ariational theory, the semi-inerse, the exp-fnction method and elliptic fnction, in an effectie way. The soltion procedre is simple if some a mathematical software is sed. This paper sggests a new approach to solitary theory.. REFERENCES. J. H. He and X. H W, Exp-fnction method for nonlinear wae eqations, Chaos Soliton. Fract. 3, pp. 7-78, 6.. M.M. Kabir and A. Khajeh, New explicit soltions for the akhnenko and a generalized form of the nonlinear heat condction eqations ia exp-fnction method, International Jornal of Nonlinear Sciences and Nmerical Simlation, 37-38, A.Esen and S. Ktlay, Application of the Exp-fnction method to the two dimensional sine-gordon eqation, International Jornal of Nonlinear Sciences and Nmerical Simlation, 3-39, 9.. A. Ebaid, Generalization of He's Exp-Fnction Method and New Exact Soltions for Brgers Eqation, Zeitschrift fr Natrforschng A 6, 6-68, 9.. S. Zhang, Exp-fnction Method: Solitary, Periodic and Rational Wae Soltions of Nonlinear Eoltion Eqations, Nonlinear Science Letters A, 3-6,. 6. Z. D. Dai, C. J. Wang, S. Q. Lin, D. L. Li and G. M, The Three-wae Method for Nonlinear Eoltion Eqations, Nonlinear Science Letters A, 77-8,. 7. H.A. Zedan, New Classes of Soltions for a System of Partial Differential Eqations by sing G /G -expansion Method, Nonlinear Science Letters A, 9-38,. 8. E. M. E. Zayed and H. M. A. Rahman, The Extended Tanh-method for Finding Traeling Wae Soltions of Nonlinear Partial Differential Eqations, Nonlinear Science Letters A., 93-,.

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