Rational homoclinic solution and rogue wave solution for the coupled long-wave short-wave system

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1 PRAMANA c Indian Academy of Sciences Vol. 86 No. journal of March 6 physics pp Rational homoclinic solution and rogue wave solution for the coupled long-wave short-wave system WEI CHEN HANLIN CHEN and ZHENGDE DAI School of Science Southwest University of Science and Technology Mianyang 6 People s Republic of China School of Mathematics and Statistics Yunnan University Kunming 659 People s Republic of China Corresponding author. chenweimy@yeah.net MS received September ; revised December ; accepted January 5 DOI:.7/s-5-8-; epublication: 7 August 5 Abstract. In this paper a rational homoclinic solution is obtained via the classical homoclinic solution for the coupled long-wave short-wave system. Based on the structures of ratinal homoclinic solution the characteristics of homoclinic solution are discussed which might provide us with useful information on the dynamics of the relevant physical fields. Keywords. Coupled long-wave short-wave system; homoclinic breather wave; rogue wave; homoclinic test. PACS Nos..Jr; 5.5.Yv; 7..+j. Introduction For the past five years rogue waves (also known as freak waves monster waves killer waves rabid-dog waves etc.) have become a hot topic in many research fields [ ]. Rogue waves were first observed in deep ocean [5]. A wave can be called a rogue wave when its height and steepness is much greater than the average crest and appears from nowhere and disappears without a trace [6]. Rogue waves are the subject of intense research in fields of oceanography [78] optical fibres [9] superfluids [] Bose Einstein condensates and related fields []. The first-order rational solution of the self-focussing nonlinear Schrödinger equation (NLS) was first introduced by Peregrine [] to describe the rogue wave phenomenon. Recently by using the Darbou dressing technique or Hirotas bilinear method rogue wave solutions were reported in some comple systems [ 7]. In the present work we use a homoclinic (heteroclinic) breather wave limit method for getting rogue wave solution to the coupled long-wave short-wave system. Pramana J. Phys. Vol. 86 No. March 6 7

2 Wei Chen Hanlin Chen and Zhengde Dai Now we consider the coupled long-wave short-wave system iu t u + u(f w) = iv t v + v(f w) = f t (uv + u v) = where u( t) and v( t) are the orthogonal components of the envelope of a rapidly varying comple field (the short wave) representing a transverse wave whose group velocity resonates with the phase velocity of a real field f(t)(the long wave) representing a longitudinal wave. The denotes comple conjugation and ω is an arbitrary real constant. The CLS equations () generalize the scalar long-wave short wave resonance equations derived by Djordjevic and Redekopp [8] for long-wave short wave interactions when the more generic nonlinear Schrödinger equation breaks down due to a singularity in the coefficient of the cubic nonlinearity. The dispersion of the short wave is balanced by the nonlinear interaction of the long wave while the self-interaction of the short wave drives the evolution of the long wave. The CLS equation () is integrable and has been studied etensively in various aspects [9]. Recently rogue wave and breather solutions were constructed for the uncoupled and coupled long-wave short-wave system by using the two-soliton method []. (). Hirota s bilinear form and rogue wave solutions In this section we concentrate on heteroclinic breather wave solutions for coupled longwave short-wave system by a suitable ansätz approach. Let b b + w. The system () can be written as iu t u + uf = iv t v + vf = () f t (uv + u v) =. We now make the dependent variable transformation u = G F v = H F f = A (ln F) () where G H are comple-valued functions and F is a real-valued function. Then system () can be rewritten as the following coupled bilinear differential equations for FG and H : (id t D )(G F)+ A(G F) = (id t D )(H F)+ A(H F) = () (D t D c)(f F)+ (G H + G H) = where C is the integration constant. H is the conjugate function of H G is the conjugate function of G. Using the etended homoclinic test approach [] we suppose G = e iat [e p( αt) + b cos(p( + αt)) + b e p( αt) ] H = e iat [e p( αt) + b cos(p( + αt)) + b e p( αt) ] (5) F = e p( αt) + b 5 cos(p( + αt))+ b 6 e p( αt) 7 Pramana J. Phys. Vol. 86 No. March 6

3 Rational homoclinic and rogue wave where aαpp b 5 and b 6 are real and b b b b are comple. Substituting eq. (5) into eq. () we can get a set of algebraic equations of e p ( αt). Then equating the coefficients of all powers of e jp ( αt) j = to zero yields the relations among A a p p b b b b b 5 b 6 as follows: a = A c = 8 p = p = α + 6 6α b = iα + iα b 5 ( iα + ) b = iα b 6 b5 = ( 9α + 6p ) b (α + 6p 6 (6) ) where α<. Substituting eq. (6) into eq. () and taking b 6 = we obtain two families of solutions of eq. (): u = v = e iat ( c cosh( pξ ln(c )) + B cos(pη)) cosh( pξ ) + D cos(pη) f = a p [ D D pξ cosh( ) cos(pη)+ D sinh( pξ ) sin(pη)] [ cosh( pξ ) + D cos(pη)] (7) u = v = e iat ( c cosh( pξ ln(c )) B cos(pη)) cosh( pξ ) D cos(pη) f = a p [ D + D pξ cosh( ) cos(pη) D sinh( pξ ) sin(pη)] [ cosh( pξ ) D cos(pη)] (8) where B = CD C = iα + iα D = ( 9α + 6p ) (α + 6p ) ξ = αt η = + αt. (9) Here we have used p = p/. Notice that η and ξ are the vertical relations. The solution represented by (u v f ) (respectively (u v f )) of eq. () is homoclinic breather solution which is a homoclinic wave to a fied cycle (e iat e iat a/) when t and (e iat+iθ e iat+iθ a/) when t of eq. () and meanwhile contain a periodic wave cos(p( + αt)) whose amplitude periodically oscillates with the evolution of time. Now we consider the limit behaviours of p as the period π/p of the periodic wave cos(p( + αt)) goes to infinity i.e. p. Note Pramana J. Phys. Vol. 86 No. March 6 75

4 Wei Chen Hanlin Chen and Zhengde Dai f f t (a) (b).5 t (c) (d) Figure a b. Spatial temporal structures of rational homoclinic wave: (a) u wave and (b) f wave with a = α =. (c) u wave for time t =. and (d) f wave for time t =.5. that (u v f ) is / type as p and by computing we obtain the following result: u = v = e iat ( t + 8t + i + it) t + 8t + f = a + t t () ( t + 8t + ). It is easy to verify that (u v f ) is a solution of eq. (). Moreover (u v f ) is also the homoclinic solution homoclinic to the fied cycle (e iat e iat a/) as t for fied. This shows (u v f ) is just a homoclinic rogue wave solution of the coupled long-wave short-wave system. The maimum amplitude of the rogue wave solution f occurs at point (t = = ) and the maimum amplitude of this rogue wave solution is equal to a +. The minimum amplitude of f occurs at two points (t = =±) 76 Pramana J. Phys. Vol. 86 No. March 6

5 Rational homoclinic and rogue wave and the minimum amplitude of this rogue wave solution is equal to a. The maimum amplitude of the rogue wave solution u occurs at point (t = = ) and the maimum amplitude of this rogue wave solution is equal to. The minimum amplitude of u occurs at point (t = / = /) and the minimum amplitude of this rogue wave solution is equal to (figure ).. Conclusion By applying homoclinic (heteroclinic) breather wave limit method to the coupled longwave short-wave system we obtained a family of homoclinic breather solutions and the rational homoclinic solution. Rational homoclinic solution obtained here is just a homoclinic rogue wave solution. In future we intend to study the interaction between the rational breather wave and the solitary wave. Acknowledgement This work was supported by Chinese Natural Science Foundation Grant No. 68. References [] P Muller C Garrett and A Osborne Oceanography 8 66 (5) [] M Onorato A R Osborne M Serio and S Bertone Phys. Rev. Lett () [] D H Peregrine J. Aust. Math. Soc. Ser. B: Appl. Math. 5 6 (98) [] K W Chow H N Chan D J Kedziora and R Grimshaw J. Phys. Soc. Jpn 8 7 () [5] C Garrett and J Gemmrich Phys. Today 6 6C (9) [6] N Akhmediev A Ankiewicz and M Taki Phys. Lett. A 7(6) 675 (9) [7] K Dysthe H E Krogstad and P M Hert Ann. Rev. Fluid Mech. 87 (8) [8] C Kharif E Pelinovsky and A Slunyaey Rogue waves in the ocean observation theories and modeling (Springer New York 9) [9] N Akhmediev A Ankiewicz and J M Soto-Crespo Phys. Rev.E8 66 (9) [] V Yu Bludov V V Konotop and N Akhmediev Opt. Lett. 5 (9) [] A N Ganshin V B Efimov G V Kolmakov L P Mezhov-Deglin and P V E McClintock Phys. Rev. Lett. 65 (8) [] V Yu Bludov V V Konotop and N Akhmediev Phys. Rev.A8 6 (9) [] C N Kumar R Gupta A Goyal S Loomba and P K Panigrahi Phys. Rev.A86 58 () [] Y Tao and J He Phys. Rev. E85() 66 () [5] L C Zhao and J Liu Phys. Rev. E87 () [6] C Li J He and K Porseizan Phys. Rev.E87 9 () [7] Z D Dai C Wang and J Liu Pramana J. Phys. 8() 7 () [8] V D Djordjevic and L G Redekopp J. Fluid Mech. 79() 7 (977) [9] D J Benney Stud. Appl. Math (977) [] O C Wright III Commun. Nonlinear. Sci. Numer. Simulat () [] K W Chow H N Chan and D J Kedziora J. Phys. Soc. Jpn 8 7 () [] C J Wang and Z D Dai Adv. Differ. Equ. 87 () Pramana J. Phys. Vol. 86 No. March 6 77