Various breathers and rogue waves for the coupled long-wave-short-wave system
© Wang and Dai; licensee Springer. 2014
Received: 18 November 2013
Accepted: 26 February 2014
Published: 14 March 2014
Explicit forms of various breathers, including inclined periodic breather, Akhmediev breather, Ma breather and rogue wave solutions, are obtained for the coupled long-wave-short-wave system by using a Hirota two-soliton method with complex frequency and complex wave number. Based on the structures of these breather solutions and figures via computer simulation, the characteristics of various breather solutions are discussed which might provide us with useful information on the dynamics of the relevant physical fields.
It is well known that solitary wave solutions of nonlinear evolution equations play an important role in nonlinear science fields, especially in nonlinear physical science, since they can provide much physical information and more insight into the physical aspects of the problem and thus lead to further applications . In recent years, rogue waves, as a special type of solitary waves, have triggered much interest in various physical branches. Rogue waves, alternatively called freak or giant waves, were first observed under circumstances of arbitrary depths of the ocean. One always has two or even more times higher amplitude than their surrounding waves and generally they form in a short time for which reason people think that it comes from nowhere [2, 3]. Rogue waves have been the subject of intensive research in oceanography , superfluid helium , Bose-Einstein condensates , optical fibers , plasma physics , financial markets and related fields [7–9]. The first-order rational solution of the self-focusing nonlinear Schödinger equation (NLS) was first found by Peregrine to describe the rogue waves phenomenon . Recently, by using the Darboux dressing technique or the Hirota bilinear method, rogue waves solutions in complex systems such as described by the Hirota equation, Sasa-Satsuma equation, Davey-Stewartson equation, coupled Gross-Pitaevskii equation, coupled NLS Maxwell-Bloch equation and so on have been demonstrated [11–18].
In the above equation, and are the orthogonal components of the envelope of a rapidly varying complex field (the short-wave) representing a transverse wave whose group velocity resonates with the phase velocity of a real field (the long wave) representing a longitudinal wave. The ∗ denotes complex conjugation and ω is an arbitrary real constant. The CLS equations (1.1) generalize the scalar long-wave-short-wave resonance equations derived by Djordjevic and Redekopp  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. Other studies of long-wave-short-wave interactions include those by Benney  and Grimshaw . The CLS equations (1.1) are integrable in the sense that they possess an equivalent scattering problem formulation as a Lax pair of commuting differential operators on a subalgebra of . Wright III  has obtained an auto-Bäklund transformation for plane-wave solutions of a system of coupled long-wave-short-wave equations by using the dressing method. The spatially periodic orbits on a homoclinic manifold of a torus of spatially independent plane waves were constructed by evaluating the auto-Bäklund transformation.
2 Hirota two-soliton method and various breathers
2.1 Hirota two-soliton method
So the solutions of original partial differential equation can be converted into the solutions of bilinear differential equations. We solve the above bilinear differential equations to get breather wave solutions by using a two-soliton method with the help of MAPLE.
where and C is the integration constant. So the system (1.1) has been bilinearized. Then the solutions of the system (1.1) can be converted into the solutions of coupled bilinear differential equations (2.2).
where , , () are complex numbers and a, , , , are real numbers. By comparing with the two-wave functions of the multiple exp-function algorithm , one concludes that they are the same.
In order to obtain the required breather solutions, we consider the case that , () are complex numbers, that is, taking wave numbers and frequencies that are complex, respectively. Indeed, let , , , , substituting (2.3) into (2.2), we can obtain various breathers by restricting the parameters suitably. They can be rewritten in terms of trigonometric and hyperbolic functions. In the following, we report the explicit forms of these breather solutions.
2.2 Various breathers of the coupled long-wave-short-wave system
2.2.1 Inclined periodic breather
where , , , , , , , .
2.2.2 Ma breather
2.2.3 Akhmediev breather
2.2.4 Rogue wave
Obviously, the center of solution (2.8) is located at a fixed point on the plane, that is, this rogue wave given by (2.8) reaches its minimum or maximum at fixed center . Indeed, we can obtain a rogue wave with a controllable center under the transformation of coordinate translation , . In this case, the rogue wave given by (2.8) reaches its minimum or maximum at point , which is a controllable center on the plane.
3 Conclusion and discussion
In this paper, the Hirota two-soliton method has been applied to the coupled long-wave-short-wave system. Maple was used to compute the inclined periodic breather, Akhmediev breather, Ma breather and rogue wave solutions. Based on the structures of these rogue wave solutions and figures via computer simulation, characteristics of rogue wave solutions are discussed which might provide us with the useful information on the dynamics of the relevant physical fields. Following these ideas in this work, further study may be needed to see whether (1.1) has another type of specially spatiotemporal structure of the solutions. Moreover, we remark that these exact breather solutions belong to the class of solutions which the multiple exp-function algorithm  produces. The multiple exp-function algorithm is a generalization of Hirota’s method. Recently, the Hirota bilinear method has also been generalized to more general bilinear equations by Ma .
This work was supported by Chinese Natural Science Foundation Grant Nos. 11361048, 11301235, and 11261049, Yunnan Natural Science Foundation Grant No. kksy201307141.
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