Abstract
A new method, homoclinic breather limit method (HBLM), for seeking rogue wave solution of nonlinear evolution equation is proposed. A new family of homoclinic breather wave solution, and rational homoclinic solution (homoclinic rogue wave) for DSI and DSII equations are obtained using the extended homoclinic test method and homoclinic breather limit method (HBLM), respectively. Moreover, rogue wave solution is exhibited as period of periodic wave in homoclinic breather wave approaches to infinite. This result shows that rogue wave can be generated by extreme behavior of homoclinic breather wave for higher dimensional nonlinear wave fields.
Highlights
In recent years, rogue waves, as a special type of solitary waves, has been triggered much interest in various physical branches, there is no exact definition up to now
Rogue waves is a kind of waves that seems abnormal which is first observed in the deep ocean, it has been the subject of intensive research in oceanography [1, 2], optical fibres [3,4,5], superfluids [6], Bose-Einstein condensates, financial markets, and related fields [7,8,9,10]
Note that solution (11) contains a periodic wave cos(p1(x − 2y − αt)), so its amplitude periodically oscillates with the evolution of time, and a solitary wave 1/ cosh(p(x−y/2+αt) +γ), which shows that interaction between a solitary wave and a periodic wave with the same velocity α and opposite propagation direction can form a new family of homoclinic solution
Summary
Rogue waves, as a special type of solitary waves, has been triggered much interest in various physical branches, there is no exact definition up to now. There are some methods to seek rogue wave such as Darboux dressing technique, Hirota bilinear method. Based on Hirota bilinear equation of nonlinear evolution equation, for Schrodinger type complex systems, there are some effective techniques such as the Peregrine breather method (PB) [11], whose representation is mathematically a ratio of two polynomials, Ma solitons [4] (MS) and Akhmediev breather methods (ABs) [3]. The main difference between these methods is the test function to Hirota bilinear equation. Φ(x, t) is real function and G(x, t), H(x, t), and D(x, t) are polynomials of (x, t), and Ei(x, t), i = 2, 3, may generate the rogue wave similar to E1 as k → 0. We propose a homoclinic (heteroclinic) breather limit method for seeking rogue wave solution. Homoclinic and heteroclinic tube solutions were obtained [29,30,31,32,33]
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