Dynamics of localized wave solutions for the coupled Higgs field equation
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ORIGINAL PAPER
Dynamics of localized wave solutions for the coupled Higgs field equation Zhaqilao
Received: 28 May 2020 / Accepted: 29 July 2020 © Springer Nature B.V. 2020
Abstract By virtue of Hirota bilinear form of the coupled Higgs field equation, some higher-order rogue wave, the Ma breather, the Akhmediev breather and the general breather solutions are constructed through symbolic computation. With the help of the contour line method, we investigate the localization characters of the first-order rogue wave, which is different from the case of its in nonlinear Schrödinger equation. The Ma breather and the Akhmediev breather solutions of the coupled Higgs field equation possess time-periodic and space-periodic, respectively. The results of this paper show us the generation and evolutions of novel nontraveling wave with several interesting structures.
here and disappear a trace [3], and it only takes second before they hit a ship. Nowadays, the studies of the phenomena of localized waves in systems governed by nonlinear evolution equation have received tremendous attention [18–27]. In mathematical physics, the rogue wave solution is a kind of interesting rational solution and it is localized in space and time. The breather solution is localized in one certain direction with periodic structure. In this work, we focus on the coupled Higgs field equation (cHFE) [28–38] u tt − u x x − βu + γ |u|2 u − 2uv = 0,
(1)
vtt + vx x − γ (|u| )x x = 0,
(2)
2
Keywords Rogue wave solution · Breather solution · Coupled Higgs field equation · Symbolic computation 1 Introduction It is well known that rogue wave appears in many scientific fields such as in oceanography [1–5], optical fibers [6,7], water wave tank [8], Bose–Einstein condensates [9], financial markets [10] and other related fields [11– 17]. Compared with tsunamis and storm associated with typhoon that can be predicted hours in advance, the oceanic rogue waves suddenly appear from now Zhaqilao (B) College of Mathematics Science, Inner Mongolia Normal University, Huhhot 010022, People’s Republic of China e-mail: [email protected]
which describes a system of conserved scalar nucleons interacting with neutral scalar mesons in particle physics. Here, the function v = v(x, t) represents a real scalar meson field and u = u(x, t) represents a complex scalar nucleon field. The subscripts t, x of u and v denote appropriate partial derivatives with respect to the time and space variables. cHFE (1)–(2) are cHFE for β > 0, γ > 0 and the coupled nonlinear Klein– Gordon equation for β < 0, γ < 0. N -soliton solutions of cHFE (1)–(2) were obtained by Hirota bilinear method [28,29]. The bright soliton, periodic wave and doubly periodic wave solutions of cHFE (1)–(2) have been shown [30–32]. The first-order rogue wave solution of cHFE (1)–(2) was discussed in [33], and other types of traveling wave solutions have been presented [33–38].
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Zhaqilao
cHFE (1)–(2) are related to the classical Higgs equation [39] u tt − u x x − βu + γ |u|2 u = 0,
(3)
where β > 0, γ >
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