Algebro-geometric constructions of quasi-periodic flows of the Newell hierarchy and applications

Geng, Xianguo; Wang, Hui

doi:10.1093/imamat/hxw008

Cited by 9 publications

(13 citation statements)

References 33 publications

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“…where ρ = ρ(λ) = ρ(x, t, λ), and = (λ) = (x, t, λ) are n-component column vectors. Then, it is not difficult that the compatibility condition of the Riccati Equation (15), ρ xt = ρ tx and xt = tx , derives also the vmYOLS Equation (8). For convenience, we denote P(λ) = (ρ(λ), (λ)), which is an n × 2 matrix.…”

Section: Lax Pair and Riccati Equationsmentioning

confidence: 99%

“…Some exact solutions for the solvable long-wave-short-wave model (2) are derived by resorting to the Darboux transformation [14]. Recently, algebro-geometric solutions of the Newell hierarchy have been obtained on the basis of the theory of trigonal curves [15]. On the other hand, the long-wave-short-wave equations are different in different physical contexts.…”

Section: Introductionmentioning

confidence: 99%

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On a Vector Modified Yajima–Oikawa Long-Wave–Short-Wave Equation

Geng

2019

Mathematics

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A vector modified Yajima–Oikawa long-wave–short-wave equation is proposed using the zero-curvature presentation. On the basis of the Riccati equations associated with the Lax pair, a method is developed to construct multi-fold classical and generalized Darboux transformations for the vector modified Yajima–Oikawa long-wave–short-wave equation. As applications of the multi-fold classical Darboux transformations and generalized Darboux transformations, various exact solutions for the vector modified long-wave–short-wave equation are obtained, including soliton, breather, and rogue wave solutions.

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Section: Lax Pair and Riccati Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On a Vector Modified Yajima–Oikawa Long-Wave–Short-Wave Equation

Geng

2019

Mathematics

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“…Geng et al. provided the algebro‐geometric constructions of quasi‐periodic flows of the DYO system and their explicit theta function representations .…”

Section: Introductionmentioning

confidence: 99%

“…By applying the dressing method, soliton solutions including cusp solution for the DYO system were derived by using the properties of Cauchy matrix [11]. Geng et al provided the algebro-geometric constructions of quasi-periodic flows of the DYO system and their explicit theta function representations [12]. The derivative nonlinear Schrödinger (DNLS) equations are the wellknown integrable models with a variety of physical applications.…”

Section: Introductionmentioning

confidence: 99%

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The Derivative Yajima–Oikawa System: Bright, Dark Soliton and Breather Solutions

et al. 2018

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In this paper, we study the derivative Yajima–Oikawa (YO) system which describes the interaction between long and short waves (SWs). It is shown that the derivative YO system is classified into three types which are similar to the ones of the derivative nonlinear Schrödinger equation. The general N‐bright and N‐dark soliton solutions in terms of Gram determinants are derived by the combination of the Hirota's bilinear method and the Kadomtsev–Petviashvili hierarchy reduction method. Particularly, it is found that for the dark soliton solution of the SW component, the magnitude of soliton can be larger than the nonzero background for some parameters, which is usually called anti‐dark soliton. The asymptotic analysis of two‐soliton solutions shows that for both kinds of soliton only elastic collision exists and each soliton results in phase shifts in the long and SWs. In addition, we derive two types of breather solutions from the different reduction, which contain the homoclinic orbit and Kuznetsov–Ma breather solutions as special cases. Moreover, we propose a new (2+1)‐dimensional derivative Yajima–Oikawa system and present its soliton and breather solutions.

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