Algebro-geometric Quasi-periodic Solutions to the Three-Wave Resonant Interaction Hierarchy

He, Guoliang; Geng, Xianguo; Wu, Lihua

doi:10.1137/130918794

Cited by 26 publications

(18 citation statements)

References 34 publications

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“…For concrete physical contexts, equation (1) can describe the propagation and mixing of waves with different frequencies in weakly nonlinear dispersive media [15,16], and it is of also elementary application in nonlinear optics [17], fluid dynamics [18], plasma physics [19], solid state physics [20], and so forth. Over recent years, there have been a substantial number of reports for equation (1), such as the inverse scattering transformation studied by Kaup [17], the Darboux transformation (DT) and multi-soliton solutions given by Zhou [21], and the finite dimensional Hamiltonian system and algebrogeometric solutions derived by Geng and Wu et al [22,23]. Especially, in recent years, by using the Darboux-dressing method and spectral techniques, the general soliton solutions, lower-order rogue wave solutions and stability problem of the solutions have been systematically investigated by Degasperis, Baronio, Conforti et al [15,16,24,25].…”

Section: Introductionmentioning

confidence: 99%

Higher-order rogue wave solutions of the three-wave resonant interaction equation via the generalized Darboux transformation

2015

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In this paper, we utilize generalized Darboux transformation to study higher-order rogue wave solutions of the three-wave resonant interaction equation, which describes the propagation and mixing of waves with different frequencies in weakly nonlinear dispersive media. A general Nthorder rogue wave solution with two characteristic velocities structural parameters and 3N independent parameters under a determined plane-wave background and a specific parameter condition is derived. As an application, we show that four fundamental rogue waves with fundamental, two kinds of line and quadrilateral patterns, or six fundamental rogue waves with fundamental, triangular, two kinds of quadrilateral and circular patterns can emerge in the secondorder rogue waves. Moreover, several important wave characteristics including the maximum values, the corresponding coordinate positions of the humps, and the stability problem for some special higher-order rogue wave solutions such as the fundamental and quadrilateral cases are discussed.

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Section: Introductionmentioning

confidence: 99%

Higher-order rogue wave solutions of the three-wave resonant interaction equation via the generalized Darboux transformation

2015

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“…In this paper we will concentrate primarily on constructing the finite genus solutions of the entire Geng hierarchy related to (1.2) based on the approaches in Refs. [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. The finite genus solutions Co-published by Atlantis Press and Taylor & Francis…”

Section: Introductionmentioning

confidence: 99%

Finite genus solutions for Geng hierarchy

Li¹

2021

JNMP

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“…Fortunately, some progress has been made. Underlying a unified framework for solving the soliton hierarchy associated with the third‐order differential operator, some famous equations have been discussed, such as the Boussinesq equation, the Kaup‐Kaupershmidt hierarchy, the 3‐wave resonant interaction hierarchy, and others …”

Section: Introductionmentioning

confidence: 99%

The trigonal curve and the integration of the Hirota‐Satsuma hierarchy

Geng

2017

Math Methods in App Sciences

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By introducing a trigonal curve  m−1 , which constructed from the characteristic polynomial of Lax matrix for the Hirota-Satsuma hierarchy, we present the associated Baker-Akhiezer function and algebraic functions carrying the data of the divisor. Then the Hirota-Satsuma equations are decomposed into the system of Dubrovin-type ordinary differential equations. Based on the theory of algebraic geometry, we obtain the explicit Riemann theta function representations of the Baker-Akhiezer function, the meromorphic function, and solutions for the Hirota-Satsuma hierarchy.−∞ f (x ′ )dx ′ under the decaying condition at infinity. As is well known, it is very important to search for quasiperiodic solutions for soliton equations, which can reveal the internal structure of the solutions and describe the quasiperiodic behavior of nonlinear phenomena in physical and engineering sciences. In a series of literatures, 11-20 several systematic methods were developed from which finite genus solutions for a lot of soliton equations associated with 2 × 2 matrix spectral problems have been obtained, such as the KdV, nonlinear Schrödinger, mKdV, and Toda lattice equations. However, the study of soliton equations associated with 3 × 3 matrix spectral problems is very few, which is also much more difficult and complicated. Fortunately, some progress has been made. Underlying a unified framework for solving the soliton hierarchy associated with the third-order differential operator, some famous equations have been discussed, such as the Boussinesq equation, 21-23 the Kaup-Kaupershmidt hierarchy, 24 the 3-wave resonant interaction hierarchy, 25 and others. [26][27][28] The outline of the present paper is as follows. In Section 2, we briefly recall the construction for the HS hierarchy associated with the 3 × 3 matrix spectral problem. In Section 3, an algebraic curve  m−1 of arithmetic genus m − 1 is introduced with the help of the characteristic polynomial of Lax matrix for the stationary HS equations, from which the stationary Baker-Akhiezer function and the associated meromorphic function on the curve are given. Next, the stationary HS equations are decomposed into the system of Dubrovin-type ordinary differential equations. Then, we present the explicit theta function representations of the stationary Baker-Akhiezer function, the meromorphic function, and the potentials u and v for the entire stationary HS hierarchy. Section 4 then extends the analyses of Section 3 to the time-dependent case.

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Algebro-geometric Quasi-periodic Solutions to the Three-Wave Resonant Interaction Hierarchy

Cited by 26 publications

References 34 publications

Higher-order rogue wave solutions of the three-wave resonant interaction equation via the generalized Darboux transformation

Higher-order rogue wave solutions of the three-wave resonant interaction equation via the generalized Darboux transformation

Finite genus solutions for Geng hierarchy

The trigonal curve and the integration of the Hirota‐Satsuma hierarchy

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