Algebro-geometric solutions of the coupled modified Korteweg–de Vries hierarchy

Geng, Xianguo; Zhai, Yunyun; Dai, Hui–Hui

doi:10.1016/j.aim.2014.06.013

Cited by 124 publications

(40 citation statements)

References 31 publications

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“…We point out that we can similarly construct algebro-geometric solutions to a linear combination of different AKNS equations in the four-component AKNS soliton hierarchy, which just increases asymptotic complexity (see, e.g., [42]). Various choices of linear combinations of Lax matrices lead to different algebro-geometric solutions to soliton hierarchies.…”

Section: Discussionmentioning

confidence: 99%

“…where {i, k, j} = {1, 2, 3} (see [40] for the Kaup-Kupershmidt case and [42] for the coupled KdV case).…”

Section: Trigonal Curves and Baker-akhiezer Functionsmentioning

confidence: 99%

“…Very recently, Geng et al successfully attempted a few 3 × 3 matrix spectral problems and constructed algebro-geometric solutions to the associated soliton hierarchies, including the modified Boussinesq hierarchy, the Kaup-Kupershmidt hierarchy and the hierarchy of three-wave resonant interaction equations (see, e.g., [24]- [42]). In this paper, we would like to propose a general framework to analyze 3 × 3 matrix spectral problems and their corresponding trigonal curves, and to generate algebro-geometric solutions to soliton hierarchies by observing asymptotic behaviors of the Baker-Akhierzer functions.…”

Section: Introductionmentioning

confidence: 99%

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Trigonal curves and algebro-geometric solutions to soliton hierarchies I

2017

Proc. R. Soc. A.

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Using linear combinations of Lax matrices of soliton hierarchies, we introduce trigonal curves by their characteristic equations, and determine Dubrovin type equations for zeros and poles of meromorphic functions defined as ratios of the Baker-Akhiezer functions. We straighten out all flows in soliton hierarchies under the Abel-Jacobi coordinates associated with Lax pairs, and generate algebro-geometric solutions to soliton hierarchies in terms of the Riemann theta functions, through observing asymptotic behaviors of the Baker-Akhiezer functions. We analyze the four-component AKNS soliton hierarchy in such a way that it leads to a general theory of trigonal curves applicable to construction of algebro-geometric solutions of an arbitrary soliton hierarchy.

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Section: Discussionmentioning

confidence: 99%

“…where {i, k, j} = {1, 2, 3} (see [40] for the Kaup-Kupershmidt case and [42] for the coupled KdV case).…”

Section: Trigonal Curves and Baker-akhiezer Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Trigonal curves and algebro-geometric solutions to soliton hierarchies I

2017

Proc. R. Soc. A.

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“…Let P = (λ, y i (λ)), i = 0, 1, 2, be an arbitrary point in the three points, then the other two points are defined as P * and P * * , respectively [20].…”

Section: The Trigonal Curvementioning

confidence: 99%

“…Based on the work of Matveev, Smirnov, Previato and McKean in the reduction theory of Riemann theta functions [30,31,36,39,32,2], Dickson and his collaborators proposed a unified framework which yields all algebro-geometric quasi-periodic solutions of the entire Boussinesq hierarchy related to the third-order operator [9,10]. Recently, this method was successfully generalized to solve soliton equations associated with 3 × 3 matrix spectral problems such as the modified Boussinesq, the Kaup-Kupershmidt, the coupled modified Korteweg-de Vries and the three-wave resonant interaction hierarchies based on the trigonal curves introduced by the characteristic polynomials of the Lax matrices [17,18,20,24]. In order to solve finite genus solutions of the coupled Sasa-Satsuma hierarchy, we shall introduce a trigonal curve with the aid of the Lax matrix.…”

Section: Introductionmentioning

confidence: 99%

The coupled Sasa–Satsuma hierarchy: Trigonal curve and finite genus solutions

Zhai

Geng

2016

Anal. Appl.

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Based on the Lenard recursion equations and the stationary zero-curvature equation, we derive the coupled Sasa-Satsuma hierarchy, in which a typical number is the coupled Sasa-Satsuma equation. The properties of the associated trigonal curve and the meromorphic functions are studied, which naturally give the essential singularities and divisors of the meromorphic functions. By comparing the asymptotic expansions for the Baker-Akhiezer function and its Riemann theta function representation, we arrive at the finite genus solutions of the whole coupled Sasa-Satsuma hierarchy in terms of the Riemann theta function.

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Algebro‐geometric constructions of the Wadati‐Konno‐Ichikawa flows and applications

Geng

Guan

2015

Math Methods in App Sciences

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Communicated by G. R. FranssensWith the aid of Lenard recursion equations, we derive the Wadati-Konno-Ichikawa hierarchy. Based on the Lax matrix, an algebraic curve K n of arithmetic genus n is introduced, from which Dubrovin-type equations and meromorphic function are established. The explicit theta function representations of solutions for the entire WKI hierarchy are given according to asymptotic properties of and the algebro-geometric characters of K n .

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Algebro-geometric solutions of the coupled modified Korteweg–de Vries hierarchy

Cited by 124 publications

References 31 publications

Trigonal curves and algebro-geometric solutions to soliton hierarchies I

Trigonal curves and algebro-geometric solutions to soliton hierarchies I

The coupled Sasa–Satsuma hierarchy: Trigonal curve and finite genus solutions

Algebro‐geometric constructions of the Wadati‐Konno‐Ichikawa flows and applications

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