Quasi-periodic solutions of mixed AKNS equations

Geng, Xianguo; Xue, Bo

doi:10.1016/j.na.2010.07.047

Cited by 23 publications

(11 citation statements)

References 20 publications

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“…One important factor in the nonlinear steepest descent method is the order of involved spectral matrices in RH problems or equivalently the inverse scattering theory. However, only 2 × 2 spectral matrices and their RH problems have been systematically considered (see, e.g., [20]), which lead to algebro-geometric solutions to integrable systems expressed by hyperelliptic functions [21]. There are very few 3 × 3 spectral matrices, whose long-time asymptotics or RH problems are considered (see, e.g., [22,23]) and whose associated inverse scattering transforms are solved (see, e.g., [24,25]).…”

Section: Introductionmentioning

confidence: 99%

Long-Time Asymptotics of a Three-Component Coupled mKdV System

2019

Mathematics

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We present an application of the nonlinear steepest descent method to a three-component coupled mKdV system associated with a 4 × 4 matrix spectral problem. An integrable coupled mKdV hierarchy with three potentials is first generated. Based on the corresponding oscillatory Riemann-Hilbert problem, the leading asympototics of the three-component mKdV system is then evaluated by using the nonlinear steepest descent method.

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Section: Introductionmentioning

confidence: 99%

Long-Time Asymptotics of a Three-Component Coupled mKdV System

2019

Mathematics

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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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“…It is known that there have been several systematic methods to obtain the finite genus solutions of soliton equations such as algebro-geometric method, the inverse scattering transformation for periodic problem, the nonlinearization approach and others [11,27,5,7,33,28,16,29,6,19,21,3]. These methods have been widely used to solve soliton equations associated with the 2 × 2 matrix spectral problems, by which finite genus solutions to a great deal of integrable models have been successfully constructed, for example, the KdV [11,27,5,7], the discrete AblowitzLadik [33,28,16], the AKNS [19,21], and the Toda lattice equations [7], etc.…”

Section: Introductionmentioning

confidence: 99%

“…These methods have been widely used to solve soliton equations associated with the 2 × 2 matrix spectral problems, by which finite genus solutions to a great deal of integrable models have been successfully constructed, for example, the KdV [11,27,5,7], the discrete AblowitzLadik [33,28,16], the AKNS [19,21], and the Toda lattice equations [7], etc. However, finite genus solutions for the soliton equations associated with the 3 × 3 matrix spectral problems, such as the coupled Sasa-Satsuma hierarchy, cannot be constructed by using these known methods because the corresponding algebraic curve has been changed from the hyperelliptic curve to the trigonal curve.…”

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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Quasi-periodic solutions of mixed AKNS equations

Cited by 23 publications

References 20 publications

Long-Time Asymptotics of a Three-Component Coupled mKdV System

Long-Time Asymptotics of a Three-Component Coupled mKdV System

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

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

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