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

Abstract

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.

Application of the Nonlinear Steepest Descent Method to the Coupled Sasa-Satsuma Equation

“…Ling [22] obtained high order solution formulas in the determinant form by using a generalised Darboux transformation and the formal series method. In [44], finite genus solutions of the Sasa-Satsuma hierarchy, associated with a 3×3 matrix spectral problem, are obtained by using asymptotic expansions of the Baker-Akhiezer function and its Riemann theta function representation [37]. The Riemann-Hilbert approach, Darboux transformation and Riccati equation are employed in investigating the integrability of multi-coupled nonlinear integrable equations and finding their exact solutions -cf.…”

Darboux transformation of a two‐component generalized Sasa–Satsuma equation and explicit solutions

“…For example, it appears in the description of ultrashort pulse propagation in optical fibers. 14,15 The Sasa-Satsuma equation has been widely studied by the Hirota bilinear method, 12,16 inverse scattering transform, 17,18 Darboux transformation, 13,[19][20][21] Riemann-Hilbert approach, [22][23][24] and others [25][26][27][28] in the past two decades. Some important explicit solutions related to the Sasa-Satsuma equation have been found.…”

Long time and Painleve-type asymptotics for the Sasa-Satsuma equation in solitonic space time regions

“…In 2007, Sergyeyev and Demskoi studied the recursion operator and nonlocal symmetries of Sasa-Satsuma and the complex sine-Gordon II equation [6]. Zhai and Geng constructed finite gap solutions of the Sasa-Satsuma equation via the algebro-geometric method [11]. The initial-boundary value problem for the Sasa-Satsuma equation on the half line was investigated by using the Fokas method [10].…”