T. C. Santiago | Searcharxiv

arXiv2026

New soliton solutions for Chen-Lee-Liu and Burgers hierarchies and its Bäcklund transformations

Positive and negative flows of the Chen-Lee-Liu model and its various reductions, including Burgers hierarchy, are formulated within the framework of Riemann-Hilbert-Birkhoff decomposition with the constant grade two generator. Two classes of vacua, namely zero vacuum and constant non-zero vacuum can be realized within a centerless Heisenberg algebra. The tau functions for soliton solutions are obtained by a dressing method and vertex operators are constructed for both types of vacua. We are able to select and classify the soliton solutions in terms of the type of vertices involved. A judicious choice of vertices yields in a closed form a particular set of multi soliton solutions for the Burgers hierarchy. We develop and analyze a class of gauge-Bäcklund transformations that generate further multi soliton solutions from those obtained by dressing method by letting them interact with various integrable defects.

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arXiv2025

Generalized Riemann-Hilbert-Birkhoff Decomposition and a New Class of Higher Grading Integrable Hierarchies

We propose a generalized Riemann-Hilbert-Birkhoff decomposition that expands the standard integrable hierarchy formalism in two fundamental ways: it allows for integer powers of Lax matrix components in the flow equations to be increased as compared to conventional models, and it incorporates constant non-zero vacuum (background) solutions. Two additional parameters control these features. The first one defines the grade of a semisimple element that underpins the algebraic construction of the hierarchy, where a grade-one semi-simple element recovers known hierarchies such as mKdV and AKNS. The second parameter distinguishes between zero and non-zero constant background (vacuum) configurations. Additionally, we introduce a third parameter associated with an ambiguity in the definition of the grade-zero component of the dressing matrices. While not affecting the decomposition itself, this parameter classifies different gauge realizations of the integrable equations (like for example, Kaup-Newell, Gerdjikov-Ivanov, Chen-Lee-Liu models). For various values of these parameters, we construct and analyze corresponding integrable models in a unified universal manner demonstrating the broad applicability and generative power of the extended formalism.

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