Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions

Abstract

We express AKNS hierarchies, admitting reductions to matrix NLS and matrix mKdV hierarchies, in terms of a bidifferential graded algebra. Application of a universal result in this framework quickly generates an infinite family of exact solutions, including e.g. the matrix solitons in the focusing NLS case. Exploiting a general Miura transformation, we recover the generalized Heisenberg magnet hierarchy and establish a corresponding solution formula for it. Simply by exchanging the roles of the two derivations of the bidifferential graded algebra, we recover "negative flows", leading to an extension of the respective hierarchy. In this way we also meet a matrix and vector version of the short pulse equation and also the sine-Gordon equation. For these equations corresponding solution formulas are also derived. In all these cases the solutions are parametrized in terms of matrix data that have to satisfy a certain Sylvester equation.

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• Hamiltonian Integrability of Two-Component Short Pulse Equations
• Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations
• A vectorial binary Darboux transformation of the first member of the negative part of the AKNS hierarchy
• Lax Representations for Matrix Short Pulse Equations
• Solitons of the (2+2)-dimensional Toda lattice
• Super Extensions of the Short Pulse Equation
• Soliton Fay identities. II. Bright soliton case
• Soliton Fay identities. I. Dark soliton case
• The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework
• Skew-selfadjoint Dirac systems with rational rectangular Weyl functions: explicit solutions of direct and inverse problems and integrable wave equations
• Discontinuous Galerkin methods for short pulse type equations via hodograph transformations
• Coupled Dispersionless and Generalized Heisenberg Ferromagnet Equations with Self-Consistent Sources: Geometry and Equivalence
• Solitons of Some Nonlinear Sigma-Like Models
• Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy