Folkert Müller-Hoissen

Using bidifferential calculus, we derive a vectorial binary Darboux transformation for an integrable matrix version of the first negative flow of the Kaup–Newell hierarchy. A reduction from the latter system to an integrable matrix version of the Fokas–Lenells equation is then shown to inherit a corresponding vectorial Darboux transformation (DT). Matrix soliton solutions are derived from the trivial seed solution. Furthermore, the DT is exploited to determine in a systematic way exact solutions of the two-component vector Fokas–Lenells equation on a plane wave background. This comprises breathers, dark solitons, rogue waves and ‘beating solitons’.

Given two tensor fields of type (1, 1) on a smooth n-dimensional manifold M , such that all their Frölicher–Nijenhuis brackets vanish, the algebra of differential forms on M becomes a bi-differential graded algebra. As a consequence, there are systems of PDEs associated with it, which arise as the integrability condition of a system of linear equations and possess a binary Darboux transformation to generate exact solutions. We recover chiral models and potential forms of the self-dual Yang–Mills equations, as well as corresponding generalizations to higher than four dimensions, and obtain new integrable non-autonomous nonlinear matrix PDEs.

Using a general result of bidifferential calculus and recent results of other authors, a vectorial binary Darboux transformation is derived for the first member of the ‘negative’ part of the potential Kaup-Newell hierarchy, which is a system of two coupled Fokas–Lenells equations. Miura transformations are found from the latter to the first member of the ‘negative’ part of the AKNS hierarchy and also to its ‘pseudodual’. The reduction to the Fokas–Lenells equation is implemented and exact solutions with a plane wave seed generated.

Using bidifferential calculus, we derive a vectorial binary Darboux transformation for the first member of the ‘negative’ part of the AKNS hierarchy. A reduction leads to the first ‘negative flow’ of the NLS hierarchy, which in turn is a reduction of a rather simple nonlinear complex PDE in two dimensions, with a leading mixed third derivative. This PDE may be regarded as describing geometric dynamics of a complex scalar field in one dimension, since it is invariant under coordinate transformations in one of the two independent variables. We exploit the correspondingly reduced vectorial binary Darboux transformation to generate multi-soliton solutions of the PDE, also with additional rational dependence on the independent variables, and on a plane wave background. This includes rogue waves.

We consider a matrix refactorization problem, i.e., a “Lax representation,” for the Yang–Baxter map that originated as the map of polarizations from the “pure” 2-soliton solution of a matrix KP equation. Using the Lax matrix and its inverse, a related refactorization problem determines another map, which is not a solution of the Yang–Baxter equation, but satisfies a mixed version of the Yang–Baxter equation together with the Yang–Baxter map. Such maps have been called “entwining Yang–Baxter maps” in recent work. In fact, the map of polarizations obtained from a pure 2-soliton solution of a matrix KP equation, and already for the matrix KdV reduction, is not in general a Yang–Baxter map, but it is described by one of the two maps or their inverses. We clarify why the weaker version of the Yang–Baxter equation holds, by exploring the pure 3-soliton solution in the “tropical limit,” where the 3-soliton interaction decomposes into 2-soliton interactions. Here, this is elaborated for pure soliton solutions, generated via a binary Darboux transformation, of matrix generalizations of the two-dimensional Toda lattice equation, where we meet the same entwining Yang–Baxter maps as in the KP case, indicating a kind of universality.

We consider a matrix refactorization problem, i.e., a "Lax representation", for the Yang-Baxter map that originated as the map of polarizations from the "pure" 2-soliton solution of a matrix KP equation. Using the Lax matrix and its inverse, a related refactorization problem determines another map, which is not a solution of the Yang-Baxter equation, but satisfies a mixed version of the Yang-Baxter equation together with the Yang-Baxter map. Such maps have been called "entwining Yang-Baxter maps" in recent work. In fact, the map of polarizations obtained from a pure 2-soliton solution of a matrix KP equation, and already for the matrix KdV reduction, is NOT in general a Yang-Baxter map, but it is described by one of the two maps or their inverses. We clarify why the weaker version of the Yang-Baxter equation holds, by exploring the pure 3-soliton solution in the "tropical limit", where the 3-soliton interaction decomposes into 2-soliton interactions. Here this is elaborated for pure soliton solutions, generated via a binary Darboux transformation, of matrix generalizations of the two-dimensional Toda lattice equation, where we meet the same entwining Yang-Baxter maps as in the KP case, indicating a kind of universality.

We study soliton solutions of matrix "good'' Boussinesq equations, generated via a binary Darboux transformation. Essential features of these solutions are revealed via their "tropical limit'', as exploited in previous work about the KP equation. This limit associates a point particle interaction picture with a soliton (wave) solution.

For the first two equations of the Volterra lattice hierarchy and the first two equations of its non-autonomous (non-isospectral) extension, we present Riccati systems for functions c _j ( t ), , such that an expression in terms of Hankel determinants built from them solves these equations on the right half of the lattice. This actually achieves a complete linearization of these equations of the extended Volterra lattice hierarchy.

For the first two equations of the Volterra lattice hierarchy and the first two equations of its non-autonomous (non-isospectral) extension, we present Riccati systems for functions c_j(t), j=0,1,..., such that an expression in terms of Hankel determinants built from them solves these equations on the right half of the lattice. This actually achieves a complete linearization of these equations of the extended Volterra lattice hierarchy.