Hamiltonian operators for integrable differential-difference equations

<p> Integrability is a concept that is challenging to define in a unique way, due to its many aspects: one of them is the existence of compatible Hamiltonian structures. This feature is not limited to partial differential equations (PDEs), but it extends to differential-difference equations (D <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:annotation encoding="application/x-tex">\Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Es), regarded as dynamical systems on a difference field. </p> <p> These notes develop the Hamiltonian theory for evolutionary D <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:annotation encoding="application/x-tex">\Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Es, introducing the algebraic formalism, with a focus on symmetries, difference operators and functionals. Later, self-contained definitions of Hamiltonian and pre-Hamiltonian operators are presented, together with a discussion of their relations with recursion and Nijenhuis operators. </p> <p>Throughout the exposition, parallels with their PDE analogues are drawn to emphasise similarities and differences between the two frameworks. The notes include examples and exercises of various difficulties designed to consolidate understanding and explore broader implications of the theory.</p>