New integrable boundary conditions for the Ablowitz-Ladik model: from Hamiltonian formalism to nonlinear mirror image method

Abstract

Using Sklyanin's classical theory of integrable boundary conditions, we use the Hamiltonian approach to derive new integrable boundary conditions for the Ablowitz-Ladik model on the finite and half infinite lattice. In the case of half infinite lattice, the special and new emphasis of this paper is to connect directly the Hamiltonian approach, based on the classical $r$-matrix, with the zero curvature representation and Bäcklund transformation approach that allows one to implement a nonlinear mirror image method and construct explicit solutions. It is shown that for our boundary conditions, which generalise (discrete) Robin boundary conditions, a nontrivial extension of the known mirror image method to what we call {\it time-dependent boundary conditions} is needed. A careful discussion of this extension is given and is facilitated by introducing the notion of intrinsic and extrinsic picture for describing boundary conditions. This gives the specific link between Sklyanin's reflection matrices and Bäcklund transformations combined with folding, {\it in the case of non-diagonal reflection matrices}. All our results reproduce the known Robin boundary conditions setup as a special case: the diagonal case. Explicit formulas for constructing multisoliton solutions on the half-lattice with our time-dependent boundary conditions are given and some examples are plotted.

32 pages. References added. Accepted authors version

References in corpus (6)

• Initial-boundary value problems associated with the Ablowitz-Ladik system
• The generalized non-linear Schrodinger model on the interval
• Integrable boundary conditions and modified Lax equations
• From Hamiltonian to zero curvature formulation for classical integrable boundary conditions
• Dressing the boundary: on soliton solutions of the nonlinear Schrödinger equation on the half-line
• Vector NLS solitons interacting with a boundary

Cited by in corpus (8)

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• Nonlinear Schrödinger equation on the half-line without a conserved number of solitons
• Discretizations of the generalized AKNS scheme
• On an inverse scattering transform for the nonlinear Schödinger equation on the half-line
• The Ablowitz-Ladik system on a graph
• Breather interactions in the integrable discrete Manakov system and trigonometric Yang-Baxter maps
• On the nonlinear Schrödinger equation with a time-dependent boundary condition