Integrable nonlinear ladder system with background-controlled intersite resonant coupling

Abstract

A new spectral problem on one-dimensional lattices is found allowing consistently to support the zero-curvature representation for a wide class of integrable nonlinear ladder systems. The modified recurrence technique for obtaining an infinite set of conservation laws is developed and some basic conserved quantities are explicitly derived. The eigenvalue problems associated with the limiting spectral operator for the special case of rapidly vanishing boundary conditions on Schrödinger-type fields and finite background condition on a concomitant field are solved and the domains of analyticity of Jost functions are presented both analytically and graphically. This particular example shows that the original auxiliary spectral problem is basically of fourth order and must generate a set of four distinct Jost functions that have to be involved in the procedure of inverse scattering transform. Moreover, there exists a critical background value of accompanying field which separates two principally different possibilities in the organization of analyticity domains of Jost functions. This crossover should inevitably lead to qualitative rearrangements in the structure of model solutions. Thus already in the limit of low-amplitude excitations we strictly observe the loss of stability regarding the linear spectrum of Schrödinger subsystem just above the critical background value of practically unexcited concomitant field, whereas in the stability region the structure of linear spectrum is essentially controlled by the magnitude of background level via effective modification of both intersite resonant coupling and self-site coupling.