Manakov system with parity symmetry on nonzero background and associated boundary value problems

Abstract

We characterize initial value problems for the defocusing Manakov system (coupled two-component nonlinear Schrödinger equation) with nonzero background and well-defined spatial parity symmetry (i.e., when each of the components of the solution is either even or odd), corresponding to boundary value problems on the half line with Dirichlet or Neumann boundary conditions at the origin. We identify the symmetries of the eigenfunctions arising from the spatial parity of the solution, and we determine the corresponding symmetries of the scattering data (reflection coefficients, discrete spectrum and norming constants). All parity induced symmetries are found to be more complicated than in the scalar (i.e., one-component) case. In particular, we show that the discrete eigenvalues giving rise to dark solitons arise in symmetric quartets, and those giving rise to dark-bright solitons in symmetric octets. We also characterize the differences between the purely even or purely odd case (in which both components are either even or odd functions of x) and the “mixed parity” cases (in which one component is even while the other is odd). Finally, we show how, in each case, the spatial symmetry yields a constraint on the possible existence of self-symmetric eigenvalues, corresponding to stationary solitons, and we study the resulting behavior of solutions.

Hydrodynamics of low-dimensional quantum systems

“…I fact, it is often useful to go back to the classical realm, in order to disentangle classical form quantum effects. In this special issue, contributions developed the hydrodynamic theory of integrable systems either quite generally [9,13], or by focussing on gases of particles [6,24], quantum spin systems [27], quantum and classical field theories [2,16,19], and even cellular automata [15,17,18,21,23] and ensembles of classical solitons of integrable partial differential equations [5,28]. Third, of particular interest in one-dimensional quantum systems is their often very peculiar or anomalous transport properties.…”