The Whitham modulation equations for the defocusing nonlinear Schrödinger (NLS) equation in two and three spatial dimensions are derived using a two-phase ansatz for the periodic traveling wave solutions and by period-averaging the conservation laws of the NLS equation. The resulting Whitham modulation equations are written in vector form, which allows one to show that they preserve the rotational invariance of the NLS equation, as well as the invariance with respect to scaling and Galilean transformations, and to immediately generalize the calculations from two spatial dimensions to three. The transformation to Riemann-type variables is described in detail; the harmonic and soliton limits of the Whitham modulation equations are explicitly written down; and the reduction of the Whitham equations to those for the radial NLS equation is explicitly carried out. Finally, the extension of the theory to higher spatial dimensions is briefly outlined. The multidimensional NLS-Whitham equations obtained here may be used to study large amplitude wavetrains in a variety of applications including nonlinear photonic and matter waves.

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We characterize the soliton solutions and their interactions for a system of coupled evolution equations of nonlinear Schrödinger (NLS) type that models the dynamics in one-dimensional repulsive Bose-Einstein condensates with spin one, taking advantage of the representation of such model as a special reduction of a 2 × 2 matrix NLS system. Specifically, we study in detail the case in which solutions tend to a non-zero background at space infinities. First we derive a compact representation for the multisoliton solutions in the system using the Inverse Scattering Transform (IST). We introduce the notion of canonical form of a solution, corresponding to the case when the background as x → ∞ is proportional to the identity. We show that solutions for which the asymptotic behavior at infinity is not proportional to the identity, referred to as being in non-canonical form, can be reduced to canonical form by unitary transformations that preserve the symmetric nature of the solution (physically corresponding to complex rotations of the quantization axes). Then we give a complete characterization of the two families of onesoliton solutions arising in this problem, corresponding to ferromagnetic and to polar states of the system, and we discuss how the physical parameters of the solitons for each family are related to the spectral data in the IST. We also show that any ferromagnetic one-soliton solution in canonical form can be reduced to a single dark soliton of the scalar NLS equation, and any polar one-soliton solution in canonical form is unitarily equivalent to a pair of oppositely polarized displaced scalar dark solitons up to a rotation of the quantization axes. Finally, we discuss two-soliton interactions and we present a complete classification of the possible scenarios that can arise depending on whether either soliton is of ferromagnetic or polar type.

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.