Waveguide solutions for a nonlinear Schrödinger equation with mixed dispersion

Abstract

In this note we provide some simple results for the 4NLS model i∂tψ + ∆ψ + |ψ| 2σψ − γ∆ 2 ψ = 0 where γ > 0. Our aim is to partially complete the discussion on waveguide solutions in Fibich (SIAM J Appl Math 62:1437–1462, 2002, Section 4.1). In particular, we show that in the model case with a Kerr nonlinearity (σ=1), the least energy waveguide solution ψ(t, x) = exp(i α t)u(x) with α > 0 is unique for small γ and qualitatively behaves like the waveguide solution of NLS. On the contrary, oscillations arise at infinity when γ is too large.