Waveguide solutions for a nonlinear Schrödinger equation with mixed dispersion

Denis Bonheure 1, 2 Robson Nascimento 1
2 MEPHYSTO - Quantitative methods for stochastic models in physics
LPP - Laboratoire Paul Painlevé - UMR 8524, ULB - Université Libre de Bruxelles [Bruxelles], Inria Lille - Nord Europe
Abstract : In this note we provide some simple results for the 4NLS model i∂tψ + ∆ψ + |ψ| 2σ ψ − γ∆ 2 ψ = 0, where γ > 0. Our aim is to partially complete the discussion on waveguide solutions in [11, 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(iαt)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.
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Denis Bonheure, Robson Nascimento. Waveguide solutions for a nonlinear Schrödinger equation with mixed dispersion. Contributions to Nonlinear Elliptic Equations and Systems, 86, Springer, 2015, Progress in Nonlinear Differential Equations and Their Applications, ⟨10.1007/978-3-319-19902-3_4⟩. ⟨hal-01182833⟩

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