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Waveguide solutions for a nonlinear Schrödinger equation with mixed dispersion

  • Denis BonheureEmail author
  • Robson Nascimento
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 86)

Abstract

In this note we provide some simple results for the 4NLS model
$$\displaystyle{i\partial _{t}\psi +\varDelta \psi +\vert \psi \vert ^{2\sigma }\psi -\gamma \varDelta ^{2}\psi = 0,}$$
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.

Keywords

Schrödinger equation mixed dispersion NLS biharmonic NLS waveguide solutions ground state solutions 

Notes

Acknowledgements

D. Bonheure is supported by INRIA - Team MEPHISTO, MIS F.4508.14 (FNRS), PDR T.1110.14F (FNRS) & ARC AUWB-2012-12/17-ULB1- IAPAS.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Département de MathématiqueUniversité Libre de BruxellesBruxellesBelgium

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