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)


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.


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



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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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Département de MathématiqueUniversité Libre de BruxellesBruxellesBelgium

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