Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities

Ola I. H. Maehlen

doi:10.3934/dcds.2020174

Article Contents

2020, Volume 40, Issue 7: 4113-4130. Doi: 10.3934/dcds.2020174

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Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities

Ola I. H. Maehlen

Department of Mathematical Sciences, NTNU – Norwegian University of Science and Technology, 7491 Trondheim, Norway

Received: February 2019

Revised: January 2020

Published online: July 2020

The author acknowledges the support from grant no. 250070 from the Research Council of Norway.

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Abstract

We prove existence of solitary-wave solutions to the equation

$ \begin{equation*} u_t+ (Lu - n(u))_x = 0\,, \end{equation*} $

for weak assumptions on the dispersion $ L $ and the nonlinearity $ n $. The symbol $ m $ of the Fourier multiplier $ L $ is allowed to be of low positive order ($ s > 0 $), while $ n $ need only be locally Lipschitz and asymptotically homogeneous at zero. We shall discover such solutions in Sobolev spaces contained in $ H^{1+s} $.

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Mathematics Subject Classification: Primary: 35A01; Secondary: 35A15, 35Q53, 76B03, 76B15.

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