Well-posedness for a coupled system of Kawahara/KdV type equations with polynomials nonlinearities

Cezar Kondo; Ronaldo Pes

doi:10.3934/cpaa.2022063

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2022, Volume 21, Issue 8: 2615-2641. Doi: 10.3934/cpaa.2022063

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Well-posedness for a coupled system of Kawahara/KdV type equations with polynomials nonlinearities

Cezar Kondo and
Ronaldo Pes^,

Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP 13565-905, Brazil

^*Corresponding author
^*Corresponding author

Received: August 31, 2021

Early access: April 2022

Published: August 2022

The first author is supported by CAPES - finance code 001

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Abstract

We consider the initial value problem (IVP) associated to a coupled system of modified Kawahara/KdV type equations with polynomials nonlinearities. For the model in question, the Cauchy problem is of interest, and is shown to be well-posed for given data in a Gevrey spaces. Our results make use of techniques presented in Grujić and Kalisch, who studied the Gevrey regularity for a class of water-wave models and the well-posedness of a IVP associated to a general equation. The proof relies on estimates in space-time norms adapted to the linear part of the equations. In particular, estimates in Bourgain spaces are proven for the linear and nonlinear terms of the system and the main result is obtained by a contraction principle. The class of system in view generalizes the system of modified Kawahara/KdV type equations studied by Kondo and Pes, which contains a number of systems arising in the modeling of waves in fluids, stability and instability of solitary waves and models for wave propagation in physical systems where both nonlinear and dispersive effects are important.

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Mathematics Subject Classification: Primary: 35Q53.

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