On the spectral stability of periodic traveling waves for the critical Korteweg-de Vries and Gardner equations

Fábio Natali (), Eleomar Cardoso () and Sabrina Amaral ()
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Fábio Natali: State University of Maringá
Eleomar Cardoso: Federal University of Santa Catarina
Sabrina Amaral: State University of Maringá

Partial Differential Equations and Applications, 2021, vol. 2, issue 3, 1-20

Abstract: Abstract In this paper, we determine spectral stability results of periodic waves for the critical Korteweg-de Vries and Gardner equations. For the first equation, we show that both positive and zero mean periodic traveling wave solutions possess a threshold value which may provides us a rupture in the spectral stability. Concerning the second equation, we establish the existence of periodic waves using a Galilean transformation on the periodic cnoidal solution for the modified Korteweg-de Vries equation and for both equations, the threshold values are the same. The main advantage presented in our paper concerns in solving some auxiliary initial value problems to obtain the spectral stability.

Keywords: Spectral stability; Spectral instability; Periodic waves; KdV type equations; 76B25; 35Q51; 35Q53 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s42985-021-00095-7

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