Multi-shockpeakons for the stochastic Degasperis-Procesi equation

Lynnyngs K. Arruda; Lynnyngs K. Arruda

doi:10.3934/era.2022117

Electronic Research Archive

2022, Volume 30, Issue 6: 2303-2320. doi: 10.3934/era.2022117

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Research article

Multi-shockpeakons for the stochastic Degasperis-Procesi equation

Lynnyngs K. Arruda ^,

Departamento de Matemática, Universidade Federal de São Carlos. CP 676, 13565-905, São Carlos - SP, Brazil

Received: 14 October 2021 Revised: 09 January 2022 Accepted: 09 January 2022 Published: 21 April 2022

The deterministic Degasperis-Procesi equation admits weak multi-shockpeakon solutions of the form

$ u(x, t) = \sum\limits_{i = 1}^nm_i(t)e^{-|x-x_i(t)|}-\sum\limits_{i = 1}^ns_i(t){\rm sgn}(x-x_i(t))e^{-|x-x_i(t)|}, $

where $ {\rm sgn}(x) $ denotes the signum function with $ {\rm sgn}(0) = 0 $, if and only if the time-dependent parameters $ x_i(t) $ (positions), $ m_i(t) $ (momenta) and $ s_i(t) $ (shock strengths) satisfy a system of $ 3n $ ordinary differential equations. We prove that a stochastic perturbation of the Degasperis-Procesi equation also has weak multi-shockpeakon solutions if and only if the positions, momenta and shock strengths obey a system of $ 3n $ stochastic differential equations.
Citation: Lynnyngs K. Arruda. Multi-shockpeakons for the stochastic Degasperis-Procesi equation[J]. Electronic Research Archive, 2022, 30(6): 2303-2320. doi: 10.3934/era.2022117

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Abstract

The deterministic Degasperis-Procesi equation admits weak multi-shockpeakon solutions of the form

$ u(x, t) = \sum\limits_{i = 1}^nm_i(t)e^{-|x-x_i(t)|}-\sum\limits_{i = 1}^ns_i(t){\rm sgn}(x-x_i(t))e^{-|x-x_i(t)|}, $

where $ {\rm sgn}(x) $ denotes the signum function with $ {\rm sgn}(0) = 0 $, if and only if the time-dependent parameters $ x_i(t) $ (positions), $ m_i(t) $ (momenta) and $ s_i(t) $ (shock strengths) satisfy a system of $ 3n $ ordinary differential equations. We prove that a stochastic perturbation of the Degasperis-Procesi equation also has weak multi-shockpeakon solutions if and only if the positions, momenta and shock strengths obey a system of $ 3n $ stochastic differential equations.

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