Multi-shockpeakons for the stochastic Degasperis-Procesi equation

Abstract

<abstract><p>The deterministic Degasperis-Procesi equation admits weak multi-shockpeakon solutions of the form</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ 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)|}, $\end{document} </tex-math></disp-formula></p> <p>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.</p></abstract>