# Hoel 200709

The Degasperis-Procesi (DP) equation

utuxxt + 4uux = 3uxuxx + uuxxx

is a shallow water equation which is an asymptotic approximation of the Euler equation. One of the main reasons for the substantial interest given this equation is its close relation to the Euler equation. That is, one hopes to discover properties of the Euler equation by studying the DP equation (and other asymptotic approximations of the Euler equation like the Camassa-Holm equation).

One striking property of the DP equation is that it admits soliton like solutions called multi-shockpeakons. Multi-shockpeakons are functions of the form

$u(x,t) = \sum_{i=1}^n(m_i(t) -\sgn{x-x_i(t)}s_i(t))e^{-|x-x_i(t)|}$

whose evolution in time is described by a dynamical system of ODEs. This makes multi-shockpeakons relatively easy to simulate numerically. In my master's thesis I created a multi-shockpeakon based numerical algorithm for solving certain Cauchy problems of the DP equation.

In this talk I will describe some general properties of the DP equation and the above mentioned numerical algorithm.