Degasperis–Procesi equation

Among the solutions of the Degasperis–Procesi equation (in the special case ${\displaystyle \kappa =0}$) are the so-called multipeakon solutions, which are functions of the form

${\displaystyle \displaystyle u(x,t)=\sum _{i=1}^{n}m_{i}(t)e^{-|x-x_{i}(t)|}}$

where the functions ${\displaystyle m_{i}}$ and ${\displaystyle x_{i}}$ satisfy[3]

${\displaystyle {\dot {x}}_{i}=\sum _{j=1}^{n}m_{j}e^{-|x_{i}-x_{j}|},\qquad {\dot {m}}_{i}=2m_{i}\sum _{j=1}^{n}m_{j}\,\operatorname {sgn} {(x_{i}-x_{j})}e^{-|x_{i}-x_{j}|}.}$

These ODEs can be solved explicitly in terms of elementary functions, using inverse spectral methods.[4]

When ${\displaystyle \kappa >0}$ the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as ${\displaystyle \kappa }$ tends to zero.[5]

The Degasperis–Procesi equation (with ${\displaystyle \kappa =0}$) is formally equivalent to the (nonlocal) hyperbolic conservation law

${\displaystyle \partial _{t}u+\partial _{x}\left[{\frac {u^{2}}{2}}+{\frac {G}{2}}*{\frac {3u^{2}}{2}}\right]=0,}$

where ${\displaystyle G(x)=\exp(-|x|)}$, and where the star denotes convolution with respect to x. In this formulation, it admits weak solutions with a very low degree of regularity, even discontinuous ones (shock waves).[6] In contrast, the corresponding formulation of the Camassa–Holm equation contains a convolution involving both ${\displaystyle u^{2}}$ and ${\displaystyle u_{x}^{2}}$, which only makes sense if u lies in the Sobolev space ${\displaystyle H^{1}=W^{1,2}}$ with respect to x. By the Sobolev embedding theorem, this means in particular that the weak solutions of the Camassa–Holm equation must be continuous with respect to x.