Degasperis–Procesi equation explained

In mathematical physics, the Degasperis–Procesi equation

\displaystyle u _{t} - u _{xxt} + 2 \kappa u _{x} + 4 u u _{x} = 3 u _{x} u _{xx} + u u _{xxx}

is one of only two exactly solvable equations in the following family of third-order, non-linear, dispersive PDEs:

\displaystyle u _{t} - u _{xxt} + 2 \kappa u _{x} + (b +1 )u u _{x} = b u _{x} u _{xx} + u u _{xxx} ,

where

\kappa and b are real parameters (b=3 for the Degasperis–Procesi equation). It was discovered by Antonio Degasperis and Michela Procesi in a search for integrable equations similar in form to the Camassa–Holm equation, which is the other integrable equation in this family (corresponding to b=2); that those two equations are the only integrable cases has been verified using a variety of different integrability tests. Although discovered solely because of its mathematical properties, the Degasperis–Procesi equation (with \kappa > 0 ) has later been found to play a similar role in water wave theory as the Camassa–Holm equation.

Soliton solutions

See main article: Peakon.

Among the solutions of the Degasperis–Procesi equation (in the special case

\kappa =0 ) are the so-called multipeakon solutions, which are functions of the form\displaystyle u(x,t) = \sum ^{n} _{i =1} m _{i(t)} e ^{- |x - x _{i(t)|}}

where the functions

m _{i} and x _{i} satisfy• x _{i} = \sum ^{n} _{j =1} m _{j} e ^{- |x _{i-x} _{j|}} , • m _{i} = 2 m _{i} \sum ^{n} _{j =1} m _{j} 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.

When

\kappa > 0 the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as \kappa tends to zero.

Discontinuous solutions

The Degasperis–Procesi equation (with

\kappa =0 ) is formally equivalent to the (nonlocal) hyperbolic conservation law\partial _{t} u + \partial _{x} \left[ (u ^{2})/(2) + (G)/(2) * (3 u ^{2})/(2) \right] = 0 ,

where

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). In contrast, the corresponding formulation of the Camassa–Holm equation contains a convolution involving both u ^{2} and u ^{2} _{x} , which only makes sense if u lies in the Sobolev space 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