In mathematical physics, the Degasperis–Procesi equation <math>\displaystyle u_t - u_{xxt} + 2\kappa u_x + 4u 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 <math>\kappa and b are real parameters (b=3 for the Degasperis–Procesi equation).

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  • In mathematical physics, the Degasperis–Procesi equation <math>\displaystyle u_t - u_{xxt} + 2\kappa u_x + 4u 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 <math>\kappa and b are real parameters (b=3 for the Degasperis–Procesi equation). It was discovered by Degasperis and 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 <math>\kappa > 0) has later been found to play a similar role in water wave theory as the Camassa–Holm equation.
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  • In mathematical physics, the Degasperis–Procesi equation <math>\displaystyle u_t - u_{xxt} + 2\kappa u_x + 4u 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 <math>\kappa and b are real parameters (b=3 for the Degasperis–Procesi equation).
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  • Degasperis–Procesi equation
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