Dr. Michela Procesi

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Degasperis-Procesi equation is a real nonlinear partial differential equation which applies to water wave propagation and is solvable by the methods of soliton theory.

The DP equation

This is the partial differential equation

<math DP>

u_t-u_{xxt}-uu_{xxx}-3u_xu_{xx}+4uu_x=0 \,\,\,,\,\,\,u=u(x,t)\,\,, [/itex] where $$u$$ is the independent variable, $$x$$ is the space co-ordinate in the direction of propagation and $$t$$is the time variable. In this notation a subscripted variable indicates partial differentiation$u_x \equiv \partial u /\partial x\,\,,\,\,u_{xx}\equiv \partial^2 u /\partial x^2\,$, etc.. Some, but not all, coefficients may be given values which are different from those appearing in (<ref>DP</ref>), and even other terms may be added, by performing the Galilei transformation $$u(x,t)\rightarrow u'(x,t)=a+u(x-ct,t)$$, $$a$$and $$c$$being arbitrary constants. However, the relative values of some coefficients cannot be changed as they are crucial to the special solvability properties of the DP equation (<ref>DP</ref>). Indeed

References

• [DP99] Degasperis A., Procesi M. Asymptotic Integrability, Symmetry and Perturbation Theory (Rome, 1998) (A. Degasperis and G. Gaeta, eds.), World Scientific Publishing, New Jersey, 1999, pp. 23-37
• [DHH02] Degasperis A., Holm D. D., Hone A. N. W. A new integrable equation with peakons solutions, Theoret. and Math. Phys. 133 1463-1474 (2002)
• [MN02] Mikhailov A. V., Novikov V. S. Perturbative symmetry approach, J. Phys. A: Math. Gen. 35 4775-4790 (2002)
• [DGH03] Dullin H. R., Gottwald G. A., Holm D. D. Camassa-Holm, Korteweg-de Vries-5and other asymptotically equivalent equations for shallow water wave, Fluid Dynamics Research 33 73-95 (2003)
• [J03] Johnson R. S. The classical problem of water waves: a reservoir of integrable and nearly-integrable equations, J. Nonlin. Math. Phys. 10(Supplement 1): 72–92 (2003)
• [LS03] Lundmark H., Szmigielski J. Multi-peakon solutions of the Degasperis-Procesi equation, Inverse Problems 19 1241-1245 (2003)