The article "The Degasperis-Procesi equation", by Degasperis and Procesi, is well-written and contains very useful informations on a distinguished integrable model arising in shallow water theory. Therefore we strongly suggest its acceptance. Suggested change: the authors should make the sentence "Moreover the initial value .. in an interval", line 13 of the "Solutions" section, more understandable.
Thanks for pointing out an unclear statement. I quite agree that a clarification is necessary and I have made the appropriate changes (see the last version of my article).
This is an excellent article on the Degasperis-Procesi equation, and provides an important service to the research community by making an accurate summary of the relevant information freely available to all. I have a few minor suggestions for changes. [I have used Latex notation for some of the mathematical terms.]
If the statement before the Contents section wants to mention that the DP equation is a "real" nonlinear p.d.e. (rather than a complex ampltude equation (e.g. like NLS) then perhaps the first line after (1) should say "for a real function $u(x,t)$". I think the 3rd sentence could be phrased better as "The DP equation has real coefficients $a,c,d,f$ with $af\neq 0$, and the form of the equation is covariant with respect to a combination of scaling, shifting and Galilean transformations (see below). The factor 4..." Then in the fourth line after (2) say "The DP equation (1) is covariant under the group of transformations $u(x,t)\rightarrow u'(x',t')=\alpha u(\beta x+\gamma t,t)+\delta$, and by a suitable choice of the parameters $\alpha, \beta, \gamma, \delta$ the coefficients can be fixed as $a=1,c=0,d=0,f=1$. " [The function should be referred to as u'(x',t')$ since really there are new independent variables x',t'.] Then in the next sentence could say "With this choice of the coefficients (after dropping the primes) the DP ..."
Earlier, on line 11 of p.2, better say "solitonic equations (that is, PDEs with infinitely many conservation laws)..."
p.3, line 2 say "while the other sequence is obtained..." Also, after (9) I would mention that "The operators B_0 and B_1 form a compatible bi-Hamiltonian pair (Hone and Wang 2003)." The paper `Prolongation algebras and Hamiltonian operators for peakon equations', A.N.W. Hone and Jing Ping Wang, Inverse Problems, 19, 129-145 (2003) should be cited here.
[Another thing that could be mentioned here (or with the Lax pair on p.2) is that the DP equation is related to one member of the hierarchy of symmetries of another well known integrable PDE, the Kaup-Kupershmidt equation, by means of a reciprocal transformation. This link is discussed in the first paper of Degasperis, Holm and Hone, and in a bit more detail in the aforementioned paper by Hone and Wang; but perhaps it is a technical fine point that should not be mentioned here.]
Bottom of p.3 better to say "even if ... is smooth, if it satisfies additional technical conditions then the corresponding..."
p.4 "so-called" (one dash)
p.5 line 2 "some of them have been constructed explicitly (Qiao 2008)..."
Thanks for the suggestions which do improve the article. I have accepted and made all the changes suggested (see the last version of my article) with the exception of one. In fact I do not agree with the notation of the reviewer regarding the transformation which covariantly changes the DP equation. The reason is that introducing the variables \(x'\) and \(t'\) is confusing because they appear in the lhs side but not in the rhs of the equation. To accept this notation one should correctly add the transformation of coordinates \((x,t)\rightarrow (x',t')\). My notation, which is quite explicit and correct, just avoids this coordinate transformation. Conclusion, I have not accepted the \(u'(x',t') \) notation.