Show that there also exist periodic traveling wave
Chapter 14, Problem 4(choose chapter or problem)
Show that there also exist periodic traveling wave solutions of KdV, as follows. Let P(f)= 2f 3 +cf2 +2af +b.(a) By solving the ODE ( f)2 = P(f), nd an implicit formula de-ning f(x). (b) Show that each simple zero of P(f) corresponds to a minimum or maximum of f(x).(c) Show that one can choose a and b in such a way that the cubic polynomial P(f) has three real zeros f1 < f2 < f3. Show that P(f) > 0 for f2 < f < f3 and for f< f1. (d) Show that there is a periodic solution f(x) with maxx f(x)=f3 and minx f(x)=f2. (e) Look up what an elliptic integral is (in [MOS], for instance) and transform the formula in (a) to an elliptic integral of the rst kind.
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