ABSTRACT
Proposition 8.83 Problem (394), (395) for ODEs (389)–(393) admits a solution g(z), which is an odd analytic function.
Uniqueness for such higher-order ODEs is a more difficult problem, which is not studied here, though it has been seen numerically. Moreover, there are some analogous results. We refer to the paper [181] (to be used later on), where uniqueness of a fourth-order semilinear ODE was established by an improved shooting argument. Notice another difficult aspect. Figures 8.40-8.42 above, which were ob-
tained by careful numerics, clearly show that the positivity holds:
g(z) > 0 for z < 0, (408)
which is also difficult to prove rigorously; see further comments below. Actually, (408) is not that important for the key convergence (376), since possible sign changes (if any) disappear in the limit, as t → T−. It seems that nothing prevents the existence of some ODEs from the family (389)–(393), with different nonlinearities, for which the shock profiles can change sign for z < 0.
Proof. As above, we consider the first ODE (389) only. We use a shooting argument, using the 2D bundle of asymptotics (406). By scaling (407), we put C = −1, so actually, we deal with the one-parameter shooting problem with the 1D family of orbits satisfying
