Multitransition Homoclinic and Heteroclinic Solutions of the Extended Fisher–Kolmogorov Equation

Shooting methods and topological transversality

“…• This last corollary is of the same flavour of a similar result known for the waterwave equation (1.1) (see [9]) and is a contribution to the understanding of the chaotic dynamics of (EFK) (see [22,23,31]). …”

“…We shall also need the following lemma. • This gives a new proof that the extended Fisher-Kolmogorov equation is chaotic for all values of 7 > | (see [22,23,31]). …”

Multiple Permanent‐wave Trains in Nonlinear Systems

“…For fourth-order systems which allow permanent waves exponentially and oscillatorily approaching a constant at infinity, we will show that countably infinite multiple permanent-wave trains exist and can be readily constructed. Thus the results in [4] and [5] are reproduced. For the coupled nonlinear Schrödinger equations, we will show that countably infinite multiple solitary-wave trains can be constructed in a large portion of the parameter space.…”

“…But in many cases, Eq. (2.52) can be cast into a self-adjoint system (see [2], [4] and [5]). Then Ψ (k) (x) and its coefficients {d (k) j } can be readily obtained from dΦ (k) /dx, and the verification of conditions (2.68) can proceed.…”

“…The permanent waves in many nonlinear wave problems are governed by fourth-order systems (2.2) (see [2], [4] and [5]). In this section, we apply theorems 1 and 2 to certain classes of such systems.…”