A novel analytic technique for the disintegration of coherent structure solutions of non-integrable and integrable nonlinear PDEs under forcing

We consider a technique for deriving exact analytic coherent structure (pulse/front/domain wall) solutions of general NLPDEs via the use of truncated invariant Painlevé expansions. Coupling this to Melnikov theory, we then consider the breakdown to chaos of such analytic coherent structure solutions of the Zakharov–Kuznetsov and reaction-diffusion equations under forcing. We also demonstrate that similar treatments are possible for integrable systems (using the well-studied forced KdV equation as an example) where the soliton/kink solutions represent the homoclinic/heteroclinic structures of the reduced ODEs. A method of treating the dynamics of the system prior to the onset of chaos by the use of intrinsic harmonic balance, multiscale or direct soliton perturbation theory is briefly discussed. It is conceivable that resummation of such perturbation series via the use of Fade approximants or other techniques may enable one to analytically follow the homoclinic or heteroclinic tangling beyond the first transversal intersection of the stable and unstable manifolds and into the chaotic regime.