Multi-bump patterns by a normal form approach. (Q5951591)

The author considers one of the simplest systems where the Ginzburg-Landau (GL) approximation breaks down, i.e., the reaction diffusion equation NEWLINE\[NEWLINEu_t = u_{xx} + r u + v + N_1 (u,v), \qquad v_t = D v_{xx} - u + s v + N_2 (u,v), NEWLINE\]NEWLINE with \(N_i\) fully nonlinear terms, \(D\) a fixed constant, and \(r,s\) bifurcation parameters; in the \(r,s\) plane there is a (codimension-two) point \(p_*\) where the GL approximation is not valid, \(p_*\) being characterized by the tangential intersection of the bifurcation curves \(rs+1 = 0\) and \(D r - s + 2 \sqrt{D} = 0\).NEWLINENEWLINEThis system was analyzed previously in [\textit{V. Rottschäfer} and \textit{A. Doelman}, On the transition from the Ginzburg-Landau equation to the extended Fisher-Kolmogorov equation, Physica D 118, 261--292 (1998)] with a modulation equation approach; it was shown there that the GL equation is valid away from \(p_*\), while as \(p_*\) is approached one has to use instead an ``extended Fisher-Kolmogorov equation''; numerical simulations showed the appearance of asymptotically stable multi-bump solutions in this region.NEWLINENEWLINEThis paper concentrates on the region near \(p_*\); the analysis is conducted via a normal form approach, through which the author reduces to study a three-dimensional system of ODEs, which is amenable to exact treatment under certain conditions. It is shown that in this case the solutions have the same multi-bump structure observed in numerical experiments, which validates the approach adopted here. A comparison of this approach with the modulation equation approach considered in [V. Rottschäfer and A. Doelman, op. cit.] is also given.