Self-similar blow-up solutions in the generalised Korteweg-de Vries equation: spectral analysis, normal form and asymptotics

In the present work we revisit the problem of the generalised Korteweg–de Vries equation parametrically, as a function of the relevant nonlinearity exponent, to examine the emergence of blow-up solutions, as traveling waveforms lose their stability past a critical point of the relevant parameter p, here at p = 5. We provide a normal form of the associated collapse dynamics, and illustrate how this captures the collapsing branch bifurcating from the unstable traveling branch. We also systematically characterise the linearisation spectrum of not only the traveling states, but importantly of the emergent collapsing waveforms in the so-called co-exploding frame where these waveforms are identified as stationary states. This spectrum, in addition to two positive real eigenvalues which are shown to be associated with the symmetries of translation and scaling invariance of the original (non-exploding) frame features complex patterns of negative eigenvalues that we also fully characterise. We show that the phenomenology of the latter is significantly affected by the boundary conditions and is far more complicated than in the corresponding symmetric Laplacian case of the nonlinear Schrödinger problem that has recently been explored. In addition, we explore the dynamics of the unstable solitary waves for p > 5 in the co-exploding frame.