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RESEARCH PRODUCT

Numerical study of blow-up and dispersive shocks in solutions to generalized Korteweg–de Vries equations

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Abstract We present a detailed numerical study of solutions to general Korteweg–de Vries equations with critical and supercritical nonlinearity, both in the context of dispersive shocks and blow-up. We study the stability of solitons and show that they are unstable against being radiated away and blow-up. In the L 2 critical case, the blow-up mechanism by Martel, Merle and Raphael can be numerically identified. In the limit of small dispersion, it is shown that a dispersive shock always appears before an eventual blow-up. In the latter case, always the first soliton to appear will blow up. It is shown that the same type of blow-up as for the perturbations of the soliton can be observed which indicates that the theory by Martel, Merle and Raphael is also applicable to initial data with a mass much larger than the soliton mass. We study the scaling of the blow-up time t ∗ in dependence of the small dispersion parameter ϵ and find an exponential dependence t ∗ ( ϵ ) and that there is a minimal blow-up time t 0 ∗ greater than the critical time of the corresponding Hopf solution for ϵ → 0 . To study the cases with blow-up in detail, we apply the first dynamic rescaling for generalized Korteweg–de Vries equations. This allows to identify the type of the singularity.