Numerical computations of self-similar blow-up solutions of the generalized Korteweg-de Vries equation
The paper deals with the initial value problem for the generalized Korteweg-de Vries equation (GKdV) \(u_t+u^pu_x+\rho u_{xxx}= 0\) with \(p \geq 1\) (integer). The authors come from the known theoretical results which are of the following spirit: For \(p=1,2,3\) the a priori estimates on the solution exist for any \(t>0\). However, when \(p \geq 4\), either the solution has a global smooth character, or it blows-up in \(L^{\infty }\)-norm in finite time. For simplicity, the authors consider the case \(p=5\) and study numerically the solution of the GKdV equation arising from a perturbed unstable solitary wave. The authors examine in details the structure of the solution near the blow-up and find the self-similar behaviour. Finally, they discuss the possible explanation how the presence of both nonlinearity and dispersion can cause the observed behaviour.
scientific article; zbMATH DE number 1453655