The Small Dispersion Limit of the Korteweg-De Vries Equation

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There are many physical systems which display shocks i.e. regions in space where the solution develops extremely large slopes. In general, such systems are too complicated to be treated by exact calculation and their properties are best studied through the proof of general theorems. A model of the formation and propagation of dispersive shocks in one space dimension, in which explicit calculation is possible, is given by the initial value problem for the Korteweg-de Vries equation: (1.1a) $$ {u_{t}} - 6u{u_{x}} + {\varepsilon ^{2}}u{u_{{xxx}}} = 0 $$ (1.1b) $$ u(x,o,\varepsilon ) = - v(x)$$ in the limit ε → 0.