Examples of the absence of a traveling wave for the generalized Korteweg-de Vries-Burgers equation

An example of convex function f(u) for which the generalized Korteweg-de Vries-Burgers equation u _t + (f(u)) _x + au _xxx − bu _xx = 0 has no solutions in the form of a traveling wave with specified limits at infinity is constructed. This example demonstrates the difficulties in analyzing asymptotic behavior of the Cauchy problem for the Korteweg-de Vries-Burgers equation that is not inherent in the type of equation for the conservation law, the Burgers-type equation, and its finite difference analog.