Self-Similar Solutions of a Burgers-Type Equation
with Quadratically Cubic Nonlinearity

Academician of the RAS O. V. Rudenko^{a, b, c, d, e} and V. A. Gusev^a

Received July 17, 2015

Abstract—Self-similar solutions are found for a quadratically cubic second-order partial differential equation
governing the behavior of nonlinear waves in various distributed systems, for example, in some metamaterials.
They are compared with self-similar solutions of the Burgers equation. One of them describing a single unipolar
pulse is shown to satisfy both equations. The other self-similar solutions of the quadratically cubic equation
behave differently from the solutions of the Burgers equation. They are constructed by matching the positive
and negative branches of the solution, so that the function itself and its first derivative are continuous. One of
these solutions corresponds to an asymmetric solitary N-wave of the sonic shock type. Self-similar solutions of
a quadratically cubic equation describing the propagation of cylindrically symmetric waves are also found.

DOI: 10.1134/S1064562416010051

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