On a class of nonlinear dispersive-dissipative interactions

We study a prototype, dissipative-dispersive equation; u_t + a(u^m)_x + (uⁿ)_xxx = μ(u^k)_xx, a, μ = consts., which represents a wide variety of interactions. At the critical value k = (m + n)/2 which separates dispersive- and dissipation-dominated phenomena, these effects are in a detailed balance and the patterns formed do not depend on the amplitude. In particular, when in = n + 2 = k + 1 the equation can be transformed into a form free of convection and dissipation, making it accessible to analysis. Both bounded and unbounded oscillations as well as solitary waves are found. A variety of exact solutions are presented, with a notable example being a solitary doublet. For n = 1 and a = (2μ/3)² the problem may be mapped into a linear equation, leading to rational, periodic or aperiodic solutions, among others.