On a class of nonlinear dispersive-dissipative interactions. (English) Zbl 0938.35172

Summary: We study a prototype dissipative-dispersive equation \[ u_t+ a(u^m)_x+ (u^n)_{xxx}= \mu(u^k)_{xx}, \] \(a,\mu=\text{consts}.\), 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 \(m=n+2=k+1\) 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 \(a=(2\mu/3)^2\) the problem may be mapped into a linear equation, leading to rational, periodic or aperiodic solutions, among others.


35Q53 KdV equations (Korteweg-de Vries equations)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
37L50 Noncompact semigroups, dispersive equations, perturbations of infinite-dimensional dissipative dynamical systems
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