# zbMATH — the first resource for mathematics

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.

##### MSC:
 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
Full Text:
##### References:
 [1] Rosenau, P.; Hyman, J.M., Phys. rev. lett., 70, 564, (1993) [2] Rosenau, P., Phys. lett. A, 211, 265, (1996) [3] Rosenau, P., Phys. lett. A, 230, 305, (1997) [4] Rosenau, P.; Kamin, S., Physica D, 8, 273, (1983) [5] Oron, A.; Rosenau, P., Phys. rev. E, 55, 1267, (1997) [6] Biler, P., Bull. Polish acad. sci, 32, 279, (1984) [7] Amick, C.J.; Bona, J.L.; Schonbek, M.E., J. diff. eqs., 81, 1, (1989) [8] Schonbek, M.E.; Rajopadhye, S.V., Ann-inst. H. poincare-anal.-non-linearie, 12, 425, (1995) [9] Bona, J.L.; Schonbek, M.E., (), 207, Sect. A [10] Jacobs, D.; McKinney, B.; Shearer, M., J. diff. eqs., 116, 448, (1995) [11] Kondoh, Y.; Van Dam, J.W., Phys. rev. E, 52, 1721, (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.