Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 073, 19 pages      arXiv:0810.4684      http://dx.doi.org/10.3842/SIGMA.2008.073

Solutions Classification to the Extended Reduced Ostrovsky Equation

Yury A. Stepanyants
Australian Nuclear Science and Technology, Organisation PMB 1, Menai (Sydney), NSW, 2234, Australia

Received July 14, 2008, in final form October 13, 2008; Published online October 26, 2008

Abstract
An alternative to the Parkes' approach [SIGMA 4 (2008), 053, 17 pages] is suggested for the solutions categorization to the extended reduced Ostrovsky equation (the exROE in Parkes' terminology). The approach is based on the application of the qualitative theory of differential equations which includes a mechanical analogy with the point particle motion in a potential field, the phase plane method, analysis of homoclinic trajectories and the like. Such an approach is seemed more vivid and free of some restrictions contained in [SIGMA 4 (2008), 053, 17 pages].

Key words: reduced Ostrovsky equation; mechanical analogy; phase plane; periodic waves; solitary waves, compactons.

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References

  1. Parkes E.J., Periodic and solitary travelling-wave solutions of an extended reduced Ostrovsky equation, SIGMA 4 (2008), 053, 17 pages, arXiv:0806.3155.
  2. Morrison A.J., Parkes E.J., The N-soliton solution of the modified generalised Vakhnenko equation (a new nonlinear evolution equation), Chaos Solitons Fractals 16 (2003), 13-26.
  3. Li J.-B., Dynamical understanding of loop soliton solution for several nonlinear wave equations, Sci. China Ser. A 50 (2007), 773-785.
  4. Stepanyants Y.A., On stationary solutions of the reduced Ostrovsky equation: Periodic waves, compactons and compound solitons, Chaos Solitons Fractals 28 (2006), 193-204.
  5. Ostrovsky L.A., Nonlinear internal waves in a rotating ocean, Okeanologiya 18 (1978), 181-191 (Engl. transl: Oceanology 18 (1978), 119-125).

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