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Discontinuity, Nonlinearity, and Complexity 3(2) (2014) 161--168 | DOI:10.5890/DNC.2014.06.005

D.V. Ruy$^{1}$; G.R. de Melo$^{2}$

$^{1}$ Instituto de Física Teórica-UNESP, Rua Dr Bento Teobaldo Ferraz 271, Bloco II, Sã Paulo, 01140-070, Brazil

$^{2}$ Núcleo Interdisciplinar em Ciência, Engenharia e Tecnologia, Centro de Ciências Exatas e Tecnológicas, Universidade Federal do Recôncavo da Bahia, Campus Universitário de Cruz das Almas Cruz das Almas, 44380-000, Bahia, Brazil

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Abstract

We discuss the relation between the self-similarity reduction of the generalized mixed mKdV-sinh-Gordon model and the fourth-order equation obtained by Kudryashov inJ. Phys. A: Math. Theor.35(2002) 93-99. Also, it is shown two particular solutions for this equation. Then, we extend the mixed AKNS-Lund-Regge model and study its self-similarity reduction. We obtain thefifth Painlevé equation as a particular case of this reduction and a fourth-order second-degree equation otherwise. The relation between a integrable model and a fourth-order second-degree equation is interesting because the general solution of this equation must have the Painlevé property due the Ablowitz, Ramani and Segur conjecture.

Acknowledgments

D. V. Ruy thanks FAPESP (2010/18110-9) for financial support. G. R. de Melo is grateful to the Instituto de Física Teórica for the hospitality. The authors are thankful to A. H. Zimerman and H. Aratyn for discussions.

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