Discontinuity, Nonlinearity, and Complexity
Extended Mixed AKNSLundRegge Model and Its Selfsimilarity Reduction
Discontinuity, Nonlinearity, and Complexity 3(2) (2014) 161168  DOI:10.5890/DNC.2014.06.005
D.V. Ruy$^{1}$; G.R. de Melo$^{2}$
$^{1}$ Instituto de Física TeóricaUNESP, Rua Dr Bento Teobaldo Ferraz 271, Bloco II, Sã Paulo, 01140070, 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, 44380000, Bahia, Brazil
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Abstract
We discuss the relation between the selfsimilarity reduction of the generalized mixed mKdVsinhGordon model and the fourthorder equation obtained by Kudryashov inJ. Phys. A: Math. Theor.35(2002) 9399. Also, it is shown two particular solutions for this equation. Then, we extend the mixed AKNSLundRegge model and study its selfsimilarity reduction. We obtain thefifth Painlevé equation as a particular case of this reduction and a fourthorder seconddegree equation otherwise. The relation between a integrable model and a fourthorder seconddegree 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 (2010/181109) 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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