
SIGMA 3 (2007), 086, 8 pages arXiv:0705.2889
https://doi.org/10.3842/SIGMA.2007.086
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics
On Transformations of the Rabelo Equations
Anton Sakovich ^{a} and Sergei Sakovich ^{b}
^{a)} National Centre of Particle and High Energy Physics, 220040 Minsk, Belarus
^{b)} Institute of Physics, National Academy of Sciences, 220072 Minsk, Belarus
Received May 28, 2007, in final form August 22, 2007; Published online September 03, 2007
Abstract
We study four distinct secondorder nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constantcharacteristic form we relate them to some wellstudied integrable equations. Two of the Rabelo equations are found to be related to the sineGordon equation. The other two are transformed into a linear equation and the Liouville equation, and in this way their general solutions are obtained.
Key words:
nonlinear PDEs; transformations; integrability.
pdf (188 kb)
ps (118 kb)
tex (9 kb)
References
 Rabelo M.L., On equations which describe pseudospherical surfaces, Stud. Appl. Math. 81 (1989), 221248.
 Beals R., Rabelo M., Tenenblat K., Bäcklund transformations and inverse scattering solutions for some pseudospherical surface equations, Stud. Appl. Math. 81 (1989), 125151.
 Wadati M., Konno K., Ichikawa Y.H., New integrable nonlinear evolution equations, J. Phys. Soc. Japan 47 (1979), 16981700.
 Calogero F., A solvable nonlinear wave equation, Stud. Appl. Math. 70 (1984), 189199.
 Pavlov M.V., The Calogero equation and Liouvilletype equations, Theoret. and Math. Phys. 128 (2001), 927932, nlin.SI/0101034.
 Schäfer T., Wayne C.E., Propagation of ultrashort optical pulses in cubic nonlinear media, Phys. D 196 (2004), 90105.
 Sakovich A., Sakovich S., The short pulse equation is integrable, J. Phys. Soc. Japan 74 (2005), 239241, nlin.SI/0409034.
 Sakovich A., Sakovich S., Solitary wave solutions of the short pulse equation, J. Phys. A: Math. Gen. 39 (2006), L361L367, nlin.SI/0601019.
 Brunelli J.C., The short pulse hierarchy, J. Math. Phys. 46 (2005), 123507, 9 pages, nlin.SI/0601015.
 Brunelli J.C., The biHamiltonian structure of the short pulse equation, Phys. Lett. A 353 (2006), 475478, nlin.SI/0601014.
 Matsuno Y., Multiloop solutions and multibreather solutions of the short pulse model equation, J. Phys. Soc. Japan 76 (2007), 084003, 6 pages.
 Parkes E.J., Some periodic and solitary travellingwave solutions of the shortpulse equation, Chaos Solitons Fractals, to appear, doi:10.1016/j.chaos.2006.10.055.
 Olver P.J., Applications of Lie groups to differential equations, Springer, New York, 1986.
 Chen H.H., General derivation of Bäcklund transformations from inverse scattering problems, Phys. Rev. Lett. 33 (1974), 925928.

