Maria V. Demina — Page 2

arXiv2006

Polygons for finding exact solutions of nonlinear differential equations

New method for finding exact solutions of nonlinear differential equations is presented. It is based on constructing the polygon corresponding to the equation studied. The algorithms of power geometry are used. The method is applied for finding one -- parameter exact solutions of the generalized Korteveg -- de Vries -- Burgers equation, the generalized Kuramoto - Sivashinsky equation, and the fifth -- order nonlinear evolution equation. All these nonlinear equations contain the term $u^mu_x$. New exact solitary waves are found.

nlin.SI↗

arXiv2005

Power and non-power expansions of the solutions for the fourth-order analogue to the second Painlevé equation

Fourth - order analogue to the second Painlevé equation is studied. This equation has its origin in the modified Korteveg - de Vries equation of the fifth order when we look for its self - similar solution. All power and non - power expansions of the solutions for the fouth - order analogue to the second Painlevé equation near points $z=0$ and $z=\infty$ are found by means of the power geometry method. The exponential additions to solutions of the equation studied are determined. Comparison of the expansions found with those of the six Painlevé equations confirm the conjecture that the fourth - order analogue to the second Painlevé equation defines new transcendental functions.

nlin.SI↗