arXiv subjects
Publications and source records attributed to Olga Yu. Efimova.
The modification of simplest equation method to look for exact solutions of nonlinear partial differential equations is presented. Using this method we obtain exact solutions of generalized Korteweg-de Vries equation with cubic source and exact solutions of third-order Kudryashov-Sinelshchikov equation describing nonlinear waves in liquids with gas bubbles.
The generalization of the simplest equation method to look for exact solutions of systems of nonlinear differential equations is presented. The exact solutions of NDE systems describing the evolution of two interacting populations in two cases (both populations have the low critical density or low critical density is typical for only one of populations) are obtained.
The fourth-order ordinary differential equation, defining new transcendents, is studied. The self-similar solutions of the Kaup-Kupershmidt and Savada-Kotera equations are shown to be found taking its solutions into account. Equation studied belongs to the class of fourth-order analogues of the Painlevé equations. All the power and non-power asymptotic forms and expansions near points $z=0$, $z=\infty$ and near arbitrary point $z=z_0$ are found by means of power geometry methods. The exponential additions to the solutions of studied equation are also determined.
The Burgers-Huxley equation is studied. All power and power-logarithmic expansions for travelling-wave solutions of this equation are presented. Using the power expansions, some exact solutions of this equation are found.
One of the fourth-order analog to the first Painlevé equation is studied. All power expansions for solutions of this equation near points $z=0$ and $z=\infty$ are found by means of the power geometry method. The exponential additions to the expansion of solution near $z=\infty$ are computed. The obtained results confirm the hypothesis that the fourth-order analog of the first Painlevé equation determines new transcendental functions.