Asymptotic Problem for Second-Order Ordinary Differential Equation with Nonlinearity Corresponding to a Butterfly Catastrophe

Abstract

For the second-order nonlinear ordinary differential equation ${u}_{xx}^{\hbox{'}\hbox{'}}={u}^5-{tu}^3-x,$ we prove the existence and uniqueness of a strictly increasing solution, which satisfies an initial condition and a limit condition at infinity and whose graph lies between the zero equation and the continuous graph of the root of the nondifferential equation u⁵ − tu³ − x = 0. For this solution, we find an asymptotics, which is uniform on the ray t ∈ (−∞,−M^t) as x → +∞; separately, we construct asymptotics on the ray s > M^s and on the segment 0 ≤ s ≤ M^s, where s = |t|^−5/2x is the variable compressed with respect to x. Using the method of matching of asymptotic expansions, we construct a composite asymptotic expansion of the solution to the Cauchy problem whose initial conditions are found from the theorem on the existence of solutions to the original problem. Finally, we construct a uniform asymptotic expansion under the restriction t ≤ 0 as x² + t² →∞.