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

For the second-order nonlinear ordinary differential equation ${u''_{xx}=u^5-tu^3-x}$, 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 ${u^5-tu^3-x=0}$. For this solution, we find an asymptotics, which is uniform on the ray ${t\in(-\infty,-M^t)}$ as $x\to+\infty$; separately, we construct asymptotics on the ray ${s>M^s}$ and on the segment ${0\leq s\leq M^s}$, where ${s=|t|^{-5/2}x}$ is the variable compressed with respect to $x$. Using the method of matching 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 ${t\leq 0}$ as ${x^2+t^2\to\infty}$.