Asymptotic problem for second-order ordinary differential equation with nonlinearity corresponding to a butterfly catastrophe (Q2214386)

For the second order nonlinear differential equation \[u''_{xx}=u^5-tu^3-x,\] the author proves 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 \(u^5-tu^3-x\). For this solution, the author constructs asymptotics on the rays \(t\in (-\infty, -M^t)\) as \(x\to \infty\) and \(s>M^s\), and on the interval \(0\le s\le M^s\), where \(s=|t|^{-\frac52}x\) is the variable compressed with respect to \(x\). In addition, the author constructs as well a composite asymptotic expansion of the solution of the Cauchy problem whose initial conditions are found from the theorem on the existence of solutions to the original problem and a uniform asymptotic expansion under the restriction \(t\le 0\) as \(x^2+t^2\to \infty\).