Resurgence in a Hamilton-Jacobi equation.
The authors study the resurgent structure associated with the Hamilton-Jacobi equation \(\partial _{\tau}\phi +H(z,\tau ,\partial _z\phi )=0\), where \(\tau \in {\mathbb R}/2\pi {\mathbb Z}\) and \(z\in {\mathbb C}\) are the independent variables, \(\mu\) is a complex parameter, and \(H(z,\tau ,p):=(-1/8)z^2p^2+2z^{-2}(1-\mu \sin \tau )\) is a Hamiltonian. The equation is obtained as the inner equation when studying the separatrix splitting problem for a perturbed pendulum via complex matching. Resurgence means that there is a divergent formal solution \(\widetilde{\phi }_0(z,\tau )=\sum _{n\geq 0}C_n(\tau )/z^{n+1}\) whose coefficients are \(2\pi\)-periodic, and whose formal Borel transform \(\widehat{\phi }_0(\zeta,\tau )=\sum _{n\geq 0}C_n(\tau )\zeta ^n/n!\) converges near the origin and defines a holomorphic function of \(\zeta\) with analytic continuation along any path of \({\mathbb C}\) which starts from the origin and avoids \(i{\mathbb Z}\). The authors derive the Bridge equation, which encompasses infinitely many resurgent relations satisfied by the formal solution and the other components of the formal integral.
scientific article; zbMATH DE number 2014675
scientific article