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Center for Nonlinear Studies |
We discuss new kinds of nonlinear eigenvalue problems, which are associated with instabilities, separatrix behavior, and hyperasymptotics. First, we consider the toy differential equation y'=cos(pi x y), which arises in several physical contexts. We show that the initial condition y(0) falls into discrete classes: a_{n-1} < y(0) < a_n (n=1,2,3,...). If y(0) is in the nth class, y(x) exhibits n oscillations. The boundaries a_n of these classes are strongly analogous to quantum-mechanical eigenvalues and calculating the large-n behavior of a n is analogous to a semiclassical (WKB) approximation in quantum mechanics. For large n, a_n is asymptotic to A\sqrt{n}, where A=2^{5/6}. The constant A is numerically close to the lower bound on the power-series constant P, which plays a fundamental role in the theory of complex variables and which is associated with the asymptotic behavior of zeros of partial sums of Taylor series. The first two Painleve transcendents P1 and P2 have a remarkable eigenvalue behavior. As n-->\infty, the nth eigenvalue for P1 grows like Bn^{3/5} and the nth eigenvalue for P2 grows like Cn^{2/3). We calculate the constants B and C analytically by reducing the Painleve transcendents to linear eigenvalue problems in PT-symmetric quantum theory.
Host: Avadh Saxena