Advancing Stability through Rigorous Computations

Computers and computational methods are an increasingly important part of the scientific endeavor, and they are changing the ways in which science progresses. One new and important such methodology is that of validated numerics. Traditional numerical methods produce approximate solutions to the equation of interest. While these methods usually produce very good approximations to the exact solutions, they typically do not have explicit bounds on the error. In validated numerics the goal is to produce an approximate solution along with an explicit guarantee that the error is no larger than some prescribed tolerance. In practice realizing such a validated numerical method requires both new mathematical analysis and new computational techniques. On the computational side, for instance, rather than doing the standard floating point arithmetic one must instead do interval arithmetic, where the result of a calculation is not a single number but an interval in which the result is guaranteed to lie. While these validated numerical calculations are much more difficult to carry out than standard numerics, the advantage is that one has a mathematical proof of the correctness of the solution. This means that there are many questions about the behavior of solutions to equations to which one can give a mathematically rigorous numerical proof. The investigators study some equations that govern nonlinear wave phenomenon, such as the propagation of a wave in the ocean or light in an optical fiber. Often one can find an exact special solution to these equations, such as a wave that propagates without changing its shape. One would like to know if this solution is stable: if solutions that begin close to this known solution remain close. Stability is an important question from the point of view of applications, as it determines whether these solutions are likely to be observed in practice. The investigators study stability via validated numerics. An important part of this proposal is training graduate students in these increasingly important techniques. The investigators address the stability of periodic traveling waves in Hamiltonian PDEs. One project establishes that the essential spectrum of the associated linearized operator to solutions of the generalized KdV and nonlinear Schrödinger equations is purely imaginary. This represents the first time that the essential spectrum has been calculated rigorously for such operators arising from non-integrable equations away from the small amplitude limit. This approach will extend to encompass other equations, including but not limited to regularized long wave type, the Benjamin-Ono and Camassa-Holm type, and two-dimensional equations. Furthermore, we aim to advance from spectral to linear stability, revealing the long-term dynamics of the solutions of the associated linearized equation. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.