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Center for Nonlinear Studies |
Continuation methods are numerical algorithmic procedures for tracing out branches of fixed points/roots to nonlinear equations as one (ormore) of the free parameters of the underlying system is varied. On top of standard continuation techniques such as the sequential and pseudo-arclength continuation, we will present a new and powerful continuation technique called the deflated continuation method which tries to find/construct undiscovered/disconnected branches of solutions by eliminating known branches. In this talk we will employ this method and apply it to the one-component Nonlinear Schr\"odinger (NLS) equation in two spatial dimensions. We will present novel nonlinear steady states that have not been reported before and discuss bifurcations involving such states. Next, we will focus on a two-component NLS system and discuss about recent developments by using the deflated continuation method where the landscape of solutions of such a system is far richer. A discussion about the challenges in the two-component setting will be offered and a summary of open problems will be emphasized.
Host: Avadh Saxena