Blowup of Multi-Peaked Waveforms in the Two-Dimensional Nonlinear Schroedinger Model
In the present work, we explore the self-focusing and resulting collapse of two-dimensional waveforms involving multiple pulses in a nonlinear Schroedinger equation with a general power-law nonlinearity. We find that a wide range of multi-peaked states bifurcate from the critical threshold of the cubic nonlinearity, thus representing ``bifurcations from infinity'', i.e., the relevant pulses start at infinite distance in the critical limit and draw nearer, as the nonlinearity exponent increases past that threshold. We identify the resulting ``interacting particle system'' as amounting to a force balance between the exponentially interacting tails (modulated by a suitable power law) and a linear phase-induced force. The equilibria emerging from this force balance are found to be in excellent agreement with the identified steady states of the partial differential equation. The spectral stability of multi-peaked configurations is analyzed, leading to the conclusion that all the relevant states are less stable than the single-peak collapsing solution whose stability was analyzed earlier. Indeed, we reveal both symmetry-breaking, as well as motion-inducing destabilizing dynamics, with the former ones among them being dominant and ultimately leading to a single dominant collapse spot. Moreover, we characterize systematically both the real and imaginary eigenvalues of multi-peaked configurations, partitioning them in groups of different sizes, described by powers of the solution's blowup rate G.