| We study various blowup phenomena associated with the vector nonlinear Schrödinger (VNLS) equation. This equation arises as a limiting case of the Zakharov system associated with plasma physics. It is characterized by a positive parameter α which is related to the mean thermal velocity of the electrons in the plasma. We are interested in studying solutions whose H1-norm blows up in finite time. We show the existence of standing wave solutions by solving a constrained minimization problem using the method of concentration compactness. These standing waves are expressed in terms of ground state solutions of an associated elliptic boundary value problem. We numerically construct ground states in both two and three dimensions and analyze their structure. In the two dimensional case we establish numerically that the limiting profile of blowup solutions of the VNLS equation near the blowup point(s) is equal, up to rescaling, to the ground state. In the three dimensional case we determine the blowup rate as a function of α. We develop a new dynamic mesh refinement method to study time evolution problems which blow up at more than one point, and apply it to study solutions of the VNLS equation in which splitting of the profile occurs. Finally, we apply this new method to study the time dispersion NLS equation, a perturbation of the focusing NLS equation which arises in nonlinear optics. |
