We consider the wave maps problem with domain R 2`1 and target S 2 in the 1-equivariant, topological degree one setting. In this setting, we recall that the soliton is a harmonic map from R 2 to S 2 , with polar angle equal to Q1prq " 2 arctanprq. By applying the scaling symmetry of the equation, Q λ prq " Q1prλq is also a harmonic map, and the family of all such Q λ are the unique minimizers of the harmonic map energy among finite energy, 1-equivariant, topological degree one maps. In this work, we construct infinite time blowup solutions along the Q λ family. More precisely, for b ą 0, and for all λ 0,0,b P C 8 pr100, 8qq satisfying, for some C l , C m,k ą 0, 5 4. Construction of the ansatz 9 4.1. Outline 9 4.2. Correcting the large r behavior of Q 1 λptq 11 4.3. The free wave correction 18 4.4. Further improvement of the soliton error term 19 4.5. The linear error terms for large r 23 4.6. The nonlinear error terms involving v 1 , v 2 , v 3 , v 4 25 4.7. The equation resulting from u ansatz 25 4.8. Choosing λptq 26 4.9. Estimates on B k r B j t F 4 131 4.10. Estimates on F 5 138 4.11. Estimates on F 6 142 4.12. Estimates on v corr -dependent quantities 142 5. Solving the final equation 144 5.1. The equation for F puq 144 5.2. Estimates on F 2 145 5.3. F 3 Estimates 147 2010 Mathematics Subject Classification. Primary 35L05, 35Q75. 1 2 MOHANDAS PILLAI 5.4. Estimates on F 4 -related oscillatory integrals 156 5.5. Setup of the final iteration 177 6. The energy of the solution, and its decomposition as in Theorem 1.1 188 Appendix A. Proof of Theorem 1.2 191 References 193

We recast the well-known Numerov method for solving Schrödinger's equation into a representation of the kinetic energy operator on a discrete lattice. With just a few lines of Mathematica code, it is simple to calculate and plot accurate eigenvalues and eigenvectors for a variety of potential problems. We illustrate the method by calculating high accuracy solutions for the |x| potential.

We consider a system consisting of a pair of D3 branes intersecting each other along a line such that half of the 16 supersymmetries are preserved. We then study the existence of magnetic monopole solutions corresponding to a D1-brane suspended between these D3 branes. We consider this problem in the zero slope limit where the tilt of the D3-branes is encoded in the uniform gradient of the adjoint scalar field. Such a system is closely related to the non-abelian flux background considered originally by van Baal. We provide three arguments supporting the existence of a single magnetic monopole solution. We also comment on the relation between our construction and a recent work by Mintun, Polchinski, and Sun.

We consider the 1-equivariant energy critical wave maps problem with two-sphere target. Using a method based on matched asymptotic expansions, we construct infinite time relaxation, blow-up, and intermediate types of solutions that have topological degree one. More precisely, for a symbol class of admissible, timedependent length scales, we construct solutions which can be decomposed as a ground state harmonic map (soliton) re-scaled by an admissible length scale, plus radiation, and small corrections which vanish (in a suitable sense) as time approaches infinity. Our class of admissible length scales includes positive and negative powers of t, with exponents sufficiently small in absolute value. In addition, we obtain solutions with soliton length scale oscillating in either a bounded or unbounded set, for all sufficiently large t.

We study the low energy effective theory of two sets of D3-branes overlapping in 1+1 dimensions, recently considered by Mintun, Polchinski, and Sun. In the original treatment by MPS, by studying the properties of magnetic solitons, the low energy effective field theory was found to require some ultraviolet completion, possibly involving full string dynamics. Recently in a companion paper, it was shown that by scaling the angle between the D3-branes and the D3 -branes in the zero slope limit in specific way, one can find simpler effective field theory which consists of a single tower of Regge trajectory states and yet is ultraviolet complete and non-singular. In this article, we study this model by further studying a limit which recovers the MPS dynamics from this non-singular construction. We approach this issue from a holographic perspective, where we consider a stack of N D3-branes overlapping with a single D3 -brane, and treat that D3 -brane as a probe in the AdS 5 × S 5 dual. In general, the D3 -brane probe supports a magnetic monopole as a non-singular soliton configuration, but in the limit where the MPS dynamics is recovered, the soliton degenerates. This is consistent with the idea that the effective dynamics in the MPS setup is incomplete, but that it can be completed with a single tower of Regge trajectory states.