Multi-soliton solutions, i.e. solutions behaving as the sum of N given solitons as t → +∞, were constructed for the L 2 critical and subcritical (NLS) and (gKdV) equations in previous works (see [23], [16] and [20]). In this paper, we extend the construction of multi-soliton solutions to the L 2 supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.

We consider 1-equivariant wave maps from R 1+2 → S 2 . For wave maps of topological degree zero we prove global existence and scattering for energies below twice the energy of harmonic map, Q, given by stereographic projection. We deduce this result via the concentration compactness/rigidity method developed by the second author and Merle. In particular, we establish a classification of equivariant wave maps with trajectories that are pre-compact in the energy space up to the scaling symmetry of the equation. Indeed, a wave map of this type can only be either 0 or Q up to a rescaling. This gives a proof in the equivariant case of a refined version of the threshold conjecture adapted to the degree zero theory where the true threshold is 2E(Q), not E(Q). The aforementioned global existence and scattering statement can also be deduced by considering the work of Sterbenz and Tataru in the equivariant setting.For wave maps of topological degree one, we establish a classification of solutions blowing up in finite time with energies less than three times the energy of Q. Under this restriction on the energy, we show that a blow-up solution of degree one is essentially the sum of a rescaled Q plus a remainder term of topological degree zero of energy less than twice the energy of Q. This result reveals the universal character of the known blow-up constructions for degree one, 1-equivariant wave maps of Krieger, the fourth author, and Tataru as well as Raphaël and Rodnianski. η αβ D α ∂ β U = 0, 1991 Mathematics Subject Classification. 35L05, 35L71.

We consider the semilinear wave equation with power nonlinearity in one space dimension. Given a blowup solution with a characteristic point, we refine the blowup behavior first derived by Merle and Zaag. We also refine the geometry of the blowup set near a characteristic point and show that, except for perhaps one exceptional situation, it is never symmetric with respect to the characteristic point. Then, we show that all blowup modalities predicted by those authors do occur. More precisely, given any integer k ≥ 2 and $\zeta _0 \in {\cal R}$, we construct a blowup solution with a characteristic point a such that the asymptotic behavior of the solution near (a,T(a)) shows a decoupled sum of k solitons with alternate signs whose centers (in the hyperbolic geometry) have ζ0 as a center of mass for all times. © 2013 Wiley Periodicals, Inc.

International audienceWe consider the radial free wave equation in all dimensions and derive asymptotic formulas for the space partition of the energy relative to a light cone, as time goes to infinity. We show that the exterior energy estimate, which Duyckaerts, Merle and the second author obtained in odd dimensions, fails in even dimensions. Positive results for restricted classes of data are obtained. This is a companion paper to our two nonlinear papers with Andrew Lawrie