Non-radial soliton resolution for the energy critical wave equation for initial data with compact support

We initiate the study of the soliton resolution conjecture for the energy critical nonlinear wave equation in the nonradial case, by considering compactly supported data in dimensions $ N = 3, 4, 5 $. Our starting point is the sequential decomposition established in [ 6 ]. Under the assumption that all the solitons in the sequential decomposition are ground state traveling waves with distinct velocities, i.e. of the form ± Wⁱ where W is the ground state, and Wⁱ is its Lorentz transform with velocity ⁱ, and where ⁱ≠ ʲ for i≠ j, we show a dichotomy. Either the soliton resolution holds for all times, or the resolution holds up to a collision on infinitely many time intervals. We then rule out this exotic scenario in the case where the number of solitons is smaller than or equal to two. Citation: