Non-radial scattering below the ground state in the low-exponent and one-dimensional regimes

Establish scattering for non-radial solutions of the focusing nonlinear fourth-order Schrödinger equation below the ground state when either the spatial dimension is one or the exponent satisfies \(1+8/d<p<1+8/(d-1)\).

Background

The paper proves scattering below the ground state for non-radial solutions of the focusing fourth-order Schrödinger equation in dimensions $d\geq 2$ under the condition $1+8/(d-1)\leq p<1+8/(d-4)_+$. The lower bound on $p$ is equivalent to the critical regularity condition $s_c\geq 1/2$, and it is required to establish positivity of the transverse virial functional associated with the direction orthogonal to the conserved momentum.

The authors explicitly identify the remaining regimes—one spatial dimension and the intercritical range $1+8/d