Null structure in a system of quadratic derivative nonlinear Schr\"odinger equations

TL;DR

This paper establishes global existence and asymptotic freedom for small solutions to a three-component quadratic derivative nonlinear Schrödinger system in two dimensions, using structural conditions and advanced analytical techniques.

Contribution

It introduces a structural condition on the nonlinearity that ensures global existence and asymptotic behavior for the system, which is a novel result for such equations.

Findings

Small data global existence under the structural condition

Solutions are asymptotically free as time goes to infinity

Utilizes commuting vector field method with smoothing effects

Abstract

We consider the initial value problem for a three-component system of quadratic derivative nonlinear Schr\"odinger equations in two space dimensions with the masses satisfying the resonance relation. We present a structural condition on the nonlinearity under which small data global existence holds. It is also shown that the solution is asymptotically free. Our proof is based on the commuting vector field method combined with smoothing effects.