Algebraic Lower Bounds on the Spatial Analyticity Radius for Higher Order Nonlinear Schrödinger Equations

We investigate the initial value problem associated to the higher order nonlinear Schrödinger equation where j ≥ 2 is any integer, u is a complex valued function, and the initial data u 0 is real analytic on ℝ and has a uniform radius of spatial analyticity σ 0 in the space variable. For such initial data, we prove that the initial value problem is locally well‐posed using space‐time estimates and show that the radius of spatial analyticity of the solution remains σ 0 till some lifespan 0 < T ≤ 1. We also show that the uniform radius of spatial analyticity σ ( t ) of solutions at time t is bounded from below by for large time t , where c > 0 is a constant. Our proof relies on standard contraction mapping argument, Plancherel’s Theorem, Hölder’s inequality, space‐time Strichartz estimates for the free Schrödinger equation, energy estimate and one‐dimensional Sobolev embedding.