On the Schrödinger equations with isotropic and anisotropic fourth-order dispersion

Villamizar-Roa, Elder J.
Banquet, Carlos
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Texas State University, Department of Mathematics
This article concerns the Cauchy problem associated with the nonlinear fourth-order Schrodinger equation with isotropic and anisotropic mixed dispersion. This model is given by the equation i∂tu + ε∆u + δAu + λ|u|αu = 0, x ∈ ℝn, t ∈ℝ, where A is either the operator ∆2 (isotropic dispersion) or ∑di=1 ∂xixixixi, 1 ≤ d < n (anisotropic dispersion), and α, ε, λ are real parameters. We obtain local and global well-posedness results in spaces of initial data with low regularity, based on weak- Lp spaces. Our analysis also includes the biharmonic and anisotropic biharmonic equation (ε = 0); in this case, we obtain the existence of self-similar solutions because of their scaling invariance property. In a second part, we analyze the convergence of solutions for the nonlinear fourth-order Schrödinger equation. i∂tu + ε∆u + δ∆2u + λ|u|αu = 0, x ∈ ℝn, t ∈ ℝ, as ε approaches zero, in the H2-norm, to the solutions of the corresponding biharmonic equation i∂tu + δ∆2u + λ|u|αu = 0.
Fourth-order Schrödinger equation, Biharmonic equation, Local and global solutions
Villamizar-Roa, E. J., & Banquet, C. (2016). On the Schrödinger equations with isotropic and anisotropic fourth-order dispersion. <i>Electronic Journal of Differential Equations, 2016</i>(13), pp. 1-20.