ORBITALLY STABLE STANDING WAVES OF A MIXED DISPERSION NONLINEAR SCHRODINGER EQUATION

Author(s)

Bonheure, Denis ^{[1, 2]} ; Casteras, Jean-Baptiste ^{[1, 2]} ; dos Santos, Ederson Moreira ^[3] ; Nascimento, Robson ^{[1, 2]}

Affiliation

^[1] Univ Libre Bruxelles, Dept Math, CP 214, B-1050 Brussels - Belgium

^[2] INRIA Team MEPHYSTO, Brussels - Belgium

^[3] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Caixa Postal 668, BR-13560970 Sao Carlos, SP - Brazil

• Document type: Journal article
• Source: SIAM JOURNAL ON MATHEMATICAL ANALYSIS; v. 50, n. 5, p. 5027-5071, 2018.
• Web of Science Citations: 4

Abstract

We study the mixed dispersion fourth order nonlinear Schrodinger equation i partial derivative(t)psi-gamma Delta(2)psi + beta Delta psi+vertical bar psi vertical bar(2 sigma )psi = 0 in R x R-N , where gamma, sigma > 0 and beta is an element of R. We focus on standing wave solutions, namely, solutions of the form psi(x,t) = e(i alpha t)u(x) for some alpha is an element of R. This ansatz yields the fourth order elliptic equation gamma Delta(2)u - beta Delta u +alpha u = vertical bar u vertical bar(2 sigma)u. We consider two associated constrained minimization problems: one with a constraint on the L-2-norm and the other on the L-2 sigma+ 2 -norm. Under suitable conditions, we establish existence of minimizers and we investigate their qualitative properties, namely, their sign, symmetry, and decay at infinity as well as their uniqueness, nondegeneracy, and orbital stability. (AU)

• FAPESP's process: 15/17096-6 - Problems on Elliptic PDEs: systems and equations
• Grantee: Ederson Moreira dos Santos
• Support Opportunities: Regular Research Grants
• FAPESP's process: 16/50453-0 - Semilinear and quasilinear elliptic partial differential equations
• Grantee: Ederson Moreira dos Santos
• Support Opportunities: Regular Research Grants