2024/11/30 by Satoshi Masaki, Masaki, Satoshi
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Photonic Systems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2412.00413
In this article, we investigate the large-time behavior of small solutions to a system of one-dimensional cubic nonlinear Schrödinger equations with two components. In previous studies, a structural condition on the nonlinearity has been employed to guarantee the existence of a coercive, mass-type conserved quantity. We identify a new class of systems that do not satisfy such a condition and thus lack a coercive conserved quantity. Nonetheless, we establish the global existence and describe the large-time behavior for small solutions in this class. In this setting, the asymptotic profile is described in terms of solutions to the corresponding system of ordinary differential equations (ODEs). A key element of our analysis is the use of a quartic conserved quantity associated with the ODE system. Moreover, for a specific example within this class, we solve the ODE system explicitly, showing that the asymptotic behavior is expressed using Jacobi elliptic functions.