报告地点：行健楼学术活动室526
邀请人：吴奕飞教授
Abstract：The one-dimensional cubic nonlinear Schrödinger (NLS) equation is known as an integrable equation. Integrable equations enjoy remarkable properties, such as the existence of infinitely many conserved quantities, and their solutions can be analyzed in great detail—for instance, via the inverse scattering method. However, equations with such structures are extremely special. The integrability of an equation is characterized by the existence of a Lax pair. However, no general method for constructing a Lax pair is known, making it difficult to determine whether a given equation is integrable.
In this talk, I will investigate the integrability of two-component (2-coupled) cubic NLS systems in one space dimension. Among such systems, several examples are known to be integrable, including the Manakov system and certain nonlocal-type NLS equations. By focusing on the existence of a fourth conserved quantity following the mass, momentum, and energy, we show that general 2-coupled NLS systems fall into one of the following three categories:
(1) Integrable case: If a system admits a nontrivial fourth conserved quantity and a conserved energy whose quadratic part is non-degenerate, then it is integrable and can be reduced to one of fifteen standard forms.
(2) If a system admits a nontrivial fourth conserved quantity, but all conserved energies have degenerate quadratic parts, then the system contains a closed single-component cubic NLS equation. The third and fourth conserved quantities arise from that single equation.
(3) The system does not possess any nontrivial fourth conserved quantity.
The latter two cases indicate a defect in the infinite sequence of conserved quantities, suggesting that the system is not integrable.
