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Generation and Coherent Control of Pulsed Quantum Frequency Combs
Published on: June 8, 2018
An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4 × 4 Lax pair on the
1Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100190, China and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China.
Abstract:
We extend the idea of the Fokas unified transform to investigate the initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4 × 4 Lax pair on the half-line. The solution of this system can be expressed in terms of the solution of a 4 × 4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. The relevant jump matrices of the RH problem can be explicitly found using the two spectral functions s(k) and S(k), which can be defined by the initial data, the Dirichlet-Neumann boundary data at x = 0. The global relation is established between the two dependent spectral functions. The general mappings between Dirichlet and Neumann boundary values are analyzed in terms of the global relation. These results may be of the potential significance in both spinor Bose-Einstein condensates and the theory of multi-component integrable systems.
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