An initial-boundary value problem for the coupled focusing-defocusing complex short pulse equation with a 4×4 Lax pair

2017/12/15 by Hu, Beibei, Xia, Tiecheng, Ma, Wen-xiu
#Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences

paper · doi:10.48550/arxiv.1712.06449

Abstract

In this paper we investigate the coupled focusing-defocusing complex short pulse equation, which describe the propagation of ultra-short optical pulses in cubic nonlinear media. Through the unified transform method, the initial-boundary value problem for the coupled focusing-defocusing complex short pulse equation with 4× 4 Lax pair on the half-line are to be analyzed. Assuming that the solution \q₁(x,t),q₂(x,t)\ of the coupled focusing-defocusing complex short pulse equation exists, we show that \q_1,x(x,t),q_2,x(x,t)\ can be expressed in terms of the unique solution of a 4× 4 matrix Riemann-Hilbert problem formulated in the complex λ-plane. Thus, the solution \q₁(x,t),q₂(x,t)\ can be obtained by integration with respect to x. Moreover, we also get that some spectral functions are not independent and satisfy the so-called global relation.

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