The derivative nonlinear Schrodinger equation on the interval

2012/05/07 by Jian Xu, Xu, Jian, Engui Fan +1
Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #nlin.SI

paper · pdf · doi:10.48550/arxiv.1205.1559

30 pages. arXiv admin note: substantial text overlap with arXiv:0808.1534, arXiv:nlin/0412008

arxiv created 2012/05/07 · arxiv updated 2012/05/09

Abstract

We use the Fokas method to analyze the derivative nonlinear Schrödinger (DNLS) equation iq_t(x,t)=-q_xx(x,t)+(r q²)_x on the interval [0,L]. Assuming that the solution q(x,t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter \x. This problem has explicit (x,t) dependence, and it has jumps across \\x ∈ \C|\im\x⁴=0 \. The relevant jump matrices are explicitly given in terms of the spectral functions \a(\x),b(\x)\,\A(\x),B(\x)\, and \\ca(\x),\cb(\x)\, which in turn are defined in terms of the initial data q₀(x)=q(x,0), the boundary data g₀(t)=q(0,t),g₁(t)=q_x(0,t), and another boundary values f₀(t)=q(L,t),f₁(t)=q_x(L,t). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.

Citations

• The derivative nonlinear Schrödinger equation on the half-line
• The Nonlinear Schrödinger Equation on the Interval

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