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Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation. (English) Zbl 1309.35147

This work provides a long-time asymptotic formula for the solutions of \[ i\frac{d}{dt}R_n+(R_{n+1}-2R_n+R_{n-1})-|R_n|^2(R_{n+1}+R_{n-1})=0 \] which is the discrete version of the defocusing NLS (the focusing NLS is likely to be more complicated because of the soliton solutions). This equation can be exactly solved by the inverse scattering transform (IST) as done by M. J. Ablowitz et al. [Discrete and continuous nonlinear Schrödinger systems. Cambridge: Cambridge University Press (2004; Zbl 1057.35058)].
The asymptotic expansion is computed by using the so-called non-linear steepest descent method [P. A. Deift et al., in: Important developments in soliton theory. Berlin: Springer-Verlag. 181–204 (1993; Zbl 0926.35132)] which consists in applying the complex stationary phase method in the Plemelj integral associated to the (IST). In the quoted reference this has been done for the continuous NLS equations for which the contour is the real axis. Here the contour is a circle with four stationary points: \[ S_1=e^{-i\pi/4}A,\quad S_2=e^{-i\pi/4}\bar A,\quad S_3=-S_1,\quad S_4=-S_2 \] where \(A=1/2(\sqrt{2+n/t}-i\sqrt{2-n/t})\).
Thus, assuming \(n < 2 t\) and that the initial datum \(R_n(0)\) decreases fast enough with respect to \(n\) the author proves that \[ R_n(t)=\frac{1}{\sqrt{t}}\sum_{j=1}^2C_je^{-i(p_jt+q_j\log(t))}+O(\log(t)/t)\qquad\text{as}\quad t\rightarrow\infty \] where \(C_j\) and \(q_j\) are defined in terms of the reflection coefficient that appears in the IST and depend on \(n/t\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q15 Riemann-Hilbert problems in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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References:

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