numerical solution of integrable systems

The problem is an open-ended research program: develop numerical methods for solving Cauchy problems for nonlinear integrable equations by computing their scattering data and then solving the associated Riemann–Hilbert problem (RHP). For the focusing nonlinear Schrödinger equation, the particular challenge is the semiclassical family on the line

$$\mathrm{i}\epsilon u_t+\frac{\epsilon^2}{2}u_{xx}+|u|^2u=0,\qquad u(x,0)=u_0(x),\qquad 0<\epsilon\downarrow0.$$

Here $u_0$ is the initial profile and $\epsilon$ is the small-dispersion parameter. In inverse scattering, the forward step maps $u_0$ to scattering data, including a reflection coefficient $r(z)$ and, when present, discrete eigenvalues and norming constants; the inverse step uses these data in an RHP to recover $u(x,t)$. The question is how to carry out both steps numerically in this small-$\epsilon$ regime, in particular for focusing NLS with general smooth initial data. The open-program formulation does not specify a precise function space for “general smooth” data. Deift (2017) The Gaussian Semiclassical Soliton Ensemble and Numerical Methods for the Focusing Nonlinear Schrödinger Equation

Background

The numerical task has two stages. The forward stage is a spectral or scattering calculation that converts the initial profile into data for the integrable evolution; the inverse stage reconstructs the solution from those data. The latter is not merely a linear Fourier inversion: the RHP jump data contain oscillatory factors involving the reflection coefficient and the equation’s phase. For large time, steepest-descent analysis can yield asymptotic formulae, but at moderate times direct numerical solution is needed. This framework is a nonlinear counterpart of Fourier methods for linear dispersive equations. Deift (2017)

The numerical RHP program developed rapidly in the early 2010s. Olver introduced a numerical method for RHPs and applied it to Painlevé II; in 2012, Deconinck, Olver, and Trogdon used the methodology for the KdV Cauchy problem, computing accurately over long times, including before steepest-descent asymptotics become effective. The broader semiclassical setting has a longer analytical history: small-dispersion KdV was studied by Lax and Levermore, followed by work of Venakides and Deift, Venakides, and Zhou. Focusing NLS is especially difficult because its Zakharov–Shabat spectral problem is non-self-adjoint, unlike the self-adjoint spectral problem underlying the KdV analysis. Deift (2017) Smooth parametric dependence of asymptotics of the semiclassical focusing NLS

Known results

For fixed-parameter whole-line NLS, Trogdon and Olver reported a numerical implementation of the full inverse-scattering transform for focusing and defocusing equations. For Schwartz initial data, they describe spectrally convergent computation of scattering data and of the solution; after contour deformation of the RHP, they also establish uniform accuracy under mild assumptions, with computational cost for a prescribed tolerance independent of $x$ and $t$. These results show that the two numerical stages can be handled effectively in important decaying-data settings, but they do not establish uniform performance as $\epsilon\downarrow0$ for general smooth focusing-NLS data. Trogdon and Olver (2012)

The semiclassical focusing problem is obstructed at the forward stage by the non-self-adjoint scattering operator. As described in the 2017 formulation, quantities relevant to its spectrum can be “beyond all orders”: approximate eigenvectors may have residuals of order $\mathcal{O}(|\epsilon|^k)$ for every $k\geq1$. Thus ordinary finite-order asymptotics may fail to resolve the spectral information needed for the numerical reconstruction. The proposed challenge is to compute the forward scattering data and then apply an RHP method in the small-dispersion limit, rather than merely to solve an RHP with already available data. Deift (2017)

There are rigorous semiclassical asymptotic analyses for particular focusing-NLS initial-data families, including descriptions of modulated waves and estimates away from break points and caustics. These results provide analytical structure for special cases, but do not amount to a numerical forward-and-inverse method for general smooth initial data. Accordingly, the fixed-parameter numerical inverse-scattering results and special-data asymptotics are important advances toward the program, not a resolution of its stated general small-dispersion challenge. Smooth parametric dependence of asymptotics of the semiclassical focusing NLS Deift (2017)