We construct fully-discrete schemes for the Benjamin–Ono, Calogero–Sutherland DNLS, and cubic Szegő equations on the torus, which are exact in time with spectral accuracy in space. We prove spectral convergence for the first two equations, of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$K^{-s+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>K</mi> <mrow> <mo>-</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norm for initial data in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^s(\mathbb {T})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$s>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, with an error constant depending linearly on the final time instead of exponentially. These schemes are based on explicit formulas, which have recently emerged in the theory of nonlinear integrable equations. Numerical simulations show the strength of the newly designed methods both at short and long time scales, thanks to the remarkable fact that the computational cost of the method is independent of the final time. These schemes open doors for the understanding of the long-time dynamics of integrable equations.

Spectrally Accurate Fully Discrete Schemes for Some Nonlocal and Nonlinear Integrable PDEs via Explicit Formulas

期刊论文
发表日期：

2026年2月2日

查看全文

Yvonne Alama Bronsard,
Xi Chen,
Matthieu Dolbeault

摘要

We construct fully-discrete schemes for the Benjamin–Ono, Calogero–Sutherland DNLS, and cubic Szegő equations on the torus, which are exact in time with spectral accuracy in space. We prove spectral convergence for the first two equations, of order $K^{-s+1}$ $K^{- s + 1}$ in $L^2$ $L^{2}$ norm for initial data in $H^s(\mathbb {T})$ $H^{s} (T)$ , $s>1$ $s > 1$ , with an error constant depending linearly on the final time instead of exponentially. These schemes are based on explicit formulas, which have recently emerged in the theory of nonlinear integrable equations. Numerical simulations show the strength of the newly designed methods both at short and long time scales, thanks to the remarkable fact that the computational cost of the method is independent of the final time. These schemes open doors for the understanding of the long-time dynamics of integrable equations.