Low regularity estimates of the Lie-Trotter time-splitting Fourier spectral method for the logarithmic Schrödinger equation

In this paper, we conduct rigorous error analysis of the Lie-Trotter time-splitting Fourier spectral scheme for the nonlinear Schrödinger equation with a logarithmic nonlinear term f ( u ) = u ln | u | 2 f(u)=u\ln \!|u|^2 (LogSE) and periodic boundary conditions on a d d -dimensional torus T d \mathbb {T}^d . Different from existing works based on regularisation of the nonlinear term f ( u ) ≈ f ε ( u ) = u ln ⁡ ( | u | + ε ) 2 f(u)\approx f^\varepsilon (u)=u\ln (|u| + \varepsilon )^2 , we directly discretize the LogSE with the understanding f ( 0 ) = 0 f(0)=0 . Remarkably, in the time-splitting scheme, the solution flow map of the nonlinear part: g ( u ) = u e − i λ t ln | u | 2 g(u)= u\,e^{-{\mathrm {i}} \lambda t \ln \!|u|^{2}} has a higher regularity than f ( u ) f(u) (which is not differentiable at u = 0 u=0 but Hölder continuous), where g ( u ) g(u) is Lipschitz continuous and possesses a certain fractional Sobolev regularity with index 0 > s > 1 0>s>1 . Accordingly, we can derive the L 2 L^2 -error estimate: O ( ( τ s / 2 + N − s ) ln N ) \mathcal {O}\big ((\tau ^{s/2} + N^{-s})\ln \! N\big ) of the proposed scheme for the LogSE with low regularity solution

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