A split-step Christov method for approximating rational PDE solutions

TL;DR

This paper introduces Christov functions as an orthogonal rational basis to overcome the slow Fourier decay when approximating rational PDE solutions, enabling spectrally accurate computations on the real line via FFT-based transforms. It develops a Christov-based, fourth-order split-step framework with spectral differentiation matrices, comparing Yoshida, partitioned Runge-Kutta, and Runge-Kutta-Nyström integrators and finding PRK and RKN to be superior in practice. The method is applied to rogue-wave dynamics in nonlinear Schrödinger equations, reproducing Peregrine-soliton instabilities and revealing rogue-wave–like structures under generic perturbations, as well as exploring non-integrable NLS variants with generalized nonlinearities and higher-order dispersion. Overall, the approach offers an efficient, high-accuracy tool for studying rational PDE solutions and their stability, with potential applications to a broad range of nonlinear wave problems.

Abstract

Rational solutions of partial differential equations (PDEs) are notoriously difficult to approximate via spectral Fourier methods due to their algebraically slow decay rate. In this work we discuss approximating rational PDE solutions in a basis of orthogonal functions known as the Fourier series, allowing for the computation of its spectrum via the fast Fourier transform. Spectral differentiation matrices are derived. Several explicit fourth-order split-step integrators are derived and their performance compared. As an application, rogue wave solutions in a family of nonlinear Schrödinger equations are explored. Perturbing the constant background is found to generate rogue wave-like structures. The effects of higher-order dispersion and generalized nonlinearities are also examined.

Figures (16)

• Figure 1: The real and imaginary parts of $\phi_n(x)$ defined in (\ref{['Ch_fcn_define']}) for $n \in \{-3, \dots, 2 \}$ indicated by blue (solid) and red (dashed) curves, respectively.
• Figure 2: (A) Numerically computed Ch coefficients for the function $f(x) = (1+x^2)^{-1}$ in blue, the exact Ch coefficients (\ref{['rational_expand_exact']}) are in red. (B) Numerically computed Ch coefficients for the function $f(x) = (1+x^4)^{-1}$ in blue, the decay rate of Ch coefficients in red. Notice that the Ch coefficients decay exponentially fast.
• Figure 3: Infinity norm error of the (A) first and (B) second Ch derivative approximations applied to the rational function $f(x) = (1 + x^2)^{-1}$ as a function of the number of Ch modes, $N$. The error approaches zero exponentially fast. All derivatives were computed using the technique described in (\ref{['Ch_diff_ap']}).
• Figure 4: Evolution of $|u(x,t)|$ corresponding to the Peregrine soliton (\ref{['peregrine']}) in the NLS equation (\ref{['nls_eq']}) for $t_0=-2$. Three different Ch split-step integrators, Ch-YSH, Ch-PRK and Ch-RKN are observed to give similar results. Computational parameters for each method are $N = 320$ Ch modes and time-step $\Delta t=0.001$.
• Figure 5: Solution modulus $|u(x,t)|$ using the same integrators and parameters as Fig. \ref{['nls_graph']} except the time-step is now $\Delta t=0.1$. This large time-step reveals a numerically induced instability in each method. Notice that the instability takes longer to appear for the PRK and RKN methods.