Spatial Analyticity of Solutions to the Korteweg–de Vries Equation and Its Applications in Numerical Modeling and Data Reconstruction

Source abstract

This paper investigates the spatial analyticity of solutions to the Korteweg–de Vries equation and exploits this property to improve the efficiency of numerical modeling and to solve inverse problems. It is theoretically proven that the solutions of the Korteweg–de Vries equation belong to Gevrey spaces, and an estimate for the analyticity radius σ(t) is derived, showing that it decays at an algebraic rate governed by σ(t)≥σ0/(1+Cσ0t). This property ensures that the Fourier coefficients decay exponentially, which is leveraged to design an efficient spectral numerical scheme with exponentially small truncation error. A detailed convergence analysis demonstrates that the spectral method achieves exponential accuracy, significantly outperforming conventional finite-difference approaches. Furthermore, a reconstruction algorithm is developed to recover the full wave field from limited sensor data, achieving an L2 error below 10−3 using only 10% of the spatial domain data. Extensive numerical experiments, including sensitivity analysis with respect to the estimated analyticity radius and comparison with finite-difference methods, confirm the theoretical predictions and demonstrate the robustness of the proposed approach. The results establish that spatial analyticity provides a powerful and practical framework for both forward modeling and data reconstruction in dispersive partial differential equations.