The Cauchy problem of coupled elliptic sine-Gordon equations with noise: Analysis of a general kernel-based regularization and reliable tools of computing

TL;DR

This paper introduces a kernel-based regularization method for solving the ill-posed coupled elliptic sine-Gordon equations with noisy Cauchy data, providing stable, convergent solutions and efficient computational tools, demonstrated through numerical examples.

Contribution

It generalizes and improves existing regularization techniques for coupled elliptic sine-Gordon equations, offering a stable, convergent solution approach and reliable computational methods.

Findings

Proposed a stable regularization method with proven error estimates.

Developed efficient techniques for approximating oscillatory integrals.

Numerical examples confirm the effectiveness and convergence of the method.

Abstract

Developments in numerical methods for problems governed by nonlinear partial differential equations underpin simulations with sound arguments in diverse areas of science and engineering. In this paper, we explore the regularization method for the coupled elliptic sine-Gordon equations along with Cauchy data. The system of equations originates from the static case of the coupled hyperbolic sine-Gordon equations modeling the coupled Josephson junctions in superconductivity, and so far it addresses the Josephson $\pi$-junctions. In general, the Cauchy problem is not well-posed, and herein the Hadamard-instability occurs drastically. Generalizing the kernel-based regularization method, we propose a stable approximate solution. Confirmed by the error estimate, this solution strongly converges to the exact solution in $L^2$-norm. The main concern of this paper is also with the way to compute the regularized solution formed by an alike integral equation. We employ the proposed techniques that successfully approximated the highly oscillatory integral, and apply the Picard-like iteration to organize an efficient and reliable tool of computations. The results are viewed as the improvement as well as the generalization of many previous works. The paper is also accompanied by a numerical example that demonstrates the potential of this idea.