  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.
