On an ill-posed problem for system of coupled sinh-Gordon equations

Abstract

The aim of this paper is considering the initial value problem for a system of coupled nonlinear sinh-Gordon equations by the association between two regularization methods: filter and truncation Fourier. Firstly, we give an example to show that the problem does not satisfy the third property which is called ill-posed in the sense of Hadamard. Secondly, under some a priori assumptions, we propose the stable regularization methods to regularize the system, i.e. the corresponding regularized solution converge to the exact solution in \(L^2\)-norm. Finally, to illustrate the proposed efficiency in the theoretical part, we show some numerical tests to check the convergence of the regularized solution and the regularized errors.