Our work focuses on the study of a class of nonlinear pseudo-differential equations on the circle \(\mathbb {S}^1\) , acting on weighted function spaces \(L_{w}^p(\mathbb {S}^1, w(\theta )d\theta )\) spaces. We consider periodic symbols \(\sigma (\theta ,k)\) on \(\mathbb {S}^1\times \mathbb {Z}\) , under some general assumptions. Throughout this work, we assume that the symbols can be expressed as functions with separable variables; that is, \(\sigma (\theta ,k)=w(\theta )b(k)\) . The variable coefficient \(w(\theta )\) is then interpreted as a weight function applied to the spaces \(L^p_w(\mathbb {S}^1)\) . The operators associated with the equations under consideration are defined via the periodic Fourier transform in the sense of distributions. The main contribution of this work is the proof of existence and regularity of the corresponding solutions. Additionally, we provide an explicit representation of the solutions for a specific class of nonlinear pseudo-differential equations.
