Up to the first two order Melnikov analysis for the exact cyclicity of planar piecewise linear vector fields with nonlinear switching curve

In this paper, we focus on providing the exact bounds for the maximum number of limit cycles $Z(3,n)$ that planar piecewise linear differential systems with two zones separated by the curve $y=x^3$ under perturbation of arbitrary polynomials of $x,y$ with degree n can have, where $n\in N$. By the first two order Melnikov functions, we achieve that $Z(3,2)=12$, $Z(3,n)=2n+1$ for $3\le n\le 88$ and $Z(3,n)\ge 2n+1$ for any n. The results are novel and improve the previous results in the literature.