Systems of polynomial equations are a common occurrence in problem formulations in
engineering, science, and mathematics. Solution sets of such systems, ie, algebraic sets, are …

We consider the Bethe equations for the isotropic spin-1/2 Heisenberg quantum spin chain
with periodic boundary conditions. We formulate a conjecture for the number of solutions …

It is quite common that a nonlinear partial differential equation (PDE) admits multiple distinct
solutions and each solution may carry a unique physical meaning. One typical approach for …

Positone and semipositone boundary value problems are semilinear elliptic partial
differential equations (PDEs) that arise in reaction–diffusion models in mathematical biology …

The homotopy continuation method has been widely used to compute multiple solutions of
nonlinear differential equations, but the computational cost grows exponentially based on …

This paper presents an innovative approach, the Adaptive Orthogonal Basis Method,
tailored for computing multiple solutions to differential equations characterized by …

Eigenfunction expansion discretization is considered for finding multiple solutions of
semilinear elliptic equations with polynomial nonlinearity. Error estimates of the …

In this paper we review applications of numerical algebraic geometry to differential
equations. The techniques we address are direct solution, bootstrapping by filtering, and …

For a function $ F: X\to Y $ between real Banach spaces, we show how continuation
methods to solve $ F (u)= g $ may improve from basic understanding of the critical set $ C …

In this paper, we discuss an efficient numerical method to obtain all solutions of fractional
Gelfand equation with Dirichlet boundary condition. More precisely, we derive a good initial …