Numerical Evaluation of 2D Ground States

Abstract

A ground state is defined as the positive radial solution of the multidimensional nonlinear problem −Δu(x)+u(x)−f(u(x))=0, x∈ℝn; lim|x|→∞|u(x)|=0,$\varepsilon \, \propto \,k_ \bot ^{1 - \xi }$ with the function f being either f(u) =a|u|p–1u or f(u) =a|u|pu+b|u|2pu. The numerical evaluation of ground states is based on the shooting method applied to an equivalent dynamical system. A combination of fourth order Runge-Kutta method and Hermite extrapolation formula is applied to solving the resulting initial value problem. The efficiency of this procedure is demonstrated in the 1D case, where the maximal difference between the exact and numerical solution is ≈ 10–11 for a discretization step 0:00025. As a major application, we evaluate numerically the critical energy constant. This constant is defined as a functional of the ground state and is used in the study of the 2D Boussinesq equations.