An analytical expression of solutions to nonlinear wave equations in higher dimensions with Robin boundary conditions

It is known that an analytical expression of solutions to nonlinear wave equations in higher dimensions may be helpful to recognizing new nonlinear physical phenomena and developing novel numerical integrators. Therefore, in this paper, we present an analytical expression for the initial–boundary value problem of higher-dimensional nonlinear wave equations Utt(X,t)−a2ΔU(X,t)=f(U(X,t),Ut(X,t))Utt(X,t)−a2ΔU(X,t)=f(U(X,t),Ut(X,t)), U(X,t0)=U0(X)U(X,t0)=U0(X), Ut(X,t0)=U1(X)Ut(X,t0)=U1(X), posed on a bounded domain Ω⊂RdΩ⊂Rd for d≥1d≥1 and equipped with the Robin boundary conditions ∇U⋅n+λU=β(X,t)∇U⋅n+λU=β(X,t), X∈∂ΩX∈∂Ω, where n is the unit outward normal vector at the boundary ∂Ω, and λ is a constant.