may be split into an infinite set of linear systems which are solved analytically, while the continuation methods require solving a discrete linear system as the homotopy parameter is varied to solve the nonlinear system. Applications In the last twenty years, the HAM has been applied to solve a growing number of nonlinear ordinary/partial differential equations in science, finance, and engineering. For example, multiple steady-state resonant waves in deep and finite water depth were found with the wave resonance criterion of arbitrary number of traveling gravity waves; this agreed with Phillips' criterion for four waves with small amplitude. Further, a unified wave model applied with the HAM, admits not only the traditional smooth progressive periodic/solitary waves, but also the progressive solitary waves with peaked crest in finite water depth. This model shows peaked solitary waves are consistent solutions along with the known smooth ones. Additionally, the HAM has been applied to many other nonlinear problems such as nonlinear heat transfer, the limit cycle of nonlinear dynamic systems, the American put option, the exact Navier–Stokes equation, the option pricing under stochastic volatility, the electrohydrodynamic flows, the Poisson–Boltzmann equation for semiconductor devices, and others. Brief mathematical description Consider a general nonlinear differential equation , where is a nonlinear operator. Let denote an auxiliary linear operator, u0(x) an initial guess of u(x), and c0 a constant (called the convergence-control parameter), respectively. Using the embedding parameter q ∈ [0,1] from homotopy theory, one may construct a family of equations, called the zeroth-order deformation equation, whose solution varies continuously with respect to the embedding parameter q ∈ [0,1]. This is the linear equation with known initial guess U(x; 0) = u0(x) when q = 0, but is equivalent to the original nonlinear equation , when q = 1, i.e. U(x; 1) = u(x)). Therefore, as q increases from 0 to 1, the solution U(x; q) of the zeroth-order deformation equation varies (or deforms) from the chosen initial guess u0(x) to the solution u(x) of the considered nonlinear equation. Expanding U(x; q) in a Taylor series about q = 0, we have the homotopy-Maclaurin series Assuming that the so-called convergence-control parameter c0 of the zeroth-order deformation equation is properly chosen that the above series is convergent at q = 1, we have the homotopy-series solution From the zeroth-order deformation equation, one can directly derive the governing equation of um(x) called the mth-order deformation equation, where and