Nucleation is usually described in the framework of the classical Gibbs theory of nucleation, which provides a simple expression for the critical radius of a growing nucleus (nanoparticles) [1]:    rc ~ γ/μ, (1) where: γ is the specific surface energy, μ  is the change in volume energy.  The Ginsburg–Landau– Cahn–Baranov theory uses this equation [2, 3, 4]: θ″ (ρ) + ((L – 1)/ ρ) θ′(ρ) – F[(a, ρ),cos{θ(ρ)}, sin{θ(ρ)}] = 0, (2)  where : L determines the space dimensionality of spinor,  and F[(a, ρ),cos{θ(ρ)}, sin{θ(ρ)}] is a polynomial (analytical function) that can be defined in each case. Application of Eq. (2) to a 2D problem was thoroughly considered in [2, 4]. Below, we consider a 1D case (L = 1), which corresponds to the Cahn–Hilliard–Hillert theory [3]. The resulting equation has the form  θ″ (ρ) = (a/2)sin{2θ(ρ)} , (3) which is similar to previously reported equations and according to which it is possible to obtain the solution  [3]: cos{θ(ρ)} = -th{ aρ)} , (4) The size of the domain wall is an important parameter determining the nucleating seed size:  rc ~ (K/μ)1/2  (5) Assuming that the domain wall size determines the critical radius of a nanoparticle, difference between Eqs. (1)  and (5)  should be noted. The simplest assumption regarding the relationship between the surface tension and the nanoparticle radius can be expressed as follows: γeff ~ γ/rrel ,  rrel   ~ rc /Δin  (6) where Δin is the thickness of the surface layer of the nanoparticle.  If rrel < 1, then the Gibbs theory is not applicable. However, it can be applied in accordance with the Cahn–Hilliard–Hillert theory. In this case, a formula similar to (1) can be written as follows:  rc  ~ γeff / μ ~ Δinγ / rc μ,       (K ~ Δinγ), (7) Thus, there is a correspondence between the Ginsburg–Landau–Cahn (see Eq. (5))  and  Gibbs (see  Eq. (1))  theories.  We will examine the nucleation of cylindrical particle (L=2). We examine nonlinear equation [2, 4]: θ″ (ρ) + θ′(ρ)/ ρ  – [a/ρ]² sin (θ(ρ)) cos (θ(ρ))  = 0.  (8)  The solution (8) is [2, 4]:  tg {θ/2}  =  1/ ρª.    (9)  This solution is termed the two-dimensional soliton (instanton) and is a rare example of the exact analytic solution of the nonlinear problem.