6 2. Equations for isolated and bound Skyrmions long-period modulated phases, the phase diagrams are expected to give qualitatively the correct results for these phenomenological models. The Euler equations for functional (8) have solutions for axisymmetric isolated structures of type ψ = ψ ( φ ), θ ( ρ ), m ( ρ ) (we introduce here spherical coordinates for the magnetization ( m, θ, ψ ) and cylindrical coordinates for the spatial variables ( ρ, φ, z )) [7]. Solutions ψ = ψ ( φ ) are known for all uniaxial and cubic noncentrosymmet- ric ferromagnets [5]. Particularly, ψ = φ − π/ 2 for cu- bic helimagnets, ψ = φ and ψ = − φ − π/ 2 for uniax- ial ferromagnets with ( nmm ) and ¯ 42 m symmetry, corre- spondingly. After substitution of solutions ψ = ψ ( φ ) into (8) and integration with respect to φ the total energy E of a Skyrmion (per unit length along z ) is E = 2 π ∫ ∞ 0 Φ( m, θ ) ρdρ with energy density where a common convention ∂f/∂x ≡ f$_{x}$ is applied. The Euler equations for the functional (9) with boundary conditions θ (0) = π , θ ( ∞ ) = 0, m ( ∞ ) = m$_{0}$ describe the structure of isolated Skyrmions (the magnetization of the homogeneous phase m$_{0}$ is derived from equation 2 am + 4 m 3 − h = 0). Typical solutions θ ( ρ ), m ( ρ ) of Eqs. (10) are plotted in Fig. 1. For present work, we have evaluated numerically the solutions for the energy functionals of type Eq. (1) within a standard approach using finite differences for gradi- ent terms and adjustable grids to accommmodate mod- ulated states with periodic boundary conditions. Search for modulated states and energy minimization was done by Monte Carlo simulated annealing. For 2-dimensional models, this approach can give converged results, as checked by comparison with analytical results. Including secondary effects, the approach is able and has been used by us to calculate thermodynamic stability of phases, and to study the complete temperature-field phase di- agrams in terms of equilbrium (mean-field) states of the energy functionals Eq. (1). As we are investigating here m$^{[}$θ$_{ρρ}$ + θ$_{ρ}$ ρ − sin θ cos θ ρ 2 − 2 sin 2 θ ρ − h sin( θ )$^{]}$ + +2 ( θ$_{ρ}$ − 1) m$_{ρ}$ = 0 , m$_{ρρ}$ + m$_{ρ}$ ρ + m$^{[}$θ 2 ρ + sin 2 θ ρ 2 − θ$_{ρ}$ − sin θ cos θ ρ ] + +2 am + 4 m 3 − h cos( θ ) = 0 (10) Φ = m 2 ρ + m 2 [ θ 2 ρ + sin 2 θ ρ 2 − θ$_{ρ}$ − sin θ cos θ ρ ] + am 2 + + m 4 − hm cos θ (9) [1] I.E. Dzyaloshinskii, Sov. Phys. JETP 19 , 960. [2] A. N. Bogdanov et al., Phys. Rev. B 66 , 214410 (2002); [3] R.M. Hornreich et al., Phys. Rev. Lett. 48 , 1404 (1982). [4] D. C Wright, N. D. Mermin, Rev. Mod. Phys. 61 , 385 (1989). [5] A.N. Bogdanov and D.A. Yablonsky, Sov. Phys. JETP 68 , 101 (1989). [6] A. Bogdanov, A. Hubert, J. Magn. Magn. Mater. 138 , 255 (1994); ibid. 195 , 182 (1999); A. Bogdanov, JETP Lett. 6 2, 247 (1995); A.N. Bogdanov et al., Physica B 359 , 1162 (2005). [7] U.K. R¨ oßler et al., Nature 442 , 797 (2006). [8] X. Z. Yu et al., Nature 465 , 901 (2010). [9] A. N. Bogdanov, U. K. R¨ oßler, Phys. Rev. Lett. 87 , 037203 (2001). [10] M. Bode et al., Nature 447 190 (2007). [11] U. Tkalec et al., Phys. Rev. Lett. 103 , 127801 (2009). [12] P.G. de Gennes, in Fluctuations, Instabilities, and Phase transitions , ed. T. Riste, NATO ASI Ser. B, vol. 2 (Plenum, New York, 1975). [13] P. Bak and M. H. Jensen, J. Phys.C: Solid State Phys. 13 , L881 (1980); O. Nakanishi et al., Solid State Comm. 35 , 995 (1980). [14] J. W. Felix et al., Phys. Rev. Lett. 57 , 2180 (1986). [15] I. Sosnowska et al., J. Phys. C 15 , 4835 (1982); A. Zhe- ludev et al. Phys. Rev. Lett. 78 , 4857 (1997). [16] B. Schaub, D. Mukamel, Phys. Rev. B 32 , 6385 (1985). [17] C. I. Gregory et al., J. Magn. Magn. Mater. 104-107 , 689 (1992); D. Lamago et al., Physica B 385-386 , 385 (2006). [18] B. Lebech et al., J. Magn. Magn. Mater. 140 , 119 (1995). [19] S. Muhlbauer et al., Science, 323 , 915 (2009). [20] C. Pfleiderer et al., Nature 427 , 227 (2004). [21] C. Pappas et al., Phys. Rev. Lett. 102 , 197202 (2009). [22] T. H. R. Skyrme, Proc. Roy. Soc. Lon. 260 , 127 (1961). [23] I. E. Dzyaloshinskii, Sov. Phys. JETP 5 , 1259 (1957), T. Moriya, Phys. Rev. 120 , 91 (1960). [24] P. G. De Gennes and J. Prost, The Physics of Liquid Crystals (Oxford University Press, Oxford, 1993), 2nd ed., D. C Wright, N. D. Mermin, Rev. Mod. Phys. 61 , 385 (1989). [25] E. H. Brandt, Rep. Prog. Phys. 58 , 1465 (1995), Phys. Rev. B, 68 , 054506 (2003). [26] See, e.g., J.-C. Tol´ edano, P. Tol´ edano, The Landau The- ory of Phase Transitions , World Scientific, Singapore 1987, chap. V and chap. VI,6. [27] Yu. A. Izyumov, Sov. Phys. Usp. 27 , 845 (1984). [28] G. G. Lonzarich, L. Taillefer, J. Phys. C. 18 , 4339 (1985), M. Yamada et al. J. Alloys Comp. 364 , 37, (2004). [29] M. L. Plumer, M. B. Walker, J. Phys. C. 14 , 4689 (1981), M. L. Plumer, J. Phys.:Cond Matt. 2 , 7503 (1990).