1. GENERAL TRAVELLING-WAVE SOLUTION OF THE LANDAU–LIFSHITS EQUATION FOR UNIAXIAL FERROMAGNET N.V. Ostrovskaya Moscow State Institute of Electronic Technology (Technical University) Zelenograd, Solnechnaya alleya, 5, Moscow, 124498, Russia 1 “SOLITONS, COLLAPSES and TURBULENCE: Achievements, Developments and Perspectives “ August 2–7, 2009, Chernogolovka, RUSSIA

2. L ITERATURE 2 1. Landau L.D., Lifshits E.M., “ To the theory of magnetic permeability of ferromagnets ”, in the Collected Papers of L.D. Landau, vol.1, Moscow, 1969. 2 Akhiezer A.I., Bar ’ jakhtar V.G., Kaganov M.I. “ Spin waves in ferromagnetcs and antiferromagnetics ”, Uspekhi Fizicheskikh Nauk, 1960, vol. XXI, no. 4, pp. 533 – 579. 3. Akhiezer A.I., Borovik A.E. “ To the theory of finite-amplitude spin waves ”, JETP, 1966, vol. 51, no 2, pp. 508 – 513. 4. Akhiezer A.I., Borovik A.E., “ On nonlinear spin waves in ferromagnetic and antiferromagnetics ”, JETP, 1967, vol. 52, no 5, pp. 1332 – 1344. 5. Kosevich A.M., Ivanov B.A., Kovalev A.S., Magnetic solitons, Physics reports, 1990, vol. 94, no 3/4, pp. 117 – 238. 6. Kamchatnov A.M. “On periodic nonlinear waves in uniaxial ferromagnets”, JETP, 1992, vol.102, no 5(11), p. 1006–1014.

3. B ASIC EQUATIONS The Landau–Lifshits equationwhere The model of uniaxial anisotropy: The Landau–Lifshits equation in Cartesian coordinates In spherical coordinates 3 3

4. T RAVELLING WAVES where From the second equation it immediately follows The resulting equation in θ is The universal trigonometric substitution : 4

5. R ATIONAL EQUATION TO BE SOLVED where The usual way for solving such equations is to put It leads to the Bernoulli equation 5

6. G ENERAL SOLUTION The solution of the Bernoulli equation is It gives the following solution for the initial equation: where are the parameters of integration. 6

7. S PECIAL CASES –I This is a standing isolate domain wall with symmetry center located in the point ξ = ξ 0. where These solutions are periodic cnoidal. 7

8. S PECIAL CASES –II This is a dark soliton solution. This is well-known Akhiezer–Borovik soliton. 8

9. H OW MANY SOLITON SOLUTIONS ARE THERE IN THIS EQUATION ? The solution can be expressed in elementary functions when the polynomial P(y ) takes the form Let us compare it with its original form Correlating the coefficients between the same powers of the polynomials, we obtain the system of four equations for three variables y 0, p, q. The system is compatible only in the case, when additional condition for parameters a, b, c is fulfilled: Φ(a,b,c)=0. [Erdelyi A., Magnus W., Oberhettinger F. and Tricomi F.G. Higher transcendental functions, vol.III, New York–Toronto– London, 1955.] 9

10. B IFURCATION MANIFOLDS –I 10 [Ostrovskaya N.V., “On travelling wave solutions of the Landau–Lifshits equation for a uniaxially anisotropic ferromagnet”, Nonlinear Dynamics, 2009 (in press). ]

11. B IFURCATION MANIFOLDS –II 11 Fig. 1 Manifold Φ 1 (a,b,c) = 0, where localized solutions exist. The manifold Φ 1 (a,b,c) = 0 represents the association of two surfaces S 1 and S 2 (Fig. 1) The manifold Φ 2 (a,b,c) = 0 consists of three surfaces Π 1, Π 2, Π 3 (Fig.2) Fig. 2 Manifold Φ 2 (a,b,c) = 0 (top view).

12. B IFURCATION MANIFOLD TURNED INSIDE OUT 12 Fig. 3 Manifold Φ 2 (a,b,c) = 0 (bottom view — turned inside out).

13. B IFURCATION MANIFOLD IN MORE DETAIL 13 Fig. 4 The central part of Φ 2 (a,b,c) = 0 (a) and one of the wedge-shaped parts (b) in more detail (turned inside out).

14. T HE SOLUTION IN THE ELEMENTARY FUNCTIONS 14 Here

15. T HE SETS OF PARAMETERS FOR LOCALIZED SOLUTIONS 15 Fig. 5 The regions in the plane (a,b), where the solution is expressed in hyperbolic functions: a) for the surface S 1, b) for the surface S 2, c) for the surface Π 3. The color symbols on first three diagrams mark the points, which correspond to the solutions given in the next figure.

16. T HE EXACT SOLUTIONS IN TERMS OF HYPERBOLIC FUNCTIONS 16 Fig. 6 Solitons from the different regions: a) S 1 (a = 2, b =0.5); b) Π 3 (a = 7, b =0.4); c) S 2 (a = 0.5 b =0.5); d) S 2 (a = 4.4, b =0.16). where

17. T HE REGIONS OF SOLITON EXISTENCE 17 Fig. 7 Here the regions of soliton existence are plotted on the corresponding surface (shaded areas). In other points of the manifolds the solution is expressed in trigonometric functions. In the points of the boundaries the solution is rational.

18. L OCALIZED SOLUTION 18

19. T YPES OF LOCALIZED DISTRIBUTIONS OF MAGNETIZATION 19 Soliton 180-degree domain wall 360-degree domain wall

20. C ONCLUSIONS 20 The general solution of the Akhiezer–Borovik problem in the form of the elliptic integral of the first kind is obtained. In the space of the parameters the manifold where the solution is expressed in the elementary functions is revealed. The regions where the solutions take the localized form are determined. It is shown that only Akhiezer-Borovik solutions have zero asymptotic.