The boundary-value problem for a half plane for the elliptic sine-Gordon equation is solved using the inverse scattering method. The solution is applied to the theory of the stationary Josephson effect.The elliptic sine-Gordon equation plays an. important role in problems of statistical physics and condensed matter physics. In dimensionless variables this equation has the form
Constancy of the abso.lute value of magnetization is an essential assumption for the phenomenologtcal consideration of weakly excited states of magnets, because then the variation of the magnetization is limited to its rotation, which is described by the Landau-Lifshits equation [1]. This model works well far from the Curie point. At moderate temperatures, one can expect spatial variation of the saturation magnetization of the sample material. Such a conjecture enables us to study the magnet within the framework of the phenomenological approach in the vicinity of the critical point (but outside the fluctuation region) and to describe a smooth spatially inhomogeneous* transition to the paramagnetic phase.1. Consider the stationary Landau-Lifshits equation for an isotropic two-dimensional ferromagnet:In (1), a > 0 is the square of the saturation magnetization of the sample and ~" = (x, y). Introduce the matrix M = 2~rY, where as are the Pauli matrices. Standard transformations of (1) implyHere I is the identity matrix and z = x + iy. The key point for further consideration is the assumption that c~ is harmonic:Thus, a two-dimensional isotropic magnet whose square of saturation magnetization is a harmonic function of the coordinates is described by the nonlinear sigma-model in a curved space. Equation (2), in view of (3), formally belongs to the same type as the gravity equations with two Killing vectors for the part of the metric g containing off-diagonal terms [3][4][5][6]. In the situation considered, however, M = M* (* denotes the Hermitian conjugation) and det M = -o~, while in gravity theory we have g = gT and det g = -cr 2. Let S = M/v~, S 2 = I, and S = ffrT, where S is the normalized magnetization vector, if2 = 1. Then (2), (3) implyi.e., we obtain the equation of the 0(3) sigma-model in a curved space. Note that similar equations occur also in other applications interesting from the physical point of view. In particular, the above-mentioned case of gravity equations is reduced to the 0(2, 1)-sigma-model in a curved space [5,6]. The dimensional reduction of axially symmetric** self-duality equations in the Yang-Mills field theory also leads to a similar model with the analogue of the matrix S equal to g'g, where g E SL(N, R) [7].*For possible integrable models of the nonstationary one-dimensional Heisenberg magnet, see [2] and the references therein.
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