Periodic weak solutions for a classical one-dimensional isotropic biquadratic Heisenberg spin chain
In the paper it is considered the equation \[ u_t = u\times u_{xxxx} +\bigg[\bigg( 1+\frac{5}{2}| u_x| ^2\bigg)u\times u_{x}\bigg]_x -\varepsilon u_{xxxx}-\epsilon \bigg(\frac{ 3}{2}| u_x| ^2_{xx} +u_x\cdot u_{xxx}\bigg)u, \] with initial conditions \(u(x, 0) = u_0\), \(u(x + 2D,t) = u(x, t)\), which relates to the one-dimensional isotropic biquadratic Heisenberg spin chain model [see \textit{M. Lakshmanan, K. Porsezian} and \textit{M. Daniel}, Phys. Lett. A 133, 483--488 (1988)]. Following the results of integrability in [\textit{K. Porsezian, M. Daniel} and \textit{M. Lakshmanan}, J. Math. Phys. 33, No. 5, 1807--1816 (1992; Zbl 1112.82309)], the authors prove the existence of weak solution \(u_\varepsilon\) of the above equation. Moreover, the uniform estimates of the solution with respect to \(\varepsilon\) is obtained.
isotropic biquadratic Heisenberg spin chain
scientific article; zbMATH DE number 5141614