Ufa Mathematical Journal
    Volume 12, Number 2, pp. 50-55

    On equivalence of one spin system and the two-component Camassa-Holm equation

    Tayshieva A.G., Nugmanova G.N., Myrzakul T.R.


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    The work is devoted to the study of the equivalence of the two-component Camassa-Holm equation (CHE) and the spin system, which is a generalization of the Heisenberg ferromagnet equation. It is known that equivalence between nonlinear integrable equations enables an advanced search for their various exact solutions. For the CHE we apply the method of the inverse scattering problem through the system of linear partial differential equations with scalar coefficients. Compared to the CHE, the coefficients of linear systems corresponding to spin equations are related to the symmetric matrix Lax representations. There-fore when establishing equivalence between the above equations, additional difficulties arise. Based on this, the matrix Lax representation for the CHE in symmetric space is proposed. Using the result, a gauge equivalence between the two-component CHE and the spin system is established. The relationship between their solutions is shown.