Physica 116A (1982) 1-33 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG North-Holland Publishing Co. LINEARIZATION OF THE NONLINEAR SCI-IRiiDINGER EQUATION AND THE ISOTROPIC HEISENBERG SPIN CHAIN F.W. NIJHOFF, J. VAN DER LINDEN, G.R.W. QLJISPEL, H.W. CAPEL and J. VELTHUIZEN Instituut-Lorentz uoor Theoretische Natuurkunde, Nieuwsteeg 18, 231 J SJ3 Leiden, The Nether- lands Received 29 April 1982 A new description in terms of one and the same linear inhomogeneous integral equation is proposed for the nonlinear Schrijdinger equation (NLS), as well as for the equation of motion for the classical isotropic Heisenberg spin chain in the continuum limit (IHSC). From the integral equa- tion which contains a two-fold integration over an arbitrary contour in the complex plane with an arbitrary measure one can obtain the various solutions of the NLS as well as of the IHSC in a direct way without going through the details of the inverse scattering formalism. Well-known properties such as the Miura transformation, the Gel’fand-Levitan equation and the Lax representations for NLS and IHSC can be derived as a corollary from the integral equation. The treatment leads also to a few more general (integrable) partial differential equations which contain the NLS and the IHSC as special cases. 1. Introduction In recent years there has been a lot of interest in the Heisenberg spin chain in the continuum limit. For the classical isotropic case explicit solutions of the equation of motion a,s= s x zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA aIs + s x B, s-s= 1, (1.1) where S = (S’, Sy, S’) denotes the spin vector and B an (external) magnetic field, were found by Lakshmanan, Ruijgrok and Thompson’) and by Tjon and Wright?. A description in terms of one real variable and the corresponding potential equation which is also valid in the case of axial symmetry have been given in refs. 3 and 4. An important development in the study of eq. (1.1) was the discovery of an inverse scattering scheme and thus the proof of complete integrability by Takhtadzhyan5). At an earlier stage the inverse scattering method had already been established for the nonlinear Schriidinger equation ia& + a:+ + 21+1*~$ = 0 (1.2) by Zakharov and Shabat6). The corresponding potential equation in terms of one real variable was given in refs. 7, 3 and 4. 0378-4371/82/0000-0000/$02.75 @ 1982 North-Holland