By Vladimir G. Makhankov

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38]. Finally, bearing in mind that the dispersion of transverse electromagnetic waves in plasmas has the form of eqn. (56) for sufficiently long waves kc« Wp we get again either the S3 equation or a system of the form of eqns. (64) and (66). 71) (the author's approximation [13]). For ion-acoustic waves in plasmas the constants a and p are small parameters proportional to the ratio m,lmj:a = 4/3m,lmj, p = 3a. Both systems of equations (64) and (71) and (64) and (70) have a very universal nature (they arise in the study of different physical models) and possess interesting mathematical properties.

J 1(l-pnllb)2+pJ o Il2b and Il = 2s(J o -cJ)(I-pnllb). If n = 1, this equation becomes the U(l,O) nonlinear Schrodinger equation derived in [49] to describe magnetic crystals. (ii) p U(O,n) = 0, q = n. For a purely 'antiferromagnetic' system we get the vector nonlinear Schrodinger equation of repulsive type: with (2) For a large number of real crystals, the interaction between colour components is altogether much weaker as compared with the interlattice interaction [50], but it is not negligible.

7), we get a system of equations describing the interaction of the Langmuir and ion-sound waves in plasmas. The derivation has a quantitative character. Detailed and strict results are presented in the original papers [6) and [13). In uniform plasmas without a magnetic field there are three oscillation modes, viz. 56) where Wp = (4'7Tne" Ime)1/2, k and c are, respectively, the Langmuir frequency, the wavevector and the velocity of light. 58) where Vs = (Tel mi)JI2 is the ion-sound velocity. The latter mode exists only in non-isothermal plasmas (Te » Ti).