## Scientific Works by Fushchych W.I., Boyko V.M. (ed.)

By Fushchych W.I., Boyko V.M. (ed.)

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The contents of this publication were used in classes given by way of the writer. the 1st was once a one-semester direction for seniors on the college of British Columbia; it was once transparent that reliable undergraduates have been completely in a position to dealing with effortless team concept and its software to uncomplicated quantum chemical difficulties.

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A, 1989, 22, 2643. 4. , Physica D, 1996, 95, 158–162. 5. , Applications of Lie groups to diﬀerential equations, New York, Springer, 1986. 6. , Group analysis of diﬀerential equations, Moscow, Nauka, 1978. 7. , 1980. I. Fushchych, Scientific Works 2004, Vol. 6, 50–55. Z. I. V. MARKO We describe all complex wave equations of the form ✷u = F (u, u∗ ) invariant under the extended Poincar´e group. As a result, we have obtained the ﬁve new classes of P (1, 3)-invariant nonlinear partial diﬀerential equations for the complex scalar ﬁeld.

Some exact solutions of a conformally invariant nonlinear Schr¨ odinger equation P. L. I. FUSHCHYCH We consider a nonlinear Schr¨ odinger equation whose symmetry algebra is the conformal algebra. Using some of these symmetries, we construct some ansatzes for solutions of the equation. This equation can be thought of as giving a wave-function description of a classical particle. 1 Introduction Many authors have proposed nonlinear generalisations of the linear equation of the following type [1, 2, 3, 4, 5, 6, 7]: iut + u= λ1 |u| |u|a |u|a u + λ2 + λ0 ln ∗ |u| |u| u u, (1) ∂u ∗ where ut = ∂u ∂t , |u|a = ∂xa , |u| = uu , a = 1, .

1996, 3, № 3–4, P. 296–301. U, (4) Unique symmetry of two nonlinear generalizations of the Schr¨odinger equation 31 It is easily seen that some nonlinear equations, which have been suggested by many authors as mathematical models of quantum mechanical, are particular cases of this nonlinear generalization of the Schr¨odinger equation. Indeed, we obtain from equation (4) (for λ0 = λ1 and λ2 = ib2 ) the following equation iUt + ∆U = λ1 ∆|U | + ib2 ln |U | U U∗ 1/2 (5) U, which was proposed in [7] for the stochastic interpretation of quantum mechanical vacuum dissipative eﬀects.