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ВИДЕОТЕКА |
Международная молодежная конференция «Геометрия и управление»
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Self-adjoint Commuting Differential Operators of Rank 2 and Their Deformations Given by the Soliton Equations Valentina Davletshina Sobolev Institute of Mathematics, Novosibirsk, Russia |
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Аннотация: In [1] and [2] I.M. Krichever and S.P. Novikov introduced a remarkable class of exact solutions of soliton equations — solutions of rank $$ \tag{1} V_{t}=\frac{1}{4}(6VV_x+6W_x+V_{xxx}),\ \ W_{t}=\frac{1}{2}(-3VW_x-W_{xxx}). $$ This system is equivalent to the commutativity condition of the self-adjoint operator $$L_{4}=(\partial_{x}^{2}+V(x,t))^{2}+W(x,t)$$ and the skew-symmetric operator $\partial_{t}-\partial_{x}^{3}-\frac{3}{2}V(x,t)\partial_{x}-\frac{3}{4}V_{x}(x,t).$ In this case “solutions of rank two” means that for every $$ dim_{{\mathbb C}}\left\{\psi: L_{4}\psi=z\psi, L_{4g+2}\psi=w\psi\right\}=2 $$ for generic eigenvalues $$w^2=F_g(z)=z^{2g+1}+c_{2g}z^{2g}+\ldots+c_{0}. $$ This curve is called spectral. There is a classification of commutative rings of ordinary differential operators of arbitrary rank obtained by Krichever [3] but in general case such operators are not found. Krichever and Novikov [1] found operators of rank two corresponding to an elliptic spectral curve. Mokhov found operators of rank three corresponding to an elliptic spectral curve. In the case of spectral curves of genus Operators $$ L_4-z=\tilde{L}_2L_2,\ \ \ L_{4g+2}-w=\tilde{L}_{4g}L_2. $$ Functions Theorem 1 [4]. If $$\chi_{0}=-\frac{Q_{xx}}{2Q}+\frac{w}{Q}-V, \ \ \ \ \ \chi_{1}=\frac{Q_{x}}{Q},$$ where $Q=z^{g}+\alpha_{g-1}(x)z^{g-1}+\ldots+\alpha_{0}(x).$ Polynomial $$ \tag{4} 4F_{g}(z)=4(z-W)Q^{2}-4V(Q_{x})^{2}+(Q_{xx})^{2}-2Q_{x}Q_{xxx} +2Q(2V_{x}Q_{x}+4VQ_{xx}+Q_{xxxx}). $$ The main aim of this paper is as follows. We study dynamics of polynomial Theorem 2. Suppose that potentials Remark 1. Similarly one can obtain the evolution equation on The following theorems are proved in [4] and [6]. Theorem 3. The operator $$ L_4^\sharp=(\partial_x^2+\alpha_3 x^3+\alpha_2 x^2+\alpha_1 x+\alpha_0)^2+\alpha_3 g(g+1)x $$ commutes with an operator Theorem 4. The operator $$ L_4^\natural=(\partial_x^2+\alpha_1\cosh(x)+\alpha_0)^2+\alpha_1g(g+1)\cosh(x), \ \ \alpha_1\neq0 $$ commutes with an operator The following theorems were proved in collaboration with E.I. Shamaev. Theorem 5. The operator Theorem 6. The operator Theorems 5 and 6 rigorously prove that Язык доклада: английский Список литературы
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