The trajectory-coherent approximation and the system of moments for the Hartree type equation
The general construction of semiclassically concentrated solutions to the Hartree type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter ℏ (ℏ→0), are constructed with a power accuracy of O(ℏ N/2...
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| Format: | Article |
| Language: | English |
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Wiley
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202112142 |
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| _version_ | 1832554618062635008 |
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| author | V. V. Belov A. Yu. Trifonov A. V. Shapovalov |
| author_facet | V. V. Belov A. Yu. Trifonov A. V. Shapovalov |
| author_sort | V. V. Belov |
| collection | DOAJ |
| description | The general construction of semiclassically concentrated solutions to the Hartree type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter ℏ (ℏ→0), are constructed with a power accuracy of O(ℏ N/2), where N is any natural number. In constructing the semiclassically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of semiclassically concentrated solutions of Hartree type equations. The results obtained are exemplified by a one-dimensional Hartree type equation with a Gaussian potential. |
| format | Article |
| id | doaj-art-23c77d611c6d4a50a5e993a265e36e97 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2002-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-23c77d611c6d4a50a5e993a265e36e972025-02-03T05:51:02ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252002-01-0132632537010.1155/S0161171202112142The trajectory-coherent approximation and the system of moments for the Hartree type equationV. V. Belov0A. Yu. Trifonov1A. V. Shapovalov2Moscow Institute of Electronics and Mathematics, 3/12 Trekhsvyatitel'sky Lane, Moscow 109028, RussiaTomsk Polytechnic University, 30 Lenin Ave., Tomsk 634034, RussiaTomsk State University, 36 Lenin Ave., Tomsk 634050, RussiaThe general construction of semiclassically concentrated solutions to the Hartree type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter ℏ (ℏ→0), are constructed with a power accuracy of O(ℏ N/2), where N is any natural number. In constructing the semiclassically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of semiclassically concentrated solutions of Hartree type equations. The results obtained are exemplified by a one-dimensional Hartree type equation with a Gaussian potential.http://dx.doi.org/10.1155/S0161171202112142 |
| spellingShingle | V. V. Belov A. Yu. Trifonov A. V. Shapovalov The trajectory-coherent approximation and the system of moments for the Hartree type equation International Journal of Mathematics and Mathematical Sciences |
| title | The trajectory-coherent approximation and the system of moments for the Hartree type equation |
| title_full | The trajectory-coherent approximation and the system of moments for the Hartree type equation |
| title_fullStr | The trajectory-coherent approximation and the system of moments for the Hartree type equation |
| title_full_unstemmed | The trajectory-coherent approximation and the system of moments for the Hartree type equation |
| title_short | The trajectory-coherent approximation and the system of moments for the Hartree type equation |
| title_sort | trajectory coherent approximation and the system of moments for the hartree type equation |
| url | http://dx.doi.org/10.1155/S0161171202112142 |
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