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The Cauchy Problem for Darboux Integrable Systems and Non-Linear d'Alembert Formulas
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2013, том 9
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The Cauchy Problem for Darboux Integrable Systems and Non-Linear d'Alembert Formulas
Інший ID:
2010 Mathematics Subject Classification: 58A15; 35L52; 58J70; 35A30; 34A26
DOI: http://dx.doi.org/10.3842/SIGMA.2013.017
DOI: http://dx.doi.org/10.3842/SIGMA.2013.017
Посилання:
The Cauchy Problem for Darboux Integrable Systems and Non-Linear d'Alembert Formulas / I.M. Anderson, M.E. Fels // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 16 назв. — англ.
Підтримка:
This paper is a contribution to the Special Issue “Symmetries of Dif ferential Equations: Frames, Invariants
and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html.
Дата:
2013
Переглядів:
486
Завантажень:
210
Анотація:
To every Darboux integrable system there is an associated Lie group G which is a fundamental invariant of the system and which we call the Vessiot group. This article shows that solving the Cauchy problem for a Darboux integrable partial differential equation can be reduced to solving an equation of Lie type for the Vessiot group G. If the Vessiot group G is solvable then the Cauchy problem can be solved by quadratures. This allows us to give explicit integral formulas, similar to the well known d'Alembert's formula for the wave equation, to the initial value problem with generic non-characteristic initial data.
