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Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications
| Content Provider | Semantic Scholar |
|---|---|
| Author | Anco, Stephen C. Bluman, George W. |
| Copyright Year | 2001 |
| Abstract | An eective algorithmic method is presented for nding the local conservation laws for partial dierential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for nding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. In the rst of two papers (Part I), examples of nonlinear wave equations are used to exhibit the method. Classication results for conservation laws of these equations are obtained. In a second paper (Part II), a general treatment of the method is given. In the study of dierential equations, conservation laws have many signicant uses, particularly with regard to integrability and linearization, constants of motion, analysis of solutions, and numerical solution methods. Consequently, an important problem is how to calculate all of the conservation laws for given dierential equations. For a dierential equation with a variational principle, Noether’s theorem [12, 4, 6, 5, 14] gives a formula for obtaining the local conservation laws by use of symmetries of the action. One usually attempts to nd these symmetries by noting that any symmetry of the action leaves invariant the extremals of the action and hence gives rise to a symmetry of the dierential equation. However, all symmetries of a dierential equation do not necessarily arise from symmetries of the action when there is a variational principle. For example, if a dierential equation is scaling invariant, then the action is often not invariant. Indeed it is often computationally awkward to determine the symmetries of the action and carry out the calculation with the formula to obtain a conservation law. Moreover, in general a dierential equation need not have a variational principle even allowing for a change of variables. Therefore, it is more eective to seek a direct, algorithmic method without involving an action principle to nd the conservation laws of a given dierential equation. |
| Starting Page | 545 |
| Ending Page | 566 |
| Page Count | 22 |
| File Format | PDF HTM / HTML |
| DOI | 10.1017/S095679250100465X |
| Volume Number | 13 |
| Alternate Webpage(s) | http://www.math.ubc.ca/~bluman/EJAM2002(1).pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/math-ph/0108023v2.pdf |
| Alternate Webpage(s) | https://arxiv.org/pdf/math-ph/0108023v2.pdf |
| Alternate Webpage(s) | https://ia801005.us.archive.org/28/items/arxiv-math-ph0108023/math-ph0108023.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/math-ph/0108023v1.pdf |
| Alternate Webpage(s) | https://doi.org/10.1017/S095679250100465X |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
