Conserved norms and related conservation laws for multi-peakon equations
| Autor: | Recio, Elena, Anco, Stephen C. |
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| Rok vydání: | 2017 |
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| Zdroj: | J. Phys. A: Math. Theor. 51 (2018) 065203 (19 pages) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1088/1751-8121/aaa0e0 |
| Popis: | All nonlinear dispersive wave equations in the general class $m_t+f(u,u_x)m+(g(u,u_x)m)_x =0$ are known to possess multi-peakon weak solutions. A classification is presented for families of multi-peakon equations in this class that possess conserved momentum; conserved $H^1$ norm of $u$; conserved $H^2$ norm of $u$; conserved $L^2$ norm of $m$; related conservation laws. The results yield, among others, two interesting wide families of equations: $m_t + 2u_x h(u,u_x)m + u (h(u,u_x)m)_x=0$ for which the $H^1$ norm of $u$ is conserved; $m_t -\tfrac{1}{2}u_x h'(u)m + (h(u)m)_x =0$ for which the $L^2$ norm of $m$ is conserved. The overlap of these two families yields a singular equation which is nevertheless found to possess both smooth solitary wave solutions and peakon travelling wave solutions. Comment: 18 pages; typos corrected and new material added giving applications of the conservation laws |
| Databáze: | arXiv |
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