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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 85, Issue 1
  • January 1979, pp. 143-160

Euler operators and conservation laws of the BBM equation

  • Peter J. Olver (a1)
  • DOI:
  • Published online: 24 October 2008

The BBM or Regularized Long Wave Equation is shown to possess only three non-trivial independent conservation laws. In order to prove this result, a new theory of Euler-type operators in the formal calculus of variations will be developed in detail.

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(1)Kh. O. Abdulloev , I. L. Bogolubsky and V. G. Makhankov One more example of inelastic soliton interaction. Phys. Lett. 56 A (1976), 427428.

(2)T. B. Benjamin , J. L. Bona and J. J. Mahony Model equations for long waves in non-linear dispersive systems. Phil. Trans. Roy. Soc. London, Ser. A272 (1972), 4778.

(4)J. C. Eilbeck and G. R. McGuire Numerical study of the regularized long wave equation. I. Numerical methods. J. Computational Phys. 19 (1975), 4357.

(5)J. C. Eilbeck and G. R. McGuire Numerical study of the regularized long wave equation. II. Interaction of solitary waves. J. Computational Phys. 23 (1977), 6373.

(8)D. J. Korteweg and G. De Vries On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. May. 39 (1895), 422443.

(9)M. D. Kruskal , R. M. Miura , C. S. Gardner and N. J. Zabusky Korteweg–de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws. J. Math. Phys. 11 (1970), 952960.

(10)P. D. Lax Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21 (1968), 467490.

(11)P. D. Lax Almost periodic solutions of the KdV equation. SIAM Rev. 18 (1976), 351375.

(12)R. M. Miura The Korteweg–de Vries equation: a survey of results. SIAM Rev. 18 (1976), 412458.

(13)R. M. Miura , C. S. Gardner and M. D. Kruskal Korteweg–de Vries equation and generalizations. II. Existence of conservation laws and constants of motion. J. Math. Phys. 9 (1968), 12041209.

(14)P. J. Olver Evolution equations possessing infinitely many symmetries. J. Math. Phys. 18 (1977), 12121215.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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