Skip to main content
    • Aa
    • Aa

Euler operators and conservation laws of the BBM equation

  • Peter J. Olver (a1)

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.

Hide All
(1)Abdulloev Kh. O., Bogolubsky I. L. and Makhankov V. G. One more example of inelastic soliton interaction. Phys. Lett. 56 A (1976), 427428.
(2)Benjamin T. B., Bona J. L. and Mahony J. J. Model equations for long waves in non-linear dispersive systems. Phil. Trans. Roy. Soc. London, Ser. A 272 (1972), 4778.
(3)Bona J. L., Pritchard W. G. and Scott L. R. Solitary-wave interaction. Univ. of Essex Fluid Mech. Research Inst. Report No. 94 (1978).
(4)Eilbeck J. C. and McGuire G. R. Numerical study of the regularized long wave equation. I. Numerical methods. J. Computational Phys. 19 (1975), 4357.
(5)Eilbeck J. C. and McGuire G. R. Numerical study of the regularized long wave equation. II. Interaction of solitary waves. J. Computational Phys. 23 (1977), 6373.
(6)Gel'fand I. M. and Dikǐ L. A. The asymptotics of the resolvent of Sturm–Liouville equations and the algebra of the Korteweg–de Vries', Equation. Uspehi Mat. Nauk 30 (1975), 67100.
(7)Gel'fand I. M. and Dikǐ L. A. A Lie algebra structure in a formal variational calculation. Funkcional Anal. i Priložen 10 (1976), 1825.
(8)Korteweg D. J. and De Vries G. 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)Kruskal M. D., Miura R. M., Gardner C. S. and Zabusky N. J. Korteweg–de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws. J. Math. Phys. 11 (1970), 952960.
(10)Lax P. D. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21 (1968), 467490.
(11)Lax P. D. Almost periodic solutions of the KdV equation. SIAM Rev. 18 (1976), 351375.
(12)Miura R. M. The Korteweg–de Vries equation: a survey of results. SIAM Rev. 18 (1976), 412458.
(13)Miura R. M., Gardner C. S. and Kruskal M. D. Korteweg–de Vries equation and generalizations. II. Existence of conservation laws and constants of motion. J. Math. Phys. 9 (1968), 12041209.
(14)Olver P. J. Evolution equations possessing infinitely many symmetries. J. Math. Phys. 18 (1977), 12121215.
(15)Olver P. J. and Shakiban C. A resolution of the Euler operator: I. Proc. Amer. Math. Soc. 66 (1978), 223229.
(16)Whitham G. B. Linear and nonlinear waves (New York, Wiley, 1974).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 13 *
Loading metrics...

Abstract views

Total abstract views: 143 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd October 2017. This data will be updated every 24 hours.