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Rational solutions of the KdV equation with damping

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Lettere al Nuovo Cimento (1971-1985)

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References

  1. S. Maxon:Cylindrical and spherical solutions, inProceedings of the Conference on the Theory and Applications of Solitons, edited byH. Flaschka andD. W. McLaughlin, Rocky Mountain Mathematics Consortium (Arizona State University) (Tempe, Ariz., 1978).

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  2. T. Taniuti:Suppl. Prog. Theor. Phys.,55, 191 (1974).

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  3. By rescalingt, x andu it is possible to multiply each term in eq. (1) by a different constant. The form (1) that we use here is chosen for convenience;c=0 corresponds to the usual KdV equation, while the term proportional toc accounts, forc<0, for (Landau) damping. Hereafter we assumec>-0.

  4. Note thatHf∃’=(Hf)∃’; here, and in the following, primes appended to functions indicate differentiation.

  5. For all the solutions considered in this paper this constant of motion vanishes.

  6. H. H. Chen, Y. C. Lee andN. R. Pereira:Algebraic internal wave solitons and the integrable Calogero-Moser-Sutherland N-body problem, preprint (to be published).

  7. H. Airault, H. McKean andJ. Moser:Comm. Pure Appl. Math.,30, 95 (1977), hereafter referred to as AMM.

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  8. D. V. Choodnovsky andG. V. Choodnovsky:Nuovo Cimento,40 B, 339 (1977).

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  9. A prime appended to a sum indicates that the singular term must be omitted when summing.

  10. Dots indicate time differentiation.

  11. Clearly the case when allx j ∃’s are in the lower half-plane is trivially related, by complex conjugation, to that treated here; while the results of AMM (see below) indicate that, at least for finiten, no solutions of (8) and (9) exists (forc≠0) unless the polesx j are all located in the same half-plane.

  12. Forc=0,i.e. in the usual KdV case, these solutions belong to the similarity classu(x, t)=(t∃-t 0)∃-2/3F[x/(t∃-t0)1/3].

  13. Of appropriately scaled size; see AMM.

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Author notes

  1. F. Calogero

    Present address: Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy

Authors and Affiliations

  1. Institute of Theoretical and Experimental Physics, 117259, Moscow, URSS

    F. Calogero, M. A. Olshanetsky & A. M. Perelomov

  2. Istituto di Fisica dell’Università, 00185, Roma, Italy

    F. Calogero

Authors
  1. F. Calogero
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  2. M. A. Olshanetsky
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  3. A. M. Perelomov
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Calogero, F., Olshanetsky, M.A. & Perelomov, A.M. Rational solutions of the KdV equation with damping. Lett. Nuovo Cimento 24, 97–100 (1979). https://doi.org/10.1007/BF02725599

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