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Integrals of nonlinear equations of evolution and solitary waves. (English) Zbl 0162.41103
Summary: In Section 1 we present a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation. A striking instance of such a procedure discovery by Gardner, Miura and Kruskal that the eigenvalues of the Schrödinger operator are integrals of the Korteweg-de Vries equation.
In Section 2 we prove the simplest case of a conjecture of Kruskal and Zabusky concerning the existence of double wave solutions of the Korteweg-de Vries equation, i.e., of solutions which for $$|t|$$ large behave as the superposition of two solitary waves travelling at different speeds. The main tool used is the first of remarkable series of integrals discovered by Kruskal and Zabusky.

##### Keywords:
partial differential equations
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##### References:
 [1] Gardner, Phys. Rev. Letters 19 pp 1095– (1967) [2] and , Similarity in the asymptotic behavior of collision-free hydromagnetic waves and water waves, New York Univ., Courant Inst. Math. Sci., Res. Rep. NYO-9082, 1960. [3] Korteweg, Philos. Mag. 39 pp 422– (1895) [4] Miura, J. Math. Phys. 9 pp 1202– (1968) [5] Miura, J. Math. Phys. 9 pp 1204– (1968) [6] On the Korteweg-de Vries equation, Uppsala Univ., Dept. of Computer Sci., Report, 1967. [7] Whitman, Proc. Roy. Soc., Ser. A. 283 pp 238– (1965) [8] A synergetic approach to problems of nonlinear dispersive wave propagation and interaction, Nonlinear Patial Differential Equations, Academic Press, New York, 1967. [9] Zabusky, Phys. Rev. Letters 15 pp 240– (1965) [10] Integrals of nonlinear equations of evolution and solitary waves, New York Univ., Courant Inst. Math. Sciences, Report NYO-1480-87, Jan. 1968.
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