Lax, Peter D. Integrals of nonlinear equations of evolution and solitary waves. (English) Zbl 0162.41103 Commun. Pure Appl. Math. 21, 467-490 (1968). 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. Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 5 ReviewsCited in 911 Documents Keywords:partial differential equations PDFBibTeX XMLCite \textit{P. D. Lax}, Commun. Pure Appl. Math. 21, 467--490 (1968; Zbl 0162.41103) Full Text: DOI 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. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.