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Extra resources for Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications: NSF Research Workshop on Contact Transformations

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W. MCLAUGHLIN, The soliton: a new concept in applied science, Proc. IEEE 61 (1973), 1443. E. S. L. M. BURDE, G. JOYCE, I. D. J. DOUCET, A. HIROSE, H. IKEZI, S. AKSORNKITTI, M. WIDNER, AND K. ESTABROOK, Properties of plasma waves defined by the dispersion 2 2 + k ~ /k 2 = 0, relation D(k,~) = 1 - ~o/~ (1974), 93, IEEE Trans. J. D. KRUSKAL, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett 15 (1965), 240. [13] E. R. M. ULAM, Studies of nonlinear problems in Collected Works of Enrico Fermi, Vol.

Fl satisfies 55 -(~-~C + ~I ffi ~x)Fl(X vlt) v2 FI(X - vlt ) + F2(x - v2t ) ' (59) (~+v I~)F z(x- v2t) Fl(X - Vlt ) + F2(x - v2t ) where we write of F1 and F 1 + F2 F2. in place of i + F1 + F2 because of the arbitrariness This form of the solution was obtained by Hashimoto [13]. The form of (59) reminds us of the Cole-Hopf transformation of Burgers equation ut - Uxx + 2uu x = 0 . In Appendix II we describe how the present method gives the Cole-Hopf transformarion [ 14]. 2. Volterra' s Equation Volterra's equation which describes the growth of conflicting populations in biological sciences is written as [15 ] dN 1 d't" = Nl(al - N2) ' (60) dN 2 dt = - N2(a2 - NI) ' where aI and NI and a2 N2 are the populations of two species, prey and predator, and are the growing and damping constants of the two species, respec- tlvely.

S. L. M. BURDE, G. JOYCE, I. D. J. DOUCET, A. HIROSE, H. IKEZI, S. AKSORNKITTI, M. WIDNER, AND K. ESTABROOK, Properties of plasma waves defined by the dispersion 2 2 + k ~ /k 2 = 0, relation D(k,~) = 1 - ~o/~ (1974), 93, IEEE Trans. J. D. KRUSKAL, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett 15 (1965), 240. [13] E. R. M. ULAM, Studies of nonlinear problems in Collected Works of Enrico Fermi, Vol. II, Univ. , 1965, p. 478. D. N. JUDICE, Recurrence of nonlinear ion acoustic waves, Phys.

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