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1
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84933120134
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Throughout this paper whenever we mention NLPDE’s we include also the possibility that these be integro-differential equations (that may, or may not, reduce to pure partial differential equations, possibly by an appropriate redefinition of the dependent variable).
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8
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84933120133
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Paper by J. Moser in ref. (3c) J. Moser, Editor: Dynamical Systems, Theory and Applications (Berlin, 1974) (see in particular the papers by M. Kruskal and by H. Flaschka and A. C. Newell); J. Moser: Adv. Math., 16, 197 (1975); F. Calogero, C. Marchioro and O. Ragnisco: Lett. Nuovo Cimento, 13, 383 (1975); F. Calogero: Lett. Nuovo Cimento, 13, 411 (1975); M. Adler: preprint (A new integrable system and a conjecture by Calogero, to be published).
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12
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84933120148
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Throughout this paper we occasionally differ, to streamline our presentation, from the notation used previously.
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14
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84933120149
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M. J. Ablowitz and R. Haberman: Phys. Rev. Lett. (in press).
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16
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84933120150
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F. Calogero and A. Degasperis: a Phys. Rev. Lett (submitted to); b Lett. Nuovo Cimento, 15, 65 (1976).
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18
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84933120152
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The difference between the integral operators I_ and I, eqs. (1.4) and (1.9), compensates exactly the sign differences between the definitions of L_ and L, eqs. (1.3) and (1.8); see the appendix.
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20
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84933120151
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After this paper was partially drafted (and the two papers of ref. 16). F. Calogero and A. Degasperis: a Phys. Rev. Lett., (submitted to); b Lett. Nuovo Cimento, 15, 65 (1976). had been submitted for publication we received a preprint by H. Flaschka and D. W. McLaughlin (Some comments on Bäcklund transformations, canonical transformations and the inverse scattering method, to be published) that takes a point of view similar to that of this paper, and reports some results that coincide with special cases of those given here.
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21
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84933120154
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A preprint by D. J. Kaup: the closure of the squared Zakharov-Shabat eigenstates (to appear in Journ. Math. Anal. Appl.) also takes a somewhat similar point of view to that of this paper.
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25
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84933120153
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A condition, that we have not, for simplicity, mentioned previously (16), F. Calogero and A. Degasperis: a Phys. Rev. Lett (submitted to); b Lett. Nuovo Cimento, 15, 65 (1976), but that is clearly implied by (4.1.10), is that γ and v have the same singularity structure in the finite part of the complex z-plane. Let us however also mention at this point that the requirement that these be entire (or ratios of entire) functions is sufficient, but not necessary, for the validity of all these results, that might indeed also hold for nonentire functions provided a suitable definition is given of the operator that obtains after replacing the argument of such a function by an operator.
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26
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84933120156
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The basic idea that is used to extract the conserved quantities is a fairly old one in potential scattering theory, that may be traced back to papers by N. Levinson, R. G. Newton and L. D. Faddeev; see, for instance, F. Calogero and A. Degasperis: Journ. Math. Phys., 9, 90 (1968).
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28
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84933120155
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Note that, since in this case both α± and α±′ vanish, we are in fact extrapolating our results to a case in which the discrete spectrum cannot be obtained by analytic continuation from the alphas. A discussion of this point is deferred to a subsequent paper, as well as a more detailed analysis of this «soliton» solution when a nontrivial γ-dependence is present (in which case in general it does not behave like a soliton at all).
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29
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84933120157
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Write explicitly the linear Bäcklund transformation for the fields (using eqs. 4.2.3) and (4.2.13 a), introduce the functionw(x)-∫x∞dx'q2(x') (and w′x), similarly related to q′(x), solve for w′−w in terms of q′+ q (choosing appropriately the sign in the solution of the second-degree equation), differentiate, simplify, and finally integrate using the asymptotic boundary conditions Q(+∞)= Q′(+∞)= q(+∞)= q′(+∞)=0.
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