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’’IST‐solvable’’ nonlinear evolution equations and existence—An extension of Lax’s method
1.C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Phys. Rev. Lett. 19, 1095 (1967);
1.C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Commun. Pure Appl. Math. XXVII, 97 (1974).
2.P. D. Lax, Commun. Pure Appl. Math. XXI, 467 (1968);
2.P. D. Lax, XXVIII, 141 (1975)., Commun. Pure Appl. Math.
3.V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971)
3.[V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972)];
3.V. E. Zakharov and A. B. Shabat, 64, 1627 (1973) , Sov. Phys. JETP
3.[V. E. Zakharov and A. B. Shabat, 37, 823 (1974)]., Sov. Phys. JETP
4.M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Stud. Appl. Math. LIII, 249 (1974). Many physically interesting cases are discussed.
5.N. Dunford and J. T. Schwartz, Linear Operators, Part II (Wiley, New York, 1963), p. 1217, Theorem 16. [See also the remark by P. D. Lax, p. 293, chapter XI in S. Leibovich and A. R. Seebass, Eds., Nonlinear Waves (Cornell U.P., Ithaca, New York, 1974).].
6.F. Calogero and A. Degasperis, Nuovo Cimento B 32, 201 (1976). (See also preprint submitted in September 1976).
7.The linear operator Λ, and the associated operators (and in the matrix case) arise here naturally without needing to be separately constructed; cf. the AKNS method.
8.Apart from the matrix case in Sec. 5, our method is showing promise for the ν‐dimensional Schrödinger case.
9.Evolution equations of a matrix type are also found and studied, and their physical application discussed in: (a) V. E. Zakharov and S. V. Manakov, Zh. Eksp. Teor. Fiz. Pis. Red. 18, 413 (1973)
9.[V. E. Zakharov and S. V. Manakov, JETP Lett. 18, 243 (1973)];
9.(b) M. J. Ablowitz and R. Haberman, J. Math. Phys. 16, 2301 (1975). With reference to our method we take note of Refs. 16 and 17 of (b).
10.D. J. Kaup, J. Math. Anal. Appl. 54, 849 (1976).
11.M. Jaulent, “Completeness of Squared Schrödinger Eigenfunctions” (unpublished, in preparation).
12.M. Jaulent and I. Miodek, Lett. Math. Phys. 1, 243 (1976).
13.I. Miodek, Preprint (PM/77/1), Lab. de Math‐Physique, Université des Sciences et Techniques du Languedoc, Montpellier, France. In this reference we show that a less restricted class of linear first order matrix differential equations can be reduced to Eq. (5.1) by invertible linear transformations; their inverse problems will thus not be independent. It can be expected that the evolution equations of the more general linear equations will not be independent from the ones obtained for Eq. (5.1).
14.In some respects this other way of obtaining our auxiliary equation is preferable because it seems to involve less manipulation and gives us other important properties in a uniform way, as we have shown. It works in both the Schrödinger and matrix cases and happens to be the way the author first found the auxiliary equation to be useful. Insofar as it is based on direct cross‐differentiation it is easily compared to the methods in Ref. 9(b) and the original method of GGKM in Ref. 1. The present presentation has the advantage of abstract generality.
15.See review by L. D. Faddeev, J. Math. Phys. 1, 72 (1973), in particular his Eq. (11.19) and his note n° 11.
16.A. Martin and P. Sabatier, “Impedence, zero energy wave function, and bound states,” Preprint (PM/77/2) Physique Mathématique U.S.T.L., Montpellier, France; to be published in J. Math. Phys.
17.M. Jaulent and I. Miodek, Lett. Nuovo Cimento (to be published).
18.(a) M. Jaulent, Ann. Inst. Henri Poincaré XVII, 363 (1972);
18.(b) M. Jaulent and C. Jean, Commun. Math. Phys. 28, 177 (1972);
18.M. Jaulent and C. Jean, Ann. Inst. Henri Poincaré XXV, 105, 119 (1976).
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