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Remarks on the local version of the inverse scattering method. (English. Russian original) Zbl 1351.35096
Proc. Steklov Inst. Math. 253, 37-50 (2006); translation from Tr. Mat. Inst. Steklova 253, 46-60 (2006).
Summary: It is very likely that all local holomorphic solutions of integrable $$(1+1)$$-dimensional parabolic-type evolution equations can be obtained from the zero solution by formal gauge transformations that belong (as formal power series) to appropriate Gevrey classes. We describe in detail the construction of solutions by means of convergent gauge transformations and prove an assertion converse to the above conjecture; namely, we suggest a simple necessary condition for the existence of a local holomorphic solution to the Cauchy problem for the evolution equations under consideration in terms of scattering data of initial conditions.

##### MSC:
 35P25 Scattering theory for PDEs 37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems 35Q53 KdV equations (Korteweg-de Vries equations) 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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##### References:
 [1] V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: The Method of Inverse Scattering (Nauka, Moscow, 1980; Plenum, New York, 1984). · Zbl 0598.35002 [2] L. D. Faddeev and L. A. Takhtadzhyan, Hamiltonian Methods in the Theory of Solitons (Nauka, Moscow, 1986; Springer, Berlin, 1987). · Zbl 0632.58003 [3] L. A. Dickey, Soliton Equations and Hamiltonian Systems (World Sci., Singapore, 1991). · Zbl 0753.35075 [4] R. Beals, P. Deift, and X. Zhou, ”The Inverse Scattering Transform on the Line,” in Important Developments of Soliton Theory, Ed. by A. S. Fokas and V. E. Zakharov (Springer, Berlin, 1993), pp. 7–32. · Zbl 0926.35129 [5] V. E. Zakharov and A. B. Shabat, ”Integration of Nonlinear Equations of Mathematical Physics by the Method of Inverse Scattering. II,” Funkts. Anal. Prilozh. 13(3), 13–22 (1979) [Funct. Anal. Appl. 13, 166–174 (1979)]. [6] I. M. Krichever, ”Nonlinear Equations and Elliptic Curves,” in Itogi Nauki Tekh., Ser.: Sovr. Probl. Mat. (VINITI, Moscow, 1983), Vol. 23, pp. 79–136 [J. Sov. Math. 28, 51–90 (1985)]. · Zbl 0595.35087 [7] G. Wilson, ”The $$\tau$$-Functions of the gAKNS Equations,” in Integrable Systems: The Verdier Memorial Conference, Ed. by O. Babelon et al. (Birkhäuser, Basel, 1993), Progr. Math. 115, pp. 131–145. [8] D. H. Sattinger and J. S. Szmigielski, ”Factorization and the Dressing Method for the Gel’fand-Dikii Hierarchy,” Physica D 64, 1–34 (1993). · Zbl 0770.34060 · doi:10.1016/0167-2789(93)90247-X [9] C.-L. Terng and K. Uhlenbeck, ”Bäcklund Transformations and Loop Group Actions,” Commun. Pure Appl. Math. 53, 1–75 (2000). · Zbl 1031.37064 · doi:10.1002/(SICI)1097-0312(200001)53:1<1::AID-CPA1>3.0.CO;2-U [10] A. V. Domrin, ”The Riemann Problem and Matrix-Valued Potentials with a Convergent Baker-Akhiezer Function,” Teor. Mat. Fiz. 144(3), 453–471 (2005) [Theor. Math. Phys. 144, 1264–1278 (2005)]. · Zbl 1178.30048 · doi:10.4213/tmf1870 [11] B. Malgrange, ”Sur les deformations isomonodromiques. I: Singularites regulieres,” in Mathematique et physique: Semin. Ec. Norm. Super. 1979–1982, Ed. by L. Boute de Monvel et al. (Birkhäuser, Basel, 1983), Progr. Math. 37, pp. 401–426. [12] R. S. Ward, ”The Painlevé Property for the Self-dual Gauge-Field Equations,” Phys. Lett. A 102, 279–282 (1984). · doi:10.1016/0375-9601(84)90680-7 [13] M. U. Schmidt, Integrable Systems and Riemann Surfaces of Infinite Genus (Am. Math. Soc., Providence, RI, 1996), Mem. AMS 581. · Zbl 0867.58038 [14] E. Goursat, A Course in Mathematical Analysis, Vol. III, Part I: Variation of Solutions. Partial Differential Equations of the Second Order (Gos. Izd. Tekh.-Teor. Literatury, Moscow, 1933; New York, Dover, 1964). [15] N. Joshi, J. A. Petersen, and L. M. Schubert, ”Nonexistence Results for the Korteweg-de Vries and Kadomtsev-Petviashvili Equations,” Stud. Appl. Math. 105, 361–374 (2000). · Zbl 1020.35085 · doi:10.1111/1467-9590.00155 [16] K. Clancey and I. Gokhberg, Factorization of Matrix Functions and Singular Integral Operators (Birkhäuser, Basel, 1981). [17] G. D. Birkhoff, ”The Generalized Riemann Problem for Linear Differential Equations and the Allied Problems for Linear Difference and q-Difference Equations,” Proc. Am. Acad. Arts Sci. 49, 521–568 (1913); reprint. in Collected Math. Papers (Dover, New York, 1950), Vol. 1, pp. 259–306. · doi:10.2307/20025482 [18] F. R. Gantmacher, The Theory of Matrices (Nauka, Moscow, 1966; Am. Math. Soc., Providence, RI, 1998). · Zbl 0136.00410 [19] Y. Sibuya, Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation (Am. Math. Soc., Providence, RI, 1990). · Zbl 1145.34378 [20] P.-F. Hsieh and Y. Sibuya, Basic Theory of Ordinary Differential Equations (Springer, Berlin, 1999). · Zbl 0924.34001 [21] A. B. Shabat, ”The Inverse Scattering Problem,” Diff. Uravn. 15, 1824–1834 (1979) [Diff. Eqns. 15, 1299–1307 (1980)]. [22] A. O. Gel’fond, Calculus of Finite Differences (Nauka, Moscow, 1967; Hindustan Publ., Delhi, 1971).
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