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Commutation methods applied to the mKdV-equation


Authors: F. Gesztesy, W. Schweiger and B. Simon
Journal: Trans. Amer. Math. Soc. 324 (1991), 465-525
MSC: Primary 35Q53; Secondary 34L25, 47E05, 58F07
DOI: https://doi.org/10.1090/S0002-9947-1991-1029000-7
MathSciNet review: 1029000
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Abstract: An explicit construction of solutions of the modified Korteweg-de Vries equation given a solution of the (ordinary) Korteweg-de Vries equation is provided. Our theory is based on commutation methods (i.e., $ N = 1$ supersymmetry) underlying Miura's transformation that links solutions of the two evolution equations.


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  • [1] M. J. Ablowitz and H. Cornille, On solutions of the Korteweg-de Vries equation, Phys. Lett. A 72 (1979), 277-280. MR 589815 (82f:35156)
  • [2] M. J. Ablowitz and J. Satsuma, Solitons and rational solutions of nonlinear evolution equations, J. Math. Phys. 19 (1978), 2180-2186. MR 507515 (80b:35121)
  • [3] M. J. Ablowitz, M. Kruskal, and H. Segur, A note on Miura's transformation, J. Math. Phys. 20 (1979), 999-1003. MR 534337 (81m:35119)
  • [4] M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974), 249-315. MR 0450815 (56:9108)
  • [5] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover, New York, 1972.
  • [6] M. Adler and J. Moser, On a class of polynomials connected with the Korteweg-de Vries equation, Comm. Math. Phys. 61 (1978), 1-30. MR 0501106 (58:18554)
  • [7] M. Antonowicz and A. Fordy, Factorization of energy dependent Schrödinger operators: Miura maps and modified systems, Comm. Math. Phys. 124 (1989), 465-486. MR 1012634 (91b:58086)
  • [8] A. Arai, Some remarks on scattering theory in supersymmetric quantum mechanics, J. Math. Phys. 28 (1987), 472-476. MR 872031 (88a:81031)
  • [9] V. A. Arkad'ev, A. K. Pogrebkov, and M. K. Polivanov, Singular solutions of the $ \operatorname{KdV} $ equation and the inverse scattering method, J. Soviet Math. 31 (1985), 3264-3279.
  • [10] T. C. Au-Yeung, C. Au, and P. C. W. Fung, One soliton Korteweg-de Vries solution with nonzero vacuum parameters obtainable from the generalized inverse scattering method, Phys. Rev. A 29 (1984), 2370-2374. MR 743862 (86b:35174)
  • [11] T. C. Au-Yeung, P. C. W. Fung, and C. Au, Modified $ KdV$ solitons with non-zero vacuum parameter obtainable from the $ ZS{\text{ - }}AKNS$ inverse method, J. Phys. A 17 (1984), 1425-1436. MR 748775 (86g:35167)
  • [12] R. Beals, P. Deift, and C. Tomei, Direct and inverse scattering on the line, Mathematical Surveys and Monographs, vol. 28, Amer. Math. Soc., Providence, R. I., 1988. MR 954382 (90a:58064)
  • [13] M. J. Bergvelt and A. P. E. ten Kroode, $ \tau $ functions and zero curvature equations of the Toda-$ AKNS$ type, J. Math. Phys. 29 (1988), 1308-1320. MR 944444 (90b:58100)
  • [14] D. Bollé, F. Gesztesy, and M. Klaus, Scattering theory for one-dimensional systems with $ \int {dxV(x) = 0} $, J. Math. Anal. Appl. 122 (1987), 496-518. MR 877834 (89k:34028a)
  • [15] D. Bollé, F. Gesztesy, and S. F. J. Wilk, A complete treatment of low-energy scattering in one dimension, J. Operator Theory 13 (1985), 3-31. MR 768299 (86f:34047)
  • [16] D. Bollé, F. Gesztesy, H. Grosse, W. Schweiger, and B. Simon, Witten index, axial anomaly, and Krein 's spectral shift function in supersymmetric quantum mechanics, J. Math. Phys. 28 (1987), 1512-1525. MR 894842 (88j:81022)
  • [17] N. V. Borisov, W. Müller, and R. Schrader, Relative index theorems and supersymmetric scattering theory, Comm. Math. Phys. 114 (1988), 475-513. MR 929141 (89j:58131)
  • [18] M. Bruschi and F. Calogero, The Lax representation for an integrable class of relativistic dynamical systems, Comm. Math. Phys. 109 (1987), 481-492. MR 882811 (88h:58052)
  • [19] J. L. Burchnall and T. W. Chaundy, Commutative ordinary differential operators, Proc. Roy. Soc. London Ser. A 118 (1928), 557-583.
  • [20] -, Commutative ordinary differential operators II. The identity $ {P^n} = {Q^m}$, Proc. Roy. Soc. London Ser. A 134 (1931-32), 471-485.
  • [21] M. Buys and A. Finkel, The inverse periodic problem for Hill's equation with a finite-gap potential, J. Differential Equations 55 (1984), 257-275. MR 764126 (86a:34052)
  • [22] F. Calogero and A. Degasperis, Spectral transforms and solitons, Vol. 1, North-Holland, Amsterdam, 1982. MR 604446 (83b:35139)
  • [23] R. Carmona, One-dimensional Schrödinger operators with random or deterministic potentials: New spectral types, J. Funct. Anal. 51 (1983), 229-258. MR 701057 (85k:34144)
  • [24] S. Chaturvedi and K. Raghunathan, Relation between the Gelfand-Levitan procedure and the method of supersymmetric partners, J. Phys. A 19 (1986), L775-L778. MR 857041 (87i:81029)
  • [25] A. Cohen, Decay and regularity in the inverse scattering problem, J. Math. Anal. Appl. 87 (1982), 395-426. MR 658022 (83j:34028)
  • [26] -, Solutions of the Korteweg-De Vries equation with steplike initial profile, Comm. Partial Differential Equations 9 (1984), 751-806. MR 748367 (86b:35175)
  • [27] A. Cohen and T. Kappeler, Scattering and inverse scattering for steplike potentials in the Schrödinger equation, Indiana Univ. Math. J. 34 (1985), 127-180. MR 773398 (86k:34017)
  • [28] M. M. Crum, Associated Sturm-Liouville systems, Quart. J. Math. Oxford (2) 6 (1955), 121-127. MR 0072332 (17:266g)
  • [29] G. Darboux, Sur une proposition relative aux équations linéaires, C.R. Acad. Sci. (Paris) 94 (1882), 1456-1459.
  • [30] E. Date and S. Tanaka, Periodic multi-soliton solutions of Korteweg-de Vries equation and Toda lattice, Suppl. Progr. Theoret. Phys. 59 (1976), 107-125. MR 0438389 (55:11303)
  • [31] E. B. Davies and B. Simon, Scattering theory for systems with different spatial asymptotics on the left and right, Comm. Math. Phys. 63 (1978), 277-301. MR 513906 (80c:81110)
  • [32] P. A. Deift, Applications of a commutation formula, Duke Math. J. 45 (1978), 267-310. MR 495676 (81g:47001)
  • [33] P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), 121-251. MR 512420 (80e:34011)
  • [34] R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and nonlinear wave equations, Academic Press, New York, 1982. MR 696935 (84j:35142)
  • [35] B. A. Dubrovin, Inverse problem for periodic finite-zoned potentials in the theory of scattering, Functional. Anal. Appl. 9 (1975), 61-62. MR 0382873 (52:3755)
  • [36] B. A. Dubrovin and S. P. Novikov, Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg-de Vries equation, Soviet Phys. JETP 40 (1975), 1058-1063. MR 0382877 (52:3759)
  • [37] N. Dunford and J. T. Schwartz, Linear operators. II, Interscience, New York, 1963. MR 0188745 (32:6181)
  • [38] M. S. P. Eastham, The spectral theory of periodic differential equations, Scottish Academic Press, Edinburgh, 1973.
  • [39] W. Eckhaus and A. Van Harten, The inverse scattering transformation and the theory of solitons, Math. Studies, no. 50, North-Holland, Amsterdam, 1981. MR 615557 (83c:35101)
  • [40] H. Flaschka, On the inverse problem of Hill's operator, Arch. Rational Mech. Anal. 59 (1975), 293-309. MR 0387711 (52:8550)
  • [41] H. Flaschka and D. W. McLaughlin, Some comments on Bäcklund transformations, canonical transformations, and the inverse scattering method, Lecture Notes in Math., vol. 516, (R. M. Miura, ed.), Springer-Verlag, Berlin, 1976, pp. 252-295. MR 0609534 (58:29425)
  • [42] H. Flaschka and A. C. Newell, Monodromy- and spectrum-preserving deformations. I, Comm. Math. Phys. 76 (1980), 65-116. MR 588248 (82g:35103)
  • [43] N. E. Firsova, Riemann surface of quasimomentum and scattering theory for the perturbed Hill operator, J. Soviet Math. 11 (1975), 487-497.
  • [44] -, An inverse scattering problem for a perturbed Hill operator, Math. Notes 18 (1975), 1085-1091.
  • [45] -, Some spectral identities for the one-dimensional Hill operator, Theoret. Math. Phys. 37 (1977), 1022-1027.
  • [46] N. E. Firsova, The direct and inverse scattering problems for the one-dimensional perturbed Hill operator, Math. USSR Sb. 58 (1987), 351-388.
  • [47] P. C. W. Fung and C. Au, Bridge between the solutions and vacuum states of the Korteweg-de Vries equation and that of the nonlinear equation $ {y_t} + {y_{xxx}} - 6{y^2}{y_x} + 6\lambda {y_x} = 0$, Phys. Rev. B 26 (1982), 4035-4038. MR 675935 (83m:35130)
  • [48] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Korteweg-de Vries equation and generalizations, VI. Methods for exact solution, Comm. Pure Appl. Math. 27 (1974), 97-133. MR 0336122 (49:898)
  • [49] F. Gesztesy, Scattering theory for one-dimensional systems with nontrivial spatial asymptotics, in Schrödinger Operators, Aarhus 1985 (E. Balslev, ed.), Lecture Notes in Math., vol. 1218, Springer-Verlag, Berlin, 1986, pp. 93-122. MR 869597 (88a:81197)
  • [50] F. Gesztesy and B. Simon, Constructing solutions of the $ \operatorname{mKdV} $-equation, J. Funct. Anal, (in print).
  • [51] J. Ginibre and Y. Tsutsumi, G. Velo, Existence and uniqueness of solutions for the generalized Korteweg-de Vries equation, SIAM J. Math. Anal. 20 (1989), 1388-1425. MR 1019307 (90i:35240)
  • [52] W. Goldberg and H. Hochstadt, On a Hill's equation with selected gaps in its spectrum, J. Differential Equations 34 (1979), 167-178. MR 550038 (81j:34043)
  • [53] D. Grecu and V. Cionga, On the mixed soliton-rational solutions of the $ \operatorname{KdV} $ equation, preprint, 1986.
  • [54] H. Grosse, Solitons of the modified $ KdV$ equation, Lett. Math. Phys. 8 (1984), 313-319. MR 759630 (86c:35138)
  • [55] -, New solitons connected to the Dirac equation, Phys. Rep. 134 (1986), 297-304. MR 832137 (87g:35209)
  • [56] P. Hartman, Ordinary differential equations Wiley, New York, 1964. MR 0171038 (30:1270)
  • [57] R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 18 (1971), 1192-1194.
  • [58] -, Exact solution of the modified Korteweg-de Vries equation for multiple collisions of solitons, J. Phys. Soc. Japan 33 (1972), 1456-1458.
  • [59] H. Hochstadt, On the determination of a Hill's equation from its spectrum, Arch. Rational. Mech. Anal. 19 (1965), 353-362. MR 0181792 (31:6019)
  • [60] A. R. Its, Inversion of hyperelliptic integrals and integration of nonlinear differential equations, Vestnik Leningrad Univ. Math. 9 (1981), 121-129. MR 0609747 (58:29453)
  • [61] A. R. Its and V. B. Matveev, Schrödinger operator with finite-gap spectrum and $ N$-soliton solutions of the Korteweg-de Vries equation, Theoret. Math. Phys. 23 (1975), 343-355. MR 0479120 (57:18570)
  • [62] K. Iwasaki, Inverse problem for Sturm-Liouville and Hill equations, Ann. Mat. Pura Appl. (4) 149 (1987), 185-206. MR 932784 (89d:34053)
  • [63] C. G. J. Jacobi, Zur Theorie der Variationsrechnung und der Differentialgleichungen, J. Reine Angew. Math. 17 (1837), 68-82.
  • [64] A. Jaffe, A. Lesniewski and M. Lewenstein, Ground state structure in supersymmetric quantum mechanics, Ann. Phys. 178 (1987), 313-329. MR 914181 (88k:81076)
  • [65] M. Jaulent and I. Miodek, Connection between Zakharov-Shabat and Schrödinger-type inverse scattering transform, Lett. Nuovo Cimento 20 (1977), 655-660.
  • [66] Y. Kametaka, Korteweg-de Vries equation III. Global existence of asymptotically periodic solutions, Proc. Japan Acad. 45 (1969), 656-660. MR 0262706 (41:7311)
  • [67] -, Korteweg-de Vries equation IV. Simplest generalization, Proc. Japan Acad. 45 (1969), 661-665. MR 0262707 (41:7312)
  • [68] T. Kato, On the Korteweg-de Vries equation, Manuscripta Math. 28 (1979), 89-99.
  • [69] -, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in Applied Mathematics, (V. Guillemin, ed.), Adv. Math. Suppl. Studies, no. 18, Academic Press, New York, 1983, pp. 93-128. MR 759907 (86f:35160)
  • [70] Y. Kato, Fredholm determinants and the Cauchy problem of a class of nonlinear evolution equations, Progr. Theoret. Phys. 78 (1987), 198-213. MR 919541 (89b:35145)
  • [71] J. Kay and H. E. Moses, Reflectionless transmission through dielectrics and scattering potentials, J. Appl. Phys. 27 (1956), 1503-1508.
  • [72] M. Klaus, Low-energy behavior of the scattering matrix for the Schrödinger equation on the line, Inverse Problems 4 (1988), 505-512. MR 954906 (89k:81185)
  • [73] I. M. Krichever, Potentials with zero coefficient of reflection on a background of finite-zone potentials, Functional Anal. Appl. 9 (1975), 161-163.
  • [74] S. N. Kruzhkov and A. V. Faminskii, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, Math. USSR Sb. 48 (1984), 391-421. MR 691986 (85c:35079)
  • [75] B. A. Kupershmidt and G. Wilson, Modifying Lax equations and the second Hamiltonian structure, Invent. Math. 62 (1981), 403-436. MR 604836 (84m:58055)
  • [76] E. A. Kuznetsov and A. V. Mikhailov, Stability of stationary waves in nonlinear weakly dispersive media, Soviet Phys. JETP 40 (1975), 855-859. MR 0387847 (52:8685)
  • [77] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467-490. MR 0235310 (38:3620)
  • [78] -, Periodic solutions of the $ KdV$ equations, Lectures in Appl. Math. 15 (1974), 85-96. MR 0344645 (49:9384)
  • [79] W. Magnus and S. Winkler, Hill's equation, Dover, New York, 1979. MR 559928 (80k:34001)
  • [80] V. A. Marchenko, Sturm-Liouville operators and application", Birkhäuser, Basel, 1986. MR 897106 (88f:34034)
  • [81] H. P. McKean, Theta functions, solitons, and singular curves, Partial Differential Equations and Geometry (C. I. Byrnes, ed.), Marcel Dekker, 1979, pp. 237-254. MR 535596 (82f:14028)
  • [82] H. P. McKean, Geometry of $ \operatorname{KdV} (1)$: Addition and the unimodular spectral classes, Rev. Mat. Iberoamericana 2 (1986), 235-261. MR 908052 (89b:58096)
  • [83] -, Geometry of $ \operatorname{KdV} (2)$: Three examples, J. Statist. Phys. 46 (1987), 1115-1143. MR 893135 (89b:58097)
  • [84] H. P. McKean and P. van Moerbeke, The spectrum of Hill's equation, Invent. Math. 30 (1975), 217-274. MR 0397076 (53:936)
  • [85] H. P. McKean and E. Trubowitz, Hills operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976), 143-226. MR 0427731 (55:761)
  • [86] -, Hill's surfaces and their theta functions, Bull. Amer. Math. Soc. 84 (1978), 1042-1085. MR 508448 (80b:30039)
  • [87] A. Menikoff, The existence of unbounded solutions of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 25 (1972), 407-432. MR 0303129 (46:2267)
  • [88] R. M. Miura, Korteweg-de Vries equation and generalizations, I. A remarkable explicit non-linear transformation, J. Math. Phys. 9 (1968), 1202-1204. MR 0252825 (40:6042a)
  • [89] H. E. Moses, A generalization of the Gelfand-Levitan equation for the one-dimensional Schrödinger equation, J. Math. Phys. 18 (1977), 2243-2250. MR 0462290 (57:2264)
  • [90] M. Murata, Structure of positive solutions to $ ( - \Delta + V)u = 0$ in $ {\mathbb{R}^n}$, Duke Math. J. 53 (1986), 869-943. MR 874676 (88f:35039)
  • [91] M. M. Nieto, Relationship between supersymmetry and the inverse method in quantum mechanics, Phys. Lett. B 145 (1984), 208-210. MR 762223 (87b:81030)
  • [92] S. P. Novikov, The periodic problem for the Korteweg-de Vries equation, Functional Anal. Appl. 8 (1974), 236-246. MR 0382878 (52:3760)
  • [93] S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of solitons, Consultants Bureau, New York, 1984. MR 779467 (86k:35142)
  • [94] M. Ohmiya, On the generalized soliton solutions of the modified Korteweg-de Vries equation, Osaka J. Math. 11 (1974), 61-71. MR 0352742 (50:5229)
  • [95] M. Reed and B. Simon, Methods of modern mathematical physics IV: Analysis of operators, Academic Press, New York, 1978. MR 0493421 (58:12429c)
  • [96] S. N. M. Ruijsenaars and P. J. M. Bongaarts, Scattering theory for one-dimensional step potentials, Ann. Inst. H. Poincaré A 26 (1977), 1-17. MR 0443712 (56:2075)
  • [97] S. N. M. Ruijsenaars and H. Schneider, A new class of integrable systems and its relation to solitons, Ann. Phys. 170 (1986), 370-405. MR 851627 (88a:58097)
  • [98] D. H. Sattinger and V. D. Zurkowski, Gauge theory of Bäcklund transformations II, Phys. D 26 (1987), 225-250. MR 892446 (89a:58121)
  • [99] U.-W. Schmincke, On Schrödinger's factorization method for Sturm-Liouville operators, Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), 67-84. MR 529570 (80f:34025)
  • [100] B. Simon, Large time behavior of the $ {L^P}$ norm of Schrödinger semigroups, J. Funct. Anal. 40 (1981), 66-83. MR 607592 (82i:81027)
  • [101] A. Sjöberg, On the Korteweg-de Vries equation: Existence and uniqueness, J. Math. Anal. Appl. 29 (1970), 569-579. MR 0410135 (53:13885)
  • [102] V. V. Sokolov and A. B. Shabat, $ (L,A)$-pairs and a Riccati type substitution, Functional Anal. Appl. 14 (1980), 148-150. MR 575223 (81k:35152)
  • [103] O. Steinmann, Äquivalente periodische Potentiale, Helv. Phys. Acta 30 (1957), 515-520. MR 0092582 (19:1130f)
  • [104] C. V. Sukumar, Supersymmetric quantum mechanics of one-dimensional systems, J. Phys. A 18 (1985), 2917-2936. MR 814636 (87i:81038a)
  • [105] S. Tanaka, Modified Korteweg-de Vries equation and scattering theory, Proc. Japan Acad. 48 (1972), 466-469. MR 0336127 (49:903)
  • [106] S. Tanaka, On the $ N$-tuple wave solutions of the Korteweg-de Vries equation, Publ. Res. Inst. Math. Soc. 8 (1972/73), 419-427. MR 0328386 (48:6728)
  • [107] S. Tanaka, Some remarks on the modified Korteweg-de Vries equation, Publ. Res. Inst. Math. Sci. 8 (1972/73), 429-437. MR 0326198 (48:4543)
  • [108] S. Tanaka, Non-linear Schrödinger equation and modified Korteweg-de Vries equation; construction of solutions in terms of scattering data, Publ. Res. Inst. Math. Soc. 10 (1975), 329-357. MR 0499846 (58:17606)
  • [109] B. Thaller, Normal forms of an abstract Dirac operator and applications to scattering theory, J. Math. Phys. 29 (1988), 249-257. MR 921790 (88m:81033)
  • [110] E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), 321-337. MR 0430403 (55:3408)
  • [111] M. Tsutsumi, On global solutions of the generalized Korteweg-de Vries equation, Publ. Res. Inst. Math. Soc. 7 (1972), 329-344. MR 0312096 (47:658)
  • [112] S. Venakides, The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data, Trans. Amer. Math. Soc. 301 (1987), 189-226. MR 879569 (88b:35188)
  • [113] M. Wadati, The exact solution of the modified Korteweg-de Vries equation, J. Phys. Soc. Japan 32 (1972), 1681.
  • [114] -, The modified Korteweg-de Vries equation, J. Phys. Soc. Japan 34 (1973), 1289-1296. MR 0371251 (51:7472)
  • [115] M. Wadati and M. Toda, The exact $ N$-soliton solution of the Korteweg-de Vries equation, J. Phys. Soc. Japan 32 (1972), 1403-1411.
  • [116] H. D. Wahlquist, Bäcklund transformation of potentials of the Korteweg-de Vries equation and the interaction of solitons with cnoidal waves, Lecture Notes in Math., vol. 515 (R. M. Miura, ed.), Springer-Verlag, Berlin, 1976, pp. 162-183. MR 0610088 (58:29470)
  • [117] H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations, J. Math. Phys. 16 (1975), 1-7. MR 0358111 (50:10576)
  • [118] G. Wilson, Infinite-dimensional Lie groups and algebraic geometry in soliton theory, Philos. Trans. Roy. Soc. London Ser. A 315 (1985), 393-404. MR 836741 (87i:58089)
  • [119] Yu-kun Zheng and W. L. Chan, Gauge transformation and the higher order Korteweg-de Vries equation, J. Math. Phys. 29 (1988), 308-314. MR 927012 (89a:35216)

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DOI: https://doi.org/10.1090/S0002-9947-1991-1029000-7
Keywords: $ \operatorname{mKdV} $-equation, commutation methods, soliton-like solutions, periodic solutions, singular solutions
Article copyright: © Copyright 1991 American Mathematical Society

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