Commutation methods applied to the mKdV-equation

Gesztesy, F.; Schweiger, W.; Simon, B.

doi:10.1090/S0002-9947-1991-1029000-7

Commutation methods applied to the mKdV-equation
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by F. Gesztesy, W. Schweiger and B. Simon PDF

Trans. Amer. Math. Soc. 324 (1991), 465-525 Request permission

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.

References

Mark J. Ablowitz and Henri Cornille, On solutions of the Korteweg-de Vries equation, Phys. Lett. A 72 (1979), no. 4-5, 277–280. MR 589815, DOI 10.1016/0375-9601(79)90467-5
M. J. Ablowitz and J. Satsuma, Solitons and rational solutions of nonlinear evolution equations, J. Math. Phys. 19 (1978), no. 10, 2180–2186. MR 507515, DOI 10.1063/1.523550
Mark J. Ablowitz, Martin Kruskal, and Harvey Segur, A note on Miura’s transformation, J. Math. Phys. 20 (1979), no. 6, 999–1003. MR 534337, DOI 10.1063/1.524197
Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 (1974), no. 4, 249–315. MR 450815, DOI 10.1002/sapm1974534249

Handbook of mathematical functions

M. Adler and J. Moser, On a class of polynomials connected with the Korteweg-de Vries equation, Comm. Math. Phys. 61 (1978), no. 1, 1–30. MR 501106
Marek Antonowicz and Allan P. Fordy, Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems, Comm. Math. Phys. 124 (1989), no. 3, 465–486. MR 1012634
Asao Arai, Some remarks on scattering theory in supersymmetric quantum systems, J. Math. Phys. 28 (1987), no. 2, 472–476. MR 872031, DOI 10.1063/1.527629

Singular solutions of the

equation and the inverse scattering method

31

T. C. Au-Yeung, C. Au, and P. C. W. Fung, One-soliton Korteweg-de Vries solutions with nonzero vacuum parameters obtainable from the generalized inverse scattering method, Phys. Rev. A (3) 29 (1984), no. 5, 2370–2374. MR 743862, DOI 10.1103/PhysRevA.29.2370
T. C. Au-Yeung, P. C. W. Fung, and C. Au, Modified KdV solitons with nonzero vacuum parameter obtainable from the ZS-AKNS inverse method, J. Phys. A 17 (1984), no. 7, 1425–1436. MR 748775
Richard Beals, Percy Deift, and Carlos Tomei, Direct and inverse scattering on the line, Mathematical Surveys and Monographs, vol. 28, American Mathematical Society, Providence, RI, 1988. MR 954382, DOI 10.1090/surv/028
M. J. Bergvelt and A. P. E. ten Kroode, $\tau$ functions and zero curvature equations of Toda-AKNS type, J. Math. Phys. 29 (1988), no. 6, 1308–1320. MR 944444, DOI 10.1063/1.527923
D. Bollé, F. Gesztesy, and M. Klaus, Scattering theory for one-dimensional systems with $\int dx\,V(x)=0$, J. Math. Anal. Appl. 122 (1987), no. 2, 496–518. MR 877834, DOI 10.1016/0022-247X(87)90281-2
D. Bollé, F. Gesztesy, and S. F. J. Wilk, A complete treatment of low-energy scattering in one dimension, J. Operator Theory 13 (1985), no. 1, 3–31. MR 768299
D. Bollé, F. Gesztesy, H. Grosse, W. Schweiger, and B. Simon, Witten index, axial anomaly, and Kreĭn’s spectral shift function in supersymmetric quantum mechanics, J. Math. Phys. 28 (1987), no. 7, 1512–1525. MR 894842, DOI 10.1063/1.527508
N. V. Borisov, W. Müller, and R. Schrader, Relative index theorems and supersymmetric scattering theory, Comm. Math. Phys. 114 (1988), no. 3, 475–513. MR 929141
M. Bruschi and F. Calogero, The Lax representation for an integrable class of relativistic dynamical systems, Comm. Math. Phys. 109 (1987), no. 3, 481–492. MR 882811

Commutative ordinary differential operators

118

Commutative ordinary differential operators

The identity

134

Mutiara Buys and Allan Finkel, The inverse periodic problem for Hill’s equation with a finite-gap potential, J. Differential Equations 55 (1984), no. 2, 257–275. MR 764126, DOI 10.1016/0022-0396(84)90083-4
F. Calogero and A. Degasperis, Reduction technique for matrix nonlinear evolution equations solvable by the spectral transform, J. Math. Phys. 22 (1981), no. 1, 23–31. MR 604446, DOI 10.1063/1.524750
René Carmona, One-dimensional Schrödinger operators with random or deterministic potentials: new spectral types, J. Funct. Anal. 51 (1983), no. 2, 229–258. MR 701057, DOI 10.1016/0022-1236(83)90027-7
S. Chaturvedi and K. Raghunathan, Relation between the Gel′fand-Levitan procedure and the method of supersymmetric partners, J. Phys. A 19 (1986), no. 13, L775–L778. MR 857041
Amy Cohen, Decay and regularity in the inverse scattering problem, J. Math. Anal. Appl. 87 (1982), no. 2, 395–426. MR 658022, DOI 10.1016/0022-247X(82)90132-9
Amy Cohen, Solutions of the Korteweg-de Vries equation with steplike initial profile, Comm. Partial Differential Equations 9 (1984), no. 8, 751–806. MR 748367, DOI 10.1080/03605308408820347
Amy Cohen and Thomas Kappeler, Scattering and inverse scattering for steplike potentials in the Schrödinger equation, Indiana Univ. Math. J. 34 (1985), no. 1, 127–180. MR 773398, DOI 10.1512/iumj.1985.34.34008
M. M. Crum, Associated Sturm-Liouville systems, Quart. J. Math. Oxford Ser. (2) 6 (1955), 121–127. MR 72332, DOI 10.1093/qmath/6.1.121

Sur une proposition relative aux équations linéaires

94

Etsur\B{o} Date and Shunichi Tanaka, Periodic multi-soliton solutions of Korteweg-de Vries equation and Toda lattice, Progr. Theoret. Phys. Suppl. 59 (1976), 107–125. MR 0438389, DOI 10.1143/PTPS.59.107
E. B. Davies and B. Simon, Scattering theory for systems with different spatial asymptotics on the left and right, Comm. Math. Phys. 63 (1978), no. 3, 277–301. MR 513906
P. A. Deift, Applications of a commutation formula, Duke Math. J. 45 (1978), no. 2, 267–310. MR 495676
P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), no. 2, 121–251. MR 512420, DOI 10.1002/cpa.3160320202
Roger K. Dodd, J. Chris Eilbeck, John D. Gibbon, and Hedley C. Morris, Solitons and nonlinear wave equations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1982. MR 696935
B. A. Dubrovin, The inverse scattering problem for periodic short-range potentials, Funkcional. Anal. i Priložen. 9 (1975), no. 1, 65–66 (Russian). MR 0382873
B. A. Dubrovin and S. P. Novikov, Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg-de Vries equation, Ž. Èksper. Teoret. Fiz. 67 (1974), no. 6, 2131–2144 (Russian, with English summary); English transl., Soviet Physics JETP 40 (1974), no. 6, 1058–1063. MR 0382877
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons, New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR 0188745

The spectral theory of periodic differential equations

Wiktor Eckhaus and Aart van Harten, The inverse scattering transformation and the theory of solitons, North-Holland Mathematics Studies, vol. 50, North-Holland Publishing Co., Amsterdam-New York, 1981. An introduction. MR 615557
H. Flaschka, On the inverse problem for Hill’s operator, Arch. Rational Mech. Anal. 59 (1975), no. 4, 293–309. MR 387711, DOI 10.1007/BF00250422
Hermann Flaschka and David W. McLaughlin, Some comments on Bäcklund transformations, canonical transformations and the inverse scattering method, Bäcklund transformations, the inverse scattering method, solitons, and their applications (Workshop Contact Transformations, Vanderbilt Univ., Nashville, Tenn., 1974) Lecture Notes in Math., Vol. 515, Springer, Berlin, 1976, pp. 253–295. MR 0609534
Hermann Flaschka and Alan C. Newell, Monodromy- and spectrum-preserving deformations. I, Comm. Math. Phys. 76 (1980), no. 1, 65–116. MR 588248

Riemann surface of quasimomentum and scattering theory for the perturbed Hill operator

11

An inverse scattering problem for a perturbed Hill operator

18

Some spectral identities for the one-dimensional Hill operator

37

The direct and inverse scattering problems for the one-dimensional perturbed Hill operator

58

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 $y_{t}+y_{xxx}-6y^{2}y_{x}+6\lambda y_{x}=0$, Phys. Rev. B (3) 26 (1982), no. 8, 4035–4038. MR 675935
Clifford S. Gardner, John M. Greene, Martin D. Kruskal, and Robert M. Miura, Korteweg-deVries equation and generalization. VI. Methods for exact solution, Comm. Pure Appl. Math. 27 (1974), 97–133. MR 336122, DOI 10.1002/cpa.3160270108
F. Gesztesy, Scattering theory for one-dimensional systems with nontrivial spatial asymptotics, Schrödinger operators, Aarhus 1985, Lecture Notes in Math., vol. 1218, Springer, Berlin, 1986, pp. 93–122. MR 869597, DOI 10.1007/BFb0073045

Constructing solutions of the

equation

J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation, SIAM J. Math. Anal. 20 (1989), no. 6, 1388–1425. MR 1019307, DOI 10.1137/0520091
Wallace Goldberg and Harry Hochstadt, On a Hill’s equation with selected gaps in its spectrum, J. Differential Equations 34 (1979), no. 2, 167–178. MR 550038, DOI 10.1016/0022-0396(79)90002-0

On the mixed soliton-rational solutions of the

equation

H. Grosse, Solitons of the modified KdV equation, Lett. Math. Phys. 8 (1984), no. 4, 313–319. MR 759630, DOI 10.1007/BF00400502
H. Grosse, New solitons connected to the Dirac equation, Phys. Rep. 134 (1986), no. 5-6, 297–304. MR 832137, DOI 10.1016/0370-1573(86)90053-0
Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038

Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons

18

Exact solution of the modified Korteweg-de Vries equation for multiple collisions of solitons

33

Harry Hochstadt, On the determination of a Hill’s equation from its spectrum, Arch. Rational Mech. Anal. 19 (1965), 353–362. MR 181792, DOI 10.1007/BF00253484
A. R. Its, Inversion of hyperelliptic integrals, and integration of nonlinear differential equations, Vestnik Leningrad. Univ. 7 Mat. Meh. Astronom. vyp. 2 (1976), 39–46, 162 (Russian, with English summary). MR 0609747
A. R. Its and V. B. Matveev, Schrödinger operators with the finite-band spectrum and the $N$-soliton solutions of the Korteweg-de Vries equation, Teoret. Mat. Fiz. 23 (1975), no. 1, 51–68 (Russian, with English summary). MR 479120
Katsunori Iwasaki, Inverse problem for Sturm-Liouville and Hill equations, Ann. Mat. Pura Appl. (4) 149 (1987), 185–206. MR 932784, DOI 10.1007/BF01773933

Zur Theorie der Variationsrechnung und der Differentialgleichungen

17

Arthur Jaffe, Andrzej Lesniewski, and Maciej Lewenstein, Ground state structure in supersymmetric quantum mechanics, Ann. Physics 178 (1987), no. 2, 313–329. MR 914181, DOI 10.1016/0003-4916(87)90018-2

Connection between Zakharov-Shabat and Schrödinger-type inverse scattering transform

20

Yoshinori Kametaka, Korteweg-de Vries equation. III. Global existence of asymptotically periodic solutions, Proc. Japan Acad. 45 (1969), 656–660. MR 262706
Yoshinori Kametaka, Korteweg-de Vries equation. IV. Simplest generalization, Proc. Japan Acad. 45 (1969), 661–665. MR 262707

On the Korteweg-de Vries equation

28

Tosio Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, Academic Press, New York, 1983, pp. 93–128. MR 759907
Yusuke Kato, Fredholm determinants and the Cauchy problem of a class of nonlinear evolution equations, Progr. Theoret. Phys. 78 (1987), no. 2, 198–213. MR 919541, DOI 10.1143/PTP.78.198

Reflectionless transmission through dielectrics and scattering potentials

27

Martin Klaus, Low-energy behaviour of the scattering matrix for the Schrödinger equation on the line, Inverse Problems 4 (1988), no. 2, 505–512. MR 954906

Potentials with zero coefficient of reflection on a background of finite-zone potentials

9

S. N. Kruzhkov and A. V. Faminskiĭ, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, Mat. Sb. (N.S.) 120(162) (1983), no. 3, 396–425 (Russian). MR 691986
B. A. Kupershmidt and George Wilson, Modifying Lax equations and the second Hamiltonian structure, Invent. Math. 62 (1981), no. 3, 403–436. MR 604836, DOI 10.1007/BF01394252
E. A. Kuznetsov and A. V. Mikhaĭlov, Stability of stationary waves in nonlinear weakly dispersive media, Ž. Èksper. Teoret. Fiz. 67 (1974), no. 5, 1717–1727 (Russian, with English summary); English transl., Soviet Physics JETP 40 (1974), no. 5, 855–859. MR 0387847
Peter D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467–490. MR 235310, DOI 10.1002/cpa.3160210503
Peter D. Lax, Periodic solutions of the KdV equations, Nonlinear wave motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972) Lectures in Appl. Math., Vol. 15, Amer. Math. Soc., Providence, R.I., 1974, pp. 85–96. MR 0344645
Wilhelm Magnus and Stanley Winkler, Hill’s equation, Dover Publications, Inc., New York, 1979. Corrected reprint of the 1966 edition. MR 559928
Vladimir A. Marchenko, Sturm-Liouville operators and applications, Operator Theory: Advances and Applications, vol. 22, Birkhäuser Verlag, Basel, 1986. Translated from the Russian by A. Iacob. MR 897106, DOI 10.1007/978-3-0348-5485-6
H. P. McKean, Theta functions, solitons, and singular curves, Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977) Lecture Notes in Pure and Appl. Math., vol. 48, Dekker, New York, 1979, pp. 237–254. MR 535596
H. P. McKean, Geometry of KdV. I. Addition and the unimodular spectral classes, Rev. Mat. Iberoamericana 2 (1986), no. 3, 253–261. MR 908052
H. P. McKean, Geometry of KdV. II. Three examples, J. Statist. Phys. 46 (1987), no. 5-6, 1115–1143. MR 893135, DOI 10.1007/BF01011159
H. P. McKean and P. van Moerbeke, The spectrum of Hill’s equation, Invent. Math. 30 (1975), no. 3, 217–274. MR 397076, DOI 10.1007/BF01425567
H. P. McKean and E. Trubowitz, Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976), no. 2, 143–226. MR 427731, DOI 10.1002/cpa.3160290203
H. P. McKean and E. Trubowitz, Hill’s surfaces and their theta functions, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1042–1085. MR 508448, DOI 10.1090/S0002-9904-1978-14542-X
A. Menikoff, The existence of unbounded solutions of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 25 (1972), 407–432. MR 303129, DOI 10.1002/cpa.3160250404
Robert M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Mathematical Phys. 9 (1968), 1202–1204. MR 252825, DOI 10.1063/1.1664700
H. E. Moses, A generalization of the Gel′fand-Levitan equation for the one-dimensional Schrödinger equation, J. Mathematical Phys. 18 (1977), no. 12, 2243–2250. MR 462290, DOI 10.1063/1.523235
Minoru Murata, Structure of positive solutions to $(-\Delta +V)u=0$ in $\textbf {R}^n$, Duke Math. J. 53 (1986), no. 4, 869–943. MR 874676, DOI 10.1215/S0012-7094-86-05347-0
Michael Martin Nieto, Relationship between supersymmetry and the inverse method in quantum mechanics, Phys. Lett. B 145 (1984), no. 3-4, 208–210. MR 762223, DOI 10.1016/0370-2693(84)90339-3
S. P. Novikov, A periodic problem for the Korteweg-de Vries equation. I, Funkcional. Anal. i Priložen. 8 (1974), no. 3, 54–66 (Russian). MR 0382878
S. Novikov, S. V. Manakov, L. P. Pitaevskiĭ, and V. E. Zakharov, Theory of solitons, Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York, 1984. The inverse scattering method; Translated from the Russian. MR 779467
Mayumi Ohmiya, On the generalized soliton solutions of the modified Korteweg-de-Vries equation, Osaka Math. J. 11 (1974), 61–71. MR 352742
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
S. N. M. Ruijsenaars and P. J. M. Bongaarts, Scattering theory for one-dimensional step potentials, Ann. Inst. H. Poincaré Sect. A (N.S.) 26 (1977), no. 1, 1–17. MR 443712
S. N. M. Ruijsenaars and H. Schneider, A new class of integrable systems and its relation to solitons, Ann. Physics 170 (1986), no. 2, 370–405. MR 851627, DOI 10.1016/0003-4916(86)90097-7
D. H. Sattinger and V. D. Zurkowski, Gauge theory of Bäcklund transformations. II, Phys. D 26 (1987), no. 1-3, 225–250. MR 892446, DOI 10.1016/0167-2789(87)90227-2
U.-W. Schmincke, On Schrödinger’s factorization method for Sturm-Liouville operators, Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), no. 1-2, 67–84. MR 529570, DOI 10.1017/S0308210500010143
Barry Simon, Large time behavior of the $L^{p}$ norm of Schrödinger semigroups, J. Functional Analysis 40 (1981), no. 1, 66–83. MR 607592, DOI 10.1016/0022-1236(81)90073-2
Anders Sjöberg, On the Korteweg-de Vries equation: existence and uniqueness, J. Math. Anal. Appl. 29 (1970), 569–579. MR 410135, DOI 10.1016/0022-247X(70)90068-5
V. V. Sokolov and A. B. Šabat, $(L,\,A)$-pairs and Riccati type substitution, Funktsional. Anal. i Prilozhen. 14 (1980), no. 2, 79–80 (Russian). MR 575223
O. Steinmann, Äquivalente periodische Potentiale, Helv. Phys. Acta 30 (1957), 515–520 (German). MR 92582
C. V. Sukumar, Supersymmetric quantum mechanics of one-dimensional systems, J. Phys. A 18 (1985), no. 15, 2917–2936. MR 814636
Shunichi Tanaka, Modified Korteweg-de Vries equation and scattering theory, Proc. Japan Acad. 48 (1972), 466–469. MR 336127
Shunichi Tanaka, On the $N$-tuple wave solutions of the Korteweg-de Vries equation, Publ. Res. Inst. Math. Sci. 8 (1972/73), 419–427. MR 0328386, DOI 10.2977/prims/1195192955
Shunichi Tanaka, Some remarks on the modified Korteweg-de Vries equations, Publ. Res. Inst. Math. Sci. 8 (1972/73), 429–437. MR 0326198, DOI 10.2977/prims/1195192956
Shunichi Tanaka, Non-linear Schrödinger equation and modified Korteweg-de Vries equation; construction of solutions in terms of scattering data, Publ. Res. Inst. Math. Sci. 10 (1974/75), 329–357. MR 0499846, DOI 10.2977/prims/1195191998
Bernd Thaller, Normal forms of an abstract Dirac operator and applications to scattering theory, J. Math. Phys. 29 (1988), no. 1, 249–257. MR 921790, DOI 10.1063/1.528182
E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), no. 3, 321–337. MR 430403, DOI 10.1002/cpa.3160300305
Masayoshi Tsutsumi, On global solutions of the generalized Kortweg-de Vries equation, Publ. Res. Inst. Math. Sci. 7 (1971/72), 329–344. MR 0312096, DOI 10.2977/prims/1195193545
Stephanos Venakides, The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data, Trans. Amer. Math. Soc. 301 (1987), no. 1, 189–226. MR 879569, DOI 10.1090/S0002-9947-1987-0879569-7

The exact solution of the modified Korteweg-de Vries equation

32

Miki Wadati, The modified Korteweg-de Vries equation, J. Phys. Soc. Japan 34 (1973), 1289–1296. MR 371251, DOI 10.1143/JPSJ.34.1289

The exact

soliton solution of the Korteweg-de Vries equation

32

Hugo D. Wahlquist, Bäcklund transformation of potentials of the Korteweg-de Vries equation and the interaction of solitons with cnoidal waves, Bäcklund transformations, the inverse scattering method, solitons, and their applications (Workshop Contact Transformations, Vanderbilt Univ., Nashville, Tenn., 1974) Lecture Notes in Math., Vol. 515, Springer, Berlin, 1976, pp. 162–183. MR 0610088
H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations, J. Mathematical Phys. 16 (1975), 1–7. MR 358111, DOI 10.1063/1.522396
G. Wilson, Infinite-dimensional Lie groups and algebraic geometry in soliton theory, Philos. Trans. Roy. Soc. London Ser. A 315 (1985), no. 1533, 393–404. New developments in the theory and application of solitons. MR 836741, DOI 10.1098/rsta.1985.0047
Yu Kun Zheng and W. L. Chan, Gauge transformation and the higher order Korteweg-de Vries equation, J. Math. Phys. 29 (1988), no. 2, 308–314. MR 927012, DOI 10.1063/1.528068