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Inverse spectral problem for the periodic Camassa-Holm equation. (English) Zbl 1069.35093

Summary: We consider the direct/inverse spectral problem for the periodic Camassa-Holm equation. In fact, we survey the direct/inverse spectral problem for the periodic weighted operator \(Ly=m^{-1}(-y''+ \frac 14 y)\) acting in the space \(L^2(\mathbb{R},m(x)dx)\), where \(m=u_{xx}-u>0\) is a 1-periodic positive function and \(u\) is the solution of the Camassa-Holm equation \[ u_t-u_{xxt}+3u u_x=2u_xu_{xx}+uu_{xxx}. \] For the operator \(L\) we describe the complete solution of the inverse spectral problem: i) uniqueness (we prove that the spectral data uniquely determines the potential), ii) characterization (we give conditions for some data to be the spectral data of some potential), iii) reconstruction (we give an algorithm for recovering the potential from the spectral data), iv) a priori estimates (we obtain two-sided a priori estimates of \(u,m\) in terms of gap lengths).

MSC:

35R30 Inverse problems for PDEs
35Q35 PDEs in connection with fluid mechanics
35P05 General topics in linear spectral theory for PDEs
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[1] Badanin A, J. Funct. Anal. 203 pp 494– (2003) · Zbl 1044.34047 · doi:10.1016/S0022-1236(03)00058-2
[2] Camassa R, Phys. Rev. Lett. 71 pp 1661– (1993) · Zbl 0972.35521 · doi:10.1103/PhysRevLett.71.1661
[3] Constantin A, J. Funct. Anal. 155 pp 352– (1998) · Zbl 0907.35009 · doi:10.1006/jfan.1997.3231
[4] Constantin A, J. Math. Anal. Appl. 210 pp 215– (1997) · Zbl 0881.35102 · doi:10.1006/jmaa.1997.5393
[5] Constantin A, Ann. Scuola Norm. Sup. Pisa 24 pp 767– (1997)
[6] Constantin A, Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. 457 pp 953– (2001) · Zbl 0999.35065 · doi:10.1098/rspa.2000.0701
[7] Constantin A, Comm. Pure Appl. Math. 52 pp 949– (1999) · Zbl 0940.35177 · doi:10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
[8] Constantin A, Acta Mathematica 181 pp 229– (1998) · Zbl 0923.76025 · doi:10.1007/BF02392586
[9] Constantin A, Comm. Pure Appl. Math. 51 pp 475– (1998) · Zbl 0934.35153 · doi:10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5
[10] Constantin A, Comm. Math. Phys. 211 pp 45– (2000) · Zbl 1002.35101 · doi:10.1007/s002200050801
[11] Constantin A, J. Phys. A 35 pp R51– (2002) · Zbl 1039.37068 · doi:10.1088/0305-4470/35/32/201
[12] Constantin A, Comment. Math. Helv. 78 pp 787– (2003) · Zbl 1037.37032 · doi:10.1007/s00014-003-0785-6
[13] Dai , H H . 1998 .Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech.Vol. 127 , 193 – 207 . · Zbl 0910.73036
[14] Dubrovin B A, Funct. Anal. Appl. 9 pp 215– (1975) · Zbl 0358.35022 · doi:10.1007/BF01075598
[15] Dullin H R, Phys. Rev. Lett. 87 pp 194501– (2001) · doi:10.1103/PhysRevLett.87.194501
[16] Garnett J, Comment. Math. Helv. 59 pp 258– (1984) · Zbl 0554.34013 · doi:10.1007/BF02566350
[17] Johnson R S, J. Fluid Mech. 455 pp 63– (2002) · Zbl 1037.76006 · doi:10.1017/S0022112001007224
[18] Klein M, Inverse Problems 16 pp 1839– (2000) · Zbl 0990.47032 · doi:10.1088/0266-5611/16/6/315
[19] Kargaev P, Invent. Math. 129 pp 567– (1997) · Zbl 0878.34011 · doi:10.1007/s002220050173
[20] Kargaev P, Comm. Math. Phys. 169 pp 597– (1995) · Zbl 0828.34076 · doi:10.1007/BF02099314
[21] Korotyaev E, J. Funct. Anal. 170 pp 188– (2000) · Zbl 0970.47021 · doi:10.1006/jfan.1999.3479
[22] Korotyaev E, J. Differential Equations 189 pp 461– (2003) · Zbl 1029.34012 · doi:10.1016/S0022-0396(02)00154-7
[23] Korotyaev E, Internat. Math. Res. Notices. 3 pp 113– (1997) · Zbl 0873.34011 · doi:10.1155/S1073792897000081
[24] Korotyaev E, Math. Z. 231 pp 345– (1999) · Zbl 0929.34016 · doi:10.1007/PL00004733
[25] Korotyaev E, Internat. Math. Res. Notices 37 pp 2019– (2003) · Zbl 1104.34059 · doi:10.1155/S1073792803209107
[26] Korotyaev E, Internat. Math. Res. Notices 38 pp 2007– (2002) · Zbl 1023.34080 · doi:10.1155/S1073792802205176
[27] Korotyaev E, J. Differential Equations 162 pp 1– (2000) · Zbl 0954.34073 · doi:10.1006/jdeq.1999.3684
[28] Korotyaev E, Asymptot. Anal. 15 pp 1– (1997)
[29] Korotyaev E, Comm. Math. Phys. 183 pp 383– (1997) · Zbl 0870.34080 · doi:10.1007/BF02506412
[30] Korotyaev E, Internat. Math. Res. Notices 10 pp 493– (1996) · Zbl 0868.30008 · doi:10.1155/S1073792896000335
[31] Krein M.G., Prikl. Mat. Meh. 21 pp 320– (1957)
[32] Levitan B M, Inverse Sturm-Liouville problems (1987)
[33] Lyapunov A, C. R. Acad. Sci. Paris 18 pp 1085– (1899)
[34] Marchenko V A, Math. USSR Sb. 26 pp 493– (1975) · Zbl 0343.34016 · doi:10.1070/SM1975v026n04ABEH002493
[35] Marchenko V A, Selecta Math. Sov. 6 pp 101– (1987)
[36] McKean H P, Asian J. Math. 2 pp 867– (1998) · Zbl 0959.35140 · doi:10.4310/AJM.1998.v2.n4.a10
[37] Misiolek G, J. Geom. Phys. 24 pp 203– (1998) · Zbl 0901.58022 · doi:10.1016/S0393-0440(97)00010-7
[38] Trubowitz E, Comm. Pure Appl. Math. 30 pp 321– (1977) · Zbl 0403.34022 · doi:10.1002/cpa.3160300305
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