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An Inverse Spectral Problem Related to the Geng–Xue Two-Component Peakon Equation
About this Title
Hans Lundmark, Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden and Jacek Szmigielski, Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, S7N 5E6, Canada
Publication: Memoirs of the American Mathematical Society
Publication Year:
2016; Volume 244, Number 1155
ISBNs: 978-1-4704-2026-0 (print); 978-1-4704-3510-3 (online)
DOI: https://doi.org/10.1090/memo/1155
Published electronically: June 17, 2016
MSC: Primary 34A55
Table of Contents
Chapters
- Acknowledgements
- 1. Introduction
- 2. Forward Spectral Problem
- 3. The Discrete Case
- 4. The Inverse Spectral Problem
- 5. Concluding Remarks
- A. Cauchy Biorthogonal Polynomials
- B. The Forward Spectral Problem on the Real Line
- C. Guide to Notation
Abstract
We solve a spectral and an inverse spectral problem arising in the computation of peakon solutions to the two-component PDE derived by Geng and Xue as a generalization of the Novikov and Degasperis–Procesi equations. Like the spectral problems for those equations, this one is of a ‘discrete cubic string’ type – a nonselfadjoint generalization of a classical inhomogeneous string – but presents some interesting novel features: there are two Lax pairs, both of which contribute to the correct complete spectral data, and the solution to the inverse problem can be expressed using quantities related to Cauchy biorthogonal polynomials with two different spectral measures. The latter extends the range of previous applications of Cauchy biorthogonal polynomials to peakons, which featured either two identical, or two closely related, measures. The method used to solve the spectral problem hinges on the hidden presence of oscillatory kernels of Gantmacher–Krein type implying that the spectrum of the boundary value problem is positive and simple. The inverse spectral problem is solved by a method which generalizes, to a nonselfadjoint case, M. G. Krein’s solution of the inverse problem for the Stieltjes string.- R. Beals, D. H. Sattinger, and J. Szmigielski, Multi-peakons and a theorem of Stieltjes, Inverse Problems 15 (1999), no. 1, L1–L4. MR 1675325, DOI 10.1088/0266-5611/15/1/001
- Richard Beals, David H. Sattinger, and Jacek Szmigielski, Multipeakons and the classical moment problem, Adv. Math. 154 (2000), no. 2, 229–257. MR 1784675, DOI 10.1006/aima.1999.1883
- Richard Beals, David H. Sattinger, and Jacek Szmigielski, The string density problem and the Camassa-Holm equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 365 (2007), no. 1858, 2299–2312. MR 2329150, DOI 10.1098/rsta.2007.2010
- M. Bertola, M. Gekhtman, and J. Szmigielski, The Cauchy two-matrix model, Comm. Math. Phys. 287 (2009), no. 3, 983–1014. MR 2486670, DOI 10.1007/s00220-009-0739-y
- M. Bertola, M. Gekhtman, and J. Szmigielski, Cubic string boundary value problems and Cauchy biorthogonal polynomials, J. Phys. A 42 (2009), no. 45, 454006, 13. MR 2556648, DOI 10.1088/1751-8113/42/45/454006
- M. Bertola, M. Gekhtman, and J. Szmigielski, Cauchy biorthogonal polynomials, J. Approx. Theory 162 (2010), no. 4, 832–867. MR 2606648, DOI 10.1016/j.jat.2009.09.008
- M. Bertola, M. Gekhtman, and J. Szmigielski, Cauchy-Laguerre two-matrix model and the Meijer-G random point field, Comm. Math. Phys. 326 (2014), no. 1, 111–144. MR 3162486, DOI 10.1007/s00220-013-1833-8
- Roberto Camassa and Darryl D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), no. 11, 1661–1664. MR 1234453, DOI 10.1103/PhysRevLett.71.1661
- A. Degasperis, D. D. Kholm, and A. N. I. Khon, A new integrable equation with peakon solutions, Teoret. Mat. Fiz. 133 (2002), no. 2, 170–183 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 133 (2002), no. 2, 1463–1474. MR 2001531, DOI 10.1023/A:1021186408422
- A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and perturbation theory (Rome, 1998) World Sci. Publ., River Edge, NJ, 1999, pp. 23–37. MR 1844104
- H. Dym and H. P. McKean, Gaussian processes, function theory, and the inverse spectral problem, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Probability and Mathematical Statistics, Vol. 31. MR 0448523
- F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Chelsea Publishing Co., New York, 1959. Translated by K. A. Hirsch. MR 0107649
- F. P. Gantmacher and M. G. Krein, Oscillation matrices and kernels and small vibrations of mechanical systems, Revised edition, AMS Chelsea Publishing, Providence, RI, 2002. Translation based on the 1941 Russian original; Edited and with a preface by Alex Eremenko. MR 1908601
- Xianguo Geng and Bo Xue, An extension of integrable peakon equations with cubic nonlinearity, Nonlinearity 22 (2009), no. 8, 1847–1856. MR 2525813, DOI 10.1088/0951-7715/22/8/004
- Daniel Gomez, Hans Lundmark, and Jacek Szmigielski, The Canada Day theorem, Electron. J. Combin. 20 (2013), no. 1, Paper 20, 16. MR 3035030, DOI 10.37236/2449
- Katelyn Grayshan, Peakon solutions of the Novikov equation and properties of the data-to-solution map, J. Math. Anal. Appl. 397 (2013), no. 2, 515–521. MR 2979590, DOI 10.1016/j.jmaa.2012.08.006
- A. Alexandrou Himonas and Curtis Holliman, The Cauchy problem for the Novikov equation, Nonlinearity 25 (2012), no. 2, 449–479. MR 2876876, DOI 10.1088/0951-7715/25/2/449
- A. Alexandrou Himonas and John Holmes, Hölder continuity of the solution map for the Novikov equation, J. Math. Phys. 54 (2013), no. 6, 061501, 11. MR 3112520, DOI 10.1063/1.4807729
- Andrew N. W. Hone, Hans Lundmark, and Jacek Szmigielski, Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa-Holm type equation, Dyn. Partial Differ. Equ. 6 (2009), no. 3, 253–289. MR 2569508, DOI 10.4310/DPDE.2009.v6.n3.a3
- Andrew N. W. Hone and Jing Ping Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A 41 (2008), no. 37, 372002, 10. MR 2430566, DOI 10.1088/1751-8113/41/37/372002
- Zaihong Jiang and Lidiao Ni, Blow-up phenomenon for the integrable Novikov equation, J. Math. Anal. Appl. 385 (2012), no. 1, 551–558. MR 2834279, DOI 10.1016/j.jmaa.2011.06.067
- Jennifer Kohlenberg, Hans Lundmark, and Jacek Szmigielski, The inverse spectral problem for the discrete cubic string, Inverse Problems 23 (2007), no. 1, 99–121. MR 2302964, DOI 10.1088/0266-5611/23/1/005
- Shaoyong Lai, Global weak solutions to the Novikov equation, J. Funct. Anal. 265 (2013), no. 4, 520–544. MR 3062535, DOI 10.1016/j.jfa.2013.05.022
- Shaoyong Lai, Nan Li, and Yonghong Wu, The existence of global strong and weak solutions for the Novikov equation, J. Math. Anal. Appl. 399 (2013), no. 2, 682–691. MR 2996746, DOI 10.1016/j.jmaa.2012.10.048
- Jonatan Lenells and Marcus Wunsch, On the weakly dissipative Camassa-Holm, Degasperis-Procesi, and Novikov equations, J. Differential Equations 255 (2013), no. 3, 441–448. MR 3053472, DOI 10.1016/j.jde.2013.04.015
- Nianhua Li and Q. P. Liu, On bi-Hamiltonian structure of two-component Novikov equation, Phys. Lett. A 377 (2013), no. 3-4, 257–281. MR 3006205, DOI 10.1016/j.physleta.2012.11.023
- Xiaochuan Liu, Yue Liu, and Changzheng Qu, Stability of peakons for the Novikov equation, J. Math. Pures Appl. (9) 101 (2014), no. 2, 172–187. MR 3158700, DOI 10.1016/j.matpur.2013.05.007
- Hans Lundmark and Jacek Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation, Inverse Problems 19 (2003), no. 6, 1241–1245. MR 2036528, DOI 10.1088/0266-5611/19/6/001
- Hans Lundmark and Jacek Szmigielski, Degasperis-Procesi peakons and the discrete cubic string, IMRP Int. Math. Res. Pap. 2 (2005), 53–116. MR 2150256
- Hans Lundmark and Jacek Szmigielski, Continuous and discontinuous piecewise linear solutions of the linearly forced inviscid Burgers equation, J. Nonlinear Math. Phys. 15 (2008), no. suppl. 3, 264–276. MR 2452438, DOI 10.2991/jnmp.2008.15.s3.26
- Yoshimasa Matsuno, Smooth multisoliton solutions and their peakon limit of Novikov’s Camassa-Holm type equation with cubic nonlinearity, J. Phys. A 46 (2013), no. 36, 365203, 27. MR 3100605, DOI 10.1088/1751-8113/46/36/365203
- Yongsheng Mi and Chunlai Mu, On the Cauchy problem for the modified Novikov equation with peakon solutions, J. Differential Equations 254 (2013), no. 3, 961–982. MR 2997360, DOI 10.1016/j.jde.2012.09.016
- Keivan Mohajer and Jacek Szmigielski, On an inverse problem associated with an integrable equation of Camassa-Holm type: explicit formulas on the real axis, Inverse Problems 28 (2012), no. 1, 015002, 13. MR 2864501, DOI 10.1088/0266-5611/28/1/015002
- J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Advances in Math. 16 (1975), 197–220. MR 375869, DOI 10.1016/0001-8708(75)90151-6
- Lidiao Ni and Yong Zhou, Well-posedness and persistence properties for the Novikov equation, J. Differential Equations 250 (2011), no. 7, 3002–3021. MR 2771253, DOI 10.1016/j.jde.2011.01.030
- Vladimir Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A 42 (2009), no. 34, 342002, 14. MR 2530232, DOI 10.1088/1751-8113/42/34/342002
- T.-J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. 8 (1894), no. 4, J1–J122 (French). MR 1508159
- F. Tığlay, The periodic Cauchy problem for Novikov’s equation, Int. Math. Res. Not. IMRN 20 (2011), 4633–4648. MR 2844933, DOI 10.1093/imrn/rnq267
- Xinglong Wu and Zhaoyang Yin, Global weak solutions for the Novikov equation, J. Phys. A 44 (2011), no. 5, 055202, 17. MR 2763454, DOI 10.1088/1751-8113/44/5/055202
- Xinglong Wu and Zhaoyang Yin, A note on the Cauchy problem of the Novikov equation, Appl. Anal. 92 (2013), no. 6, 1116–1137. MR 3197924, DOI 10.1080/00036811.2011.649735
- Wei Yan, Yongsheng Li, and Yimin Zhang, The Cauchy problem for the integrable Novikov equation, J. Differential Equations 253 (2012), no. 1, 298–318. MR 2917410, DOI 10.1016/j.jde.2012.03.015
- Wei Yan, Yongsheng Li, and Yimin Zhang, The Cauchy problem for the Novikov equation, NoDEA Nonlinear Differential Equations Appl. 20 (2013), no. 3, 1157–1169. MR 3057170, DOI 10.1007/s00030-012-0202-1