Opuscula Math. 37, no. 1 (2017), 167-187
http://dx.doi.org/10.7494/OpMath.2017.37.1.167

Opuscula Mathematica

# The inverse scattering transform in the form of a Riemann-Hilbert problem for the Dullin-Gottwald-Holm equation

Dmitry Shepelsky
Lech Zielinski

Abstract. The Cauchy problem for the Dullin-Gottwald-Holm (DGH) equation $u_t-\alpha^2 u_{xxt}+2\omega u_x +3uu_x+\gamma u_{xxx}=\alpha^2 (2u_x u_{xx} + uu_{xxx})$ with zero boundary conditions (as $$|x|\to\infty$$) is treated by the Riemann-Hilbert approach to the inverse scattering transform method. The approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used for further studying the properties of the solution, particularly, in studying its long-time behavior. Using the proposed formalism, smooth solitons as well as non-smooth cuspon solutions are presented.

Keywords: Dullin-Gottwald-Holm equation, Camassa-Holm equation, inverse scattering transform, Riemann-Hilbert problem.

Mathematics Subject Classification: 35Q53, 37K15, 35Q15, 35B40, 35Q51, 37K40.

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1. X. Ai, G. Gui, On the inverse scattering problem and the low regularity solutions for the Dullin-Gottwald-Holm equation, Nonlinear Analysis: Real World Applications 11 (2010), 888-894.
2. R. Beals, R.R. Coifman, Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math. 37 (1984), 39-90.
3. R. Beals, P. Deift, C. Tomei, Direct and Inverse Scattering on the Line, AMS, Providence, Rhode Island, 1988.
4. A. Boutet de Monvel, A. Its, D. Shepelsky, Painlevé-type asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal. 42 (2010), 1854-1873.
5. A. Boutet de Monvel, A. Kostenko, D. Shepelsky, G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal. 41 (2009), 1559-1588.
6. A. Boutet de Monvel, D. Shepelsky, Riemann-Hilbert approach for the Camassa-Holm equation on the line, C. R. Math. Acad. Sci. Paris 343 (2006) 10, 627-632.
7. A. Boutet de Monvel, D. Shepelsky, Long-time asymptotics of the Camassa-Holm equation on the line, [in:] Integrable systems and random matrices, Contemp. Math. 458, Amer. Math. Soc., Providence, RI, 2008, 99-116.
8. A. Boutet de Monvel, D. Shepelsky, Riemann-Hilbert problem in the inverse scattering for the Camassa-Holm equation on the line, [in:] Probability, geometry and integrable systems, Math. Sci. Res. Inst. Publ. 55, Cambridge Univ. Press, Cambridge, 2008, 53-75.
9. A. Boutet de Monvel, D. Shepelsky, The Camassa-Holm equation on the half-line: a Riemann-Hilbert approach, J. Geom. Anal. 18 (2008), 285-323.
10. A. Boutet de Monvel, D. Shepelsky, Long time asymptotics of the Camassa-Holm equation in the half-line, Ann. Inst. Fourier (Grenoble) 59 (2009) 7, 3015-3056.
11. A. Boutet de Monvel, D. Shepelsky, L. Zielinski, The short-wave model for the Camassa-Holm equation: a Riemann-Hilbert approach, Inverse Problems 27 (2011), 105006.
12. R. Camassa, D.D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 11, 1661-1664.
13. O. Christov, S. Hakkaev, On the inverse scattering approach and action-angle variables for the Dullin-Gottwald-Holm equation, Physica D 238 (2009), 9-19.
14. A. Constantin, On the scattering problem for the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), 953-970.
15. A. Constantin, J. Lenells, On the inverse scattering approach to the Camassa-Holm equation, J. Nonlinear Math. Phys. 10 (2003) 3, 252-255.
16. A. Constantin, V.S. Gerdjikov, R.I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems 22 (2006) 6, 2197-2207.
17. R. Dullin, G. Gottwald, D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett. 87 (2001) 9, 4501-4504.
18. A. Fokas, B. Fuchssteiner, Symplectic structures, their Backlund transform and hereditary symmetries, Physica D 4 (1981), 47-66.
19. J. Lenells, The scattering approach for the Camassa-Holm equation, J. Nonlinear Math. Phys. 9 (2002) 4, 389-393.
20. L. Tian, G. Gui, Y. Liu, On the well-posedness problem and the scattering problem for the Dullin-Gottwald-Holm equation, Comm. Math. Phys. 257 (2005), 667-701.
21. Z. Yin, Well-posedness, global existence and blowup phenomena for an integrable shallow water equation, Discrete Contin. Dynam. Systems 10 (2004), 393-411.
• Dmitry Shepelsky
• Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkiv, Ukraine
• V. N. Karazin Kharkiv National University, 4 Svobody Square, 61022 Kharkiv, Ukraine
• Lech Zielinski
• LMPA, Université du Littoral Côte d'Opale, 50 rue F. Buisson, CS 80699, 62228 Calais, France
• Communicated by Alexander Gomilko.
• Accepted: 2016-09-24.
• Published online: 2016-12-14.