A singular limit problem  for the Ibragimov-Shabat equation

Giuseppe Maria Coclite; Lorenzo di  Ruvo

doi:10.3934/dcdss.2016020

Article Contents

2016, Volume 9, Issue 3: 661-673. Doi: 10.3934/dcdss.2016020

This issue Previous Article Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations Next Article $1$-dimensional Harnack estimates

A singular limit problem for the Ibragimov-Shabat equation

Giuseppe Maria Coclite^1, and
Lorenzo di Ruvo^2,

1.
Department of Mathematics, University of Bari, Via E. Orabona 4, I--70125 Bari
2.
Department of Science and Methods for Engineering, University of Modena and Reggio Emilia, via G. Amendola 2, 42122 Reggio Emilia

Received: November 2014

Revised: September 2015

Published: June 2016

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Abstract

We consider the Ibragimov-Shabat equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.

Keywords:

Mathematics Subject Classification: Primary: 35G25, 35L65; Secondary: 35L05.

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References

[1]	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.doi: 10.1002/sapm1974534249.
[2]	R. Beals, M. Rabelo and K. Tenenblat, Bäcklund transformations and inverse scattering solutions for some pseudospherical surface equations, Stud. Appl. Math., 81 (1989), 125-151.doi: 10.1002/sapm1989812125.
[3]	G. M. Coclite and L. di Ruvo, Well-posedness results for the short pulse equation, Z. Angew. Math. Phys., 66 (2015), 1529-1557.doi: 10.1007/s00033-014-0478-6.
[4]	G. M. Coclite and L. di Ruvo, On the Wellposedness of the exp-Rabelo equation, Ann. Mat. Pura Appl., to appear. doi: 10.1007/s10231-015-0497-8.
[5]	G. M. Coclite, L. di Ruvo, J. Ernest and S. Mishra, Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes, Netw. Heterog. Media., 8 (2013), 969-984.doi: 10.3934/nhm.2013.8.969.
[6]	G. M. Coclite and K. H. Karlsen, A singular limit problem for conservation laws related to the Camassa-Holm shallow water equation, Comm. Partial Differential Equations, 31 (2006), 1253-1272.doi: 10.1080/03605300600781600.
[7]	R. K. Dodd and R. K. Bullough, Bäcklund transformations for the A.K.N.S. inverse method, Phys. Lett. A, 62 (1977), 70-74.doi: 10.1016/0375-9601(77)90952-5.
[8]	L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Dover, New York, 1960.
[9]	E. Goursat, Le Problème de Bäcklund, Mémorial des Sciences Mathématiques, Fasc. VI, Gauthier-Villars, Paris, 1925.
[10]	A. H. Khater, D. K. Callebaut, A. A. Abdalla and S. M. Sayed, Exact solutions for self-dual Yang-Mills equations, Chaos Solitons Fractals, 10 (1999), 1309-1320.doi: 10.1016/S0960-0779(98)00155-6.
[11]	A. H. Khater, D. K. Callebaut and R. S. Ibrahim, Bäcklund transformations and Painlevé analysis: Exact solutions for the unstable nonlinear Schrödinger equation modelling electron-beam plasma, Phys. Plasmas, 5 (1998), 395-400.doi: 10.1063/1.872723.
[12]	A. H. Khater, D. K. Callebaut and S. M. Sayed, Conservation laws for some nonlinear evolution equations which describe pseudo-spherical surfaces, J. Geom. Phys., 51 (2004), 332-352.doi: 10.1016/j.geomphys.2003.11.009.
[13]	A. H. Khater, D. K. Callebaut and S. M. Sayed, Bäcklund transformations for some nonlinear evolution equations which describe pseudospherical surfaces, submitted.
[14]	A. H. Khater, D. K. Callebaut and S. M. Sayed, Exact solutions for some nonlinear evolution equations which describe pseudo-spherical surfaces, J. Comp. and Appl. Math., 189 (2006), 387-411.doi: 10.1016/j.cam.2005.10.007.
[15]	A. H. Khater, M. A. Helal and O. H. El-Kalaawy, Two new classes of exact solutions for the KdV equation via Bäcklund transformations, Choas Solitons Fractals, 8 (1997), 1901-1909.doi: 10.1016/S0960-0779(97)00090-8.
[16]	A. H. Khater, A. M. Shehata, D. K. Callebaut and S. M. Sayed, Self-dual solutions for $SU(2)$ and $SU(3)$ gauge fields one Euclidean space, J. Theoret. Phys., 43 (2004), 151-159.doi: 10.1023/B:IJTP.0000028857.57274.cd.
[17]	K. Konno and M. Wadati, Simple derivation of Bäcklund transformation from Riccati form of inverse method, Progr. Theoret. Phys., 53 (1975), 1652-1656.doi: 10.1143/PTP.53.1652.
[18]	M. G. Lamb, Bäcklund transformations for certain nonlinear evolution equations, J. Math. Phys., 15 (1974), 2157-2165.doi: 10.1063/1.1666595.
[19]	P. G. LeFloch and R. Natalini, Conservation laws with vanishing nonlinear diffusion and dispersion, Nonlinear Anal., 36 (1999), 213-230.doi: 10.1016/S0362-546X(98)00012-1.
[20]	Y. G. Lu, Convergence of solutions to nonlinear dispersive equations without convexity conditions, Appl. Anal., 31 (1989), 239-246.doi: 10.1080/00036818908839828.
[21]	F. Murat, L'injection du cône positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$, J. Math. Pures Appl. (9), 60 (1981), 309-322.
[22]	M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000.doi: 10.1080/03605308208820242.
[23]	M. Rabelo, On equations which describe pseudospherical surfaces, Stud. Appl. Math., 81 (1989), 221-248.doi: 10.1002/sapm1989813221.
[24]	C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations, in Geometry and Modern Applications in Soliton Theory, Cambridge University Press, Cambridge, 2002.doi: 10.1017/CBO9780511606359.
[25]	C. Rogers and W. K. Schief, Bäcklund Transformations and Their Applications, Academic Press, New York, 1982.
[26]	S. M. Sayed, A. M. Elkholy and G. M. Gharib, Exact solutions and conservation laws for Ibragimov-Shabat equation which describe pseudo-spherical surface, Comput. & Appl. Math., 27 (2008), 305-318.doi: 10.1590/S0101-82052008000300005.
[27]	V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP, 34 (1972), 62-69.