June  2016, 9(3): 661-673. doi: 10.3934/dcdss.2016020

A singular limit problem for the Ibragimov-Shabat equation

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  April 2016

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.
Citation: Giuseppe Maria Coclite, Lorenzo di Ruvo. A singular limit problem for the Ibragimov-Shabat equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 661-673. doi: 10.3934/dcdss.2016020
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.

show all references

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.

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