American Institute of Mathematical Sciences

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

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  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.
Keywords: entropy condition., compensated compactness, Singular limit, Ibragimov-Shabat equation.
Mathematics Subject Classification: Primary: 35G25, 35L65; Secondary: 35L0.
Citation: Giuseppe Maria Coclite, Lorenzo di Ruvo. A singular limit problem for the Ibragimov-Shabat equation. Discrete & 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.  doi: 10.1002/sapm1974534249.  Google Scholar

[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.  doi: 10.1002/sapm1989812125.  Google Scholar

[3]

G. M. Coclite and L. di Ruvo, Well-posedness results for the short pulse equation,, Z. Angew. Math. Phys., 66 (2015), 1529.  doi: 10.1007/s00033-014-0478-6.  Google Scholar

[4]

G. M. Coclite and L. di Ruvo, On the Wellposedness of the exp-Rabelo equation,, Ann. Mat. Pura Appl., ().  doi: 10.1007/s10231-015-0497-8.  Google Scholar

[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.  doi: 10.3934/nhm.2013.8.969.  Google Scholar

[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.  doi: 10.1080/03605300600781600.  Google Scholar

[7]

R. K. Dodd and R. K. Bullough, Bäcklund transformations for the A.K.N.S. inverse method,, Phys. Lett. A, 62 (1977), 70.  doi: 10.1016/0375-9601(77)90952-5.  Google Scholar

[8]

L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces,, Dover, (1960).   Google Scholar

[9]

E. Goursat, Le Problème de Bäcklund,, Mémorial des Sciences Mathématiques, (1925).   Google Scholar

[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.  doi: 10.1016/S0960-0779(98)00155-6.  Google Scholar

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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.  doi: 10.1063/1.872723.  Google Scholar

[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.  doi: 10.1016/j.geomphys.2003.11.009.  Google Scholar

[13]

A. H. Khater, D. K. Callebaut and S. M. Sayed, Bäcklund transformations for some nonlinear evolution equations which describe pseudospherical surfaces,, submitted., ().   Google Scholar

[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.  doi: 10.1016/j.cam.2005.10.007.  Google Scholar

[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.  doi: 10.1016/S0960-0779(97)00090-8.  Google Scholar

[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.  doi: 10.1023/B:IJTP.0000028857.57274.cd.  Google Scholar

[17]

K. Konno and M. Wadati, Simple derivation of Bäcklund transformation from Riccati form of inverse method,, Progr. Theoret. Phys., 53 (1975), 1652.  doi: 10.1143/PTP.53.1652.  Google Scholar

[18]

M. G. Lamb, Bäcklund transformations for certain nonlinear evolution equations,, J. Math. Phys., 15 (1974), 2157.  doi: 10.1063/1.1666595.  Google Scholar

[19]

P. G. LeFloch and R. Natalini, Conservation laws with vanishing nonlinear diffusion and dispersion,, Nonlinear Anal., 36 (1999), 213.  doi: 10.1016/S0362-546X(98)00012-1.  Google Scholar

[20]

Y. G. Lu, Convergence of solutions to nonlinear dispersive equations without convexity conditions,, Appl. Anal., 31 (1989), 239.  doi: 10.1080/00036818908839828.  Google Scholar

[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.   Google Scholar

[22]

M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations,, Comm. Partial Differential Equations, 7 (1982), 959.  doi: 10.1080/03605308208820242.  Google Scholar

[23]

M. Rabelo, On equations which describe pseudospherical surfaces,, Stud. Appl. Math., 81 (1989), 221.  doi: 10.1002/sapm1989813221.  Google Scholar

[24]

C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations, in Geometry and Modern Applications in Soliton Theory,, Cambridge University Press, (2002).  doi: 10.1017/CBO9780511606359.  Google Scholar

[25]

C. Rogers and W. K. Schief, Bäcklund Transformations and Their Applications,, Academic Press, (1982).   Google Scholar

[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.  doi: 10.1590/S0101-82052008000300005.  Google Scholar

[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.   Google Scholar

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.  doi: 10.1002/sapm1974534249.  Google Scholar

[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.  doi: 10.1002/sapm1989812125.  Google Scholar

[3]

G. M. Coclite and L. di Ruvo, Well-posedness results for the short pulse equation,, Z. Angew. Math. Phys., 66 (2015), 1529.  doi: 10.1007/s00033-014-0478-6.  Google Scholar

[4]

G. M. Coclite and L. di Ruvo, On the Wellposedness of the exp-Rabelo equation,, Ann. Mat. Pura Appl., ().  doi: 10.1007/s10231-015-0497-8.  Google Scholar

[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.  doi: 10.3934/nhm.2013.8.969.  Google Scholar

[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.  doi: 10.1080/03605300600781600.  Google Scholar

[7]

R. K. Dodd and R. K. Bullough, Bäcklund transformations for the A.K.N.S. inverse method,, Phys. Lett. A, 62 (1977), 70.  doi: 10.1016/0375-9601(77)90952-5.  Google Scholar

[8]

L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces,, Dover, (1960).   Google Scholar

[9]

E. Goursat, Le Problème de Bäcklund,, Mémorial des Sciences Mathématiques, (1925).   Google Scholar

[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.  doi: 10.1016/S0960-0779(98)00155-6.  Google Scholar

[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.  doi: 10.1063/1.872723.  Google Scholar

[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.  doi: 10.1016/j.geomphys.2003.11.009.  Google Scholar

[13]

A. H. Khater, D. K. Callebaut and S. M. Sayed, Bäcklund transformations for some nonlinear evolution equations which describe pseudospherical surfaces,, submitted., ().   Google Scholar

[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.  doi: 10.1016/j.cam.2005.10.007.  Google Scholar

[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.  doi: 10.1016/S0960-0779(97)00090-8.  Google Scholar

[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.  doi: 10.1023/B:IJTP.0000028857.57274.cd.  Google Scholar

[17]

K. Konno and M. Wadati, Simple derivation of Bäcklund transformation from Riccati form of inverse method,, Progr. Theoret. Phys., 53 (1975), 1652.  doi: 10.1143/PTP.53.1652.  Google Scholar

[18]

M. G. Lamb, Bäcklund transformations for certain nonlinear evolution equations,, J. Math. Phys., 15 (1974), 2157.  doi: 10.1063/1.1666595.  Google Scholar

[19]

P. G. LeFloch and R. Natalini, Conservation laws with vanishing nonlinear diffusion and dispersion,, Nonlinear Anal., 36 (1999), 213.  doi: 10.1016/S0362-546X(98)00012-1.  Google Scholar

[20]

Y. G. Lu, Convergence of solutions to nonlinear dispersive equations without convexity conditions,, Appl. Anal., 31 (1989), 239.  doi: 10.1080/00036818908839828.  Google Scholar

[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.   Google Scholar

[22]

M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations,, Comm. Partial Differential Equations, 7 (1982), 959.  doi: 10.1080/03605308208820242.  Google Scholar

[23]

M. Rabelo, On equations which describe pseudospherical surfaces,, Stud. Appl. Math., 81 (1989), 221.  doi: 10.1002/sapm1989813221.  Google Scholar

[24]

C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations, in Geometry and Modern Applications in Soliton Theory,, Cambridge University Press, (2002).  doi: 10.1017/CBO9780511606359.  Google Scholar

[25]

C. Rogers and W. K. Schief, Bäcklund Transformations and Their Applications,, Academic Press, (1982).   Google Scholar

[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.  doi: 10.1590/S0101-82052008000300005.  Google Scholar

[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.   Google Scholar

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