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October  2020, 13(10): 2691-2701. doi: 10.3934/dcdss.2020222

Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion

María-Santos Bruzón , , Elena Recio , Tamara-María Garrido and  Rafael de la Rosa

Department of Mathematics, University of Cádiz, PO.BOX 40, 11510 Puerto Real, Cádiz, Spain

* Corresponding author: María-Santos Bruzón

Received  February 2019 Revised  July 2019 Published  December 2019

We provide a complete classification of point symmetries and low-order local conservation laws of the generalized quasilinear KdV equation in terms of the arbitrary function. The corresponding interpretation of symmetry transformation groups are given. In addition, a physical description of the conserved quantities is included. Finally, few travelling wave solutions have been obtained.

Keywords: Quasilinear KdV, Lie point symmetries, conservation laws, conserved quantities and travelling wave solutions.
Mathematics Subject Classification: Primary: 35B06, 35L65, 35C07; Secondary: 35Q53.
Citation: María-Santos Bruzón, Elena Recio, Tamara-María Garrido, Rafael de la Rosa. Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2691-2701. doi: 10.3934/dcdss.2020222
References:
[1]

L. M. Alonso, On the Noether map, Lett. Math. Phys., 3 (1979), 419-424.  doi: 10.1007/BF00397216.  Google Scholar

[2]

D. M. Ambrose and J. D. Wright, Traveling waves and weak solutions for an equation with degenerate dispersion, Proc. Amer. Math. Soc., 141 (2013), 3825-3838.  doi: 10.1090/S0002-9939-2013-12070-8.  Google Scholar

[3]

S. C. Anco, Symmetry properties of conservation laws, Internat. J. Modern Phys. B, 30 (2016), 28-29.  doi: 10.1142/S0217979216400038.  Google Scholar

[4]

S. C. Anco, Generalization of noether's theorem in modern form to non-variational partial differential equations, Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, Fields Inst. Commun., Springer, New York, 79 (2017), 119-182.  Google Scholar

[5]

S. C. Anco and G. Bluman, Direct construction of conservation laws from field equations, Phys. Rev. Lett., 78 (1997), 2869-2873.  doi: 10.1103/PhysRevLett.78.2869.  Google Scholar

[6]

S. C. Anco and G. Bluman, Integrating factors and first integrals for ordinary differential equations, European J. Appl. Math., 9 (1998), 245-259.  doi: 10.1017/S0956792598003477.  Google Scholar

[7]

S. C. Anco and G. Bluman, Direct constrution method for conservation laws of partial differential equations part 1: Examples of conservation law classifications, European J. Appl. Math., 13 (2002), 545-566.  doi: 10.1017/S095679250100465X.  Google Scholar

[8]

S. C. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations part 2: General treatment, European J. Appl. Math., 13 (2002), 567-585.  doi: 10.1017/S0956792501004661.  Google Scholar

[9]

G. W. Bluman, S. C. Anco, Symmetry and Integration Methods for Differential Equations, Applied Mathematical Sciences, 154. Springer-Verlag, New York, 2002.  Google Scholar

[10]

G. W. Bluman, A. F. Cheviakov and S. C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Applied Mathematical Sciences, 168. Springer, New York, 2010. doi: 10.1007/978-0-387-68028-6.  Google Scholar

[11]

G. W. Bluman and J. D. Cole, General similarity solution of the heat equation, J. Math. Mech., 18 (1968/69), 1025-1042.   Google Scholar

[12]

G. Bluman and S. Kumei, On the remarkable nonlinear diffusion equation $(\partial /\partial x)[a(u+b)^{-2}(\partial u/\partial x)] -(\partial u/\partial t) = 0$., J. Math. Phys., 21 (1980), 1019-1023.  doi: 10.1063/1.524550.  Google Scholar

[13]

G. W. Bluman, S. Kumei and G. J. Reid, New classes of symmetries for partial differential equations, J. Math. Phys., 29 (1988), 806-811.  doi: 10.1063/1.527974.  Google Scholar

[14]

M. S. Bruzón and T. M. Garrido, Symmetries and conservation laws of a KdV6 equation, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 631-641.  doi: 10.3934/dcdss.2018038.  Google Scholar

[15]

M. S. Bruzón, E. Recio, T. M. Garrido, A. P. Márquez and R. de la Rosa, On the similarity solutions and conservation laws of the Cooper-Shepard-Sodano equation, Math. Methods Appl. Sci., 41 (2018), 7325-7332.  doi: 10.1002/mma.4829.  Google Scholar

[16]

M. S. Bruzón, M. L. Gandarias and R. de la Rosa, An overview of the generalized Gardner equation: Symmetry groups and conservation laws, A Mathematical Modeling Approach from Nonlinear Dynamics to Complex Systems, Nonlinear Syst. Complex., Springer, Cham, 22 (2019), 7-26.  Google Scholar

[17]

F. Cooper, H. Shepard and P. Sodano, Solitary waves in a class of generalized Korteweg-de Vries equations, Phys. Rev. E (3), 48 (1993), 4027-4032.  doi: 10.1103/PhysRevE.48.4027.  Google Scholar

[18]

P. Germain, B. Harrop-Griffiths and J. M. Marzuola, Existence and uniqueness of solutions for a quasilinear KdV equation with degenerate dispersion, Comm. Pure Appl. Math., 72 (2019), 2449-2484.  doi: 10.1002/cpa.21828.  Google Scholar

[19]

A. Karasu-Kalkanli, A. Karasu, A. Sakovich, S. Sakovich and R. Turhan, A new integrable generalization of the Korteweg de Vries equation, Journal of Mathematical Physics, 49 (2008), 10 pp.  doi: 10.1063/1.2953474.  Google Scholar

[20]

N. A. Kudryashov, On 'new travelling wave solutions' of the KdV and the KdV Burgers equations, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 1891-1900.  doi: 10.1016/j.cnsns.2008.09.020.  Google Scholar

[21]

P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4684-0274-2.  Google Scholar

[22]

E. Recio and S. C. Anco, Conservation laws and symmetries of radial generalized nonlinear p-Laplacian evolution equations, J. Math. Anal. Appl., 452 (2017), 1229-1261.  doi: 10.1016/j.jmaa.2017.03.050.  Google Scholar

[23]

P. Rosenau and J. M. Hyman, Compactons: Solitons with finite wavelength, Physical Review Letters, 70 (1993), 564-567.   Google Scholar

[24]

J. Satsuma and R. Hirota, A coupled KdV equation is one case of the four-reduction of the KP hierarchy, J. Phys. Soc. Japan, 51 (1982), 3390-3397.  doi: 10.1143/JPSJ.51.3390.  Google Scholar

[25]

K. Sawada and T. Kotera, A method for finding N-soliton solutions of the KdV equation and KdV-like equation, Theoret. Phys., 51 (1974), 1355-1367.  doi: 10.1143/PTP.51.1355.  Google Scholar

[26]

J. Weiss, M. Tabor and G. Carnevale, The painlevè property for partial differential equations, J. Math. Phys., 24 (1983), 522-526.  doi: 10.1063/1.525721.  Google Scholar

[27]

G. A. Zakeri and E. Yomba, Exact solutions of a generalized autonomous Duffing-type equation, Appl. Math. Model., 39 (2015), 4607-4616.  doi: 10.1016/j.apm.2015.04.027.  Google Scholar

show all references

References:
[1]

L. M. Alonso, On the Noether map, Lett. Math. Phys., 3 (1979), 419-424.  doi: 10.1007/BF00397216.  Google Scholar

[2]

D. M. Ambrose and J. D. Wright, Traveling waves and weak solutions for an equation with degenerate dispersion, Proc. Amer. Math. Soc., 141 (2013), 3825-3838.  doi: 10.1090/S0002-9939-2013-12070-8.  Google Scholar

[3]

S. C. Anco, Symmetry properties of conservation laws, Internat. J. Modern Phys. B, 30 (2016), 28-29.  doi: 10.1142/S0217979216400038.  Google Scholar

[4]

S. C. Anco, Generalization of noether's theorem in modern form to non-variational partial differential equations, Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, Fields Inst. Commun., Springer, New York, 79 (2017), 119-182.  Google Scholar

[5]

S. C. Anco and G. Bluman, Direct construction of conservation laws from field equations, Phys. Rev. Lett., 78 (1997), 2869-2873.  doi: 10.1103/PhysRevLett.78.2869.  Google Scholar

[6]

S. C. Anco and G. Bluman, Integrating factors and first integrals for ordinary differential equations, European J. Appl. Math., 9 (1998), 245-259.  doi: 10.1017/S0956792598003477.  Google Scholar

[7]

S. C. Anco and G. Bluman, Direct constrution method for conservation laws of partial differential equations part 1: Examples of conservation law classifications, European J. Appl. Math., 13 (2002), 545-566.  doi: 10.1017/S095679250100465X.  Google Scholar

[8]

S. C. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations part 2: General treatment, European J. Appl. Math., 13 (2002), 567-585.  doi: 10.1017/S0956792501004661.  Google Scholar

[9]

G. W. Bluman, S. C. Anco, Symmetry and Integration Methods for Differential Equations, Applied Mathematical Sciences, 154. Springer-Verlag, New York, 2002.  Google Scholar

[10]

G. W. Bluman, A. F. Cheviakov and S. C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Applied Mathematical Sciences, 168. Springer, New York, 2010. doi: 10.1007/978-0-387-68028-6.  Google Scholar

[11]

G. W. Bluman and J. D. Cole, General similarity solution of the heat equation, J. Math. Mech., 18 (1968/69), 1025-1042.   Google Scholar

[12]

G. Bluman and S. Kumei, On the remarkable nonlinear diffusion equation $(\partial /\partial x)[a(u+b)^{-2}(\partial u/\partial x)] -(\partial u/\partial t) = 0$., J. Math. Phys., 21 (1980), 1019-1023.  doi: 10.1063/1.524550.  Google Scholar

[13]

G. W. Bluman, S. Kumei and G. J. Reid, New classes of symmetries for partial differential equations, J. Math. Phys., 29 (1988), 806-811.  doi: 10.1063/1.527974.  Google Scholar

[14]

M. S. Bruzón and T. M. Garrido, Symmetries and conservation laws of a KdV6 equation, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 631-641.  doi: 10.3934/dcdss.2018038.  Google Scholar

[15]

M. S. Bruzón, E. Recio, T. M. Garrido, A. P. Márquez and R. de la Rosa, On the similarity solutions and conservation laws of the Cooper-Shepard-Sodano equation, Math. Methods Appl. Sci., 41 (2018), 7325-7332.  doi: 10.1002/mma.4829.  Google Scholar

[16]

M. S. Bruzón, M. L. Gandarias and R. de la Rosa, An overview of the generalized Gardner equation: Symmetry groups and conservation laws, A Mathematical Modeling Approach from Nonlinear Dynamics to Complex Systems, Nonlinear Syst. Complex., Springer, Cham, 22 (2019), 7-26.  Google Scholar

[17]

F. Cooper, H. Shepard and P. Sodano, Solitary waves in a class of generalized Korteweg-de Vries equations, Phys. Rev. E (3), 48 (1993), 4027-4032.  doi: 10.1103/PhysRevE.48.4027.  Google Scholar

[18]

P. Germain, B. Harrop-Griffiths and J. M. Marzuola, Existence and uniqueness of solutions for a quasilinear KdV equation with degenerate dispersion, Comm. Pure Appl. Math., 72 (2019), 2449-2484.  doi: 10.1002/cpa.21828.  Google Scholar

[19]

A. Karasu-Kalkanli, A. Karasu, A. Sakovich, S. Sakovich and R. Turhan, A new integrable generalization of the Korteweg de Vries equation, Journal of Mathematical Physics, 49 (2008), 10 pp.  doi: 10.1063/1.2953474.  Google Scholar

[20]

N. A. Kudryashov, On 'new travelling wave solutions' of the KdV and the KdV Burgers equations, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 1891-1900.  doi: 10.1016/j.cnsns.2008.09.020.  Google Scholar

[21]

P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4684-0274-2.  Google Scholar

[22]

E. Recio and S. C. Anco, Conservation laws and symmetries of radial generalized nonlinear p-Laplacian evolution equations, J. Math. Anal. Appl., 452 (2017), 1229-1261.  doi: 10.1016/j.jmaa.2017.03.050.  Google Scholar

[23]

P. Rosenau and J. M. Hyman, Compactons: Solitons with finite wavelength, Physical Review Letters, 70 (1993), 564-567.   Google Scholar

[24]

J. Satsuma and R. Hirota, A coupled KdV equation is one case of the four-reduction of the KP hierarchy, J. Phys. Soc. Japan, 51 (1982), 3390-3397.  doi: 10.1143/JPSJ.51.3390.  Google Scholar

[25]

K. Sawada and T. Kotera, A method for finding N-soliton solutions of the KdV equation and KdV-like equation, Theoret. Phys., 51 (1974), 1355-1367.  doi: 10.1143/PTP.51.1355.  Google Scholar

[26]

J. Weiss, M. Tabor and G. Carnevale, The painlevè property for partial differential equations, J. Math. Phys., 24 (1983), 522-526.  doi: 10.1063/1.525721.  Google Scholar

[27]

G. A. Zakeri and E. Yomba, Exact solutions of a generalized autonomous Duffing-type equation, Appl. Math. Model., 39 (2015), 4607-4616.  doi: 10.1016/j.apm.2015.04.027.  Google Scholar

Figure 1.  Solution $ u(x,t) = {\rm sech}^2(x-t) $ of Eq. (2)
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Figure 2.  Solution $ u(x,t) = {\rm sinh}(x-t) $ of Eq. (2)
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Figure 3.  Solution $ u(x,t) = {\rm sin}(x-t) $ of Eq. (2)
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