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Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion
October  2020, 13(10): 2703-2717. doi: 10.3934/dcdss.2020218

## Symmetries and conservation laws of a time dependent nonlinear reaction-convection-diffusion equation

 1 Department of Mathematics, College of Science, China Three Gorges University, Yichang, Hubei 443002, China 2 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China 3 African Institute for Mathematical Sciences, Muizenberg, Cape Town, South Africa

* Corresponding author: Lijun Zhang

Received  January 2019 Revised  June 2019 Published  October 2020 Early access  December 2019

Fund Project: The first author is funded by the China Scholarship Fund (No.201908420198). The second author is supported by NSF grant No. 11672270. The revision was done during L Zhang's research visit to African Institute for Mathematical Sciences South Africa

In this paper, we study a nonlinear diffusion-convection-reaction equation with a variable coefficient which has applications in many fields. The Lie point symmetries of this equation are derived, according to which this equation is classified into four different kinds. Conservation laws for this equation are constructed by using the conservation theorem of Ibragimov.

Citation: Zhijie Cao, Lijun Zhang. Symmetries and conservation laws of a time dependent nonlinear reaction-convection-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2703-2717. doi: 10.3934/dcdss.2020218
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show all references

##### References:
  S. C. Anco and G. W. Bluman, Direct construction of conservation laws from field equations, Phys. Rev. Lett., 78 (1997), 2869-2873.  doi: 10.1103/PhysRevLett.78.2869. Google Scholar  E. D. Avdonina, N. H. Ibragimov and R. Khamitova, Exact solutions of gasdynamic equaitons obtained by the method of conservation laws, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 2359-2366.  doi: 10.1016/j.cnsns.2012.12.023. Google Scholar  G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations, Applications mathematics series, 154. Springer, 2002. Google Scholar  G. W. Bluman, Te muerchaolu and S. C. Anco, New conservation laws obtained directly from symmetry action on a known conservation law, J. Math. Anal. Appl., 322 (2006), 233-250.  doi: 10.1016/j.jmaa.2005.08.092. Google Scholar  I. L. Freire, Conservation laws for self-adjoint first order evolution equation, J. Nonlin. Math. Phys., 18 (2011), 279-290.  doi: 10.1142/S1402925111001453.  Google Scholar  I. L. Freire, New conservation laws for inviscid Burgers equation, Comp. Appl. Math., 31 (2012), 559-567.  doi: 10.1590/S1807-03022012000300007.  Google Scholar  I. L. Freire and J. C. S. Sampaio, On the nonlinear self-adjointness and local conservation laws for a class of evolution equations unifying many models, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 350-360.  doi: 10.1016/j.cnsns.2013.06.010. Google Scholar  N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311-328.  doi: 10.1016/j.jmaa.2006.10.078.  Google Scholar  N. H. Ibragimov, Integrating factors, adjoint equations and Lagrangians, J. Math. Anal. Appl., 318 (2006), 742-757.  doi: 10.1016/j.jmaa.2005.11.012.  Google Scholar  N. H.Ibragimov, Method of conservation laws for constructing solutions to systems of PDEs, Disc. Nonlinearity compl., 1 (2012), 353-362.   Google Scholar  N. H. Ibragimov and T. Kolsrud, Lagrangian approach to evolution equations: Symmetries and conservation laws, Nonlinear Dyn., 36 (2004), 29-40.  doi: 10.1023/B:NODY.0000034644.82259.1f.  Google Scholar  A. H. Kara and F. M. Mahomed, Relationship between symmetries and conservation laws, Internat. J. Theoret. Phys., 39 (2000), 23-40.  doi: 10.1023/A:1003686831523.  Google Scholar  A. H. Kara and F. M. Mahomed, Noether-type symmetries and conservation laws via partial Lagrangians, Nonlinear Dyn., 45 (2006), 367-383.  doi: 10.1007/s11071-005-9013-9.  Google Scholar  A. Mishra and R. Kumar, Exact solutions of variable coefficient nonlinear diffusion-reaction equations with a nonliear convective term, Phys. Lett. A, 374 (2010), 2921-2924.  doi: 10.1016/j.physleta.2010.03.039.  Google Scholar  B. Muatjetjeja and C. M. Khalique, First integrals for a generalized coupled Lane-Emden system, Nonlinear Anal. Real World Appl., 12 (2011), 1202-1212.  doi: 10.1016/j.nonrwa.2010.09.013.  Google Scholar  J. D. Murray, Mathematical Biology, Springer-Verlag, New York, 1993. Google Scholar  P. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986. Google Scholar  R. Popovych, Direct methods of construction of conservation laws, Physics AUC, 16 (2006), 81-947.   Google Scholar  J. C. S. Sampaio and I. L. Freire, Nonlinear self-adjoint classification of a Burgers-KdV family of equations, Abs. Appl. Anal., 2014 (2014), 1-7.  doi: 10.1155/2014/804703.  Google Scholar  H. Steudel, Noether's theorem and the conservation laws of the Korteweg-de Vries equation, Ann. Physik, 32 (1975), 445-455.  doi: 10.1002/andp.19754870605.  Google Scholar
The results for symmetries of equation (1.1)
 Cases Sol. of determining eq. Infinitesimal generator Restriction on $k$ $\xi=\frac{1}{2}c_{1}x+$ $X=$ $a=0$ $c_4\int{k(t)dt}+c_{3},$ $c_1(\frac12 x\partial_x+t\partial_t)$ $b=0$ $\tau=c_{1}t+c_{2},$ $+c_2\partial_t+c_3\partial_x+c_5u\partial_u$ $(c_5+\frac{1}{2}c_1)k+$ $\phi=c_{5}u+c_{4}.$ $+c_4(\int{k(t)dt}\partial_x+\partial_u)$ $(c_1t+c_2)k'=0$ $\xi=\frac{1}{2}c_{1}x+c_{3},$ X= $a=0$ $\tau=c_{1}t+c_{2},$ $c_1(\frac12x\partial_x+t\partial_t-u\partial_u)$ $(c_1t+c_2)k'-$ $b\neq 0$ $\phi=-c_{1}u$ $+c_2\partial_t+c_3\partial_x.$ $\frac{1}{2}c_1k(t)=0$ $\xi=\frac{1}{2}c_{1}x+$ $X=$ $a\neq 0$ $c_4\int{e^{at}kdt}+c_{3},$ $c_1(\frac12x\partial_x+t\partial_t+atu\partial_u)$ $b=0$ $\tau=c_{1}t+c_{2},$ $+c_2\partial_t+ c_3\partial_x+c_4(e^{at}\partial_u$ $(c_1(at+\frac12)+c_5)k$ $\phi=(c_{1}at+c_5)u+c_4e^{at}$ $+\int{e^{at}kdt}\partial_x)+c_5u\partial_u$ $+(c_1t+c_2)k'=0$ ${\xi=c_{3}},$ $X=$ ${c_2k' = 0}$ $a\neq 0$ ${\tau = c_{2}},$ ${c_2\partial_t+c_3\partial_x}.$ $b\neq0$ ${\phi = 0}$
 Cases Sol. of determining eq. Infinitesimal generator Restriction on $k$ $\xi=\frac{1}{2}c_{1}x+$ $X=$ $a=0$ $c_4\int{k(t)dt}+c_{3},$ $c_1(\frac12 x\partial_x+t\partial_t)$ $b=0$ $\tau=c_{1}t+c_{2},$ $+c_2\partial_t+c_3\partial_x+c_5u\partial_u$ $(c_5+\frac{1}{2}c_1)k+$ $\phi=c_{5}u+c_{4}.$ $+c_4(\int{k(t)dt}\partial_x+\partial_u)$ $(c_1t+c_2)k'=0$ $\xi=\frac{1}{2}c_{1}x+c_{3},$ X= $a=0$ $\tau=c_{1}t+c_{2},$ $c_1(\frac12x\partial_x+t\partial_t-u\partial_u)$ $(c_1t+c_2)k'-$ $b\neq 0$ $\phi=-c_{1}u$ $+c_2\partial_t+c_3\partial_x.$ $\frac{1}{2}c_1k(t)=0$ $\xi=\frac{1}{2}c_{1}x+$ $X=$ $a\neq 0$ $c_4\int{e^{at}kdt}+c_{3},$ $c_1(\frac12x\partial_x+t\partial_t+atu\partial_u)$ $b=0$ $\tau=c_{1}t+c_{2},$ $+c_2\partial_t+ c_3\partial_x+c_4(e^{at}\partial_u$ $(c_1(at+\frac12)+c_5)k$ $\phi=(c_{1}at+c_5)u+c_4e^{at}$ $+\int{e^{at}kdt}\partial_x)+c_5u\partial_u$ $+(c_1t+c_2)k'=0$ ${\xi=c_{3}},$ $X=$ ${c_2k' = 0}$ $a\neq 0$ ${\tau = c_{2}},$ ${c_2\partial_t+c_3\partial_x}.$ $b\neq0$ ${\phi = 0}$
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