# American Institute of Mathematical Sciences

August  2018, 11(4): 747-757. doi: 10.3934/dcdss.2018047

## Differential invariants of a generalized variable-coefficient Gardner equation

 Departamento de Matemáticas, Universidad de Cádiz, P.O. Box 40, Puerto Real 11510, Cádiz, Spain

* Corresponding author: M.S. Bruzón

Received  December 2016 Revised  May 2017 Published  November 2017

In this paper, we consider a generalized variable-coefficient Gardner equation. By using the equivalence group of this equation, we derive the differential invariants of first order and the corresponding invariant equations. We employ these differential invariants and invariant equations to find the most general subclass of variable-coefficient Gardner equations which can be mapped into a specific constant-coefficient equation by means of an equivalence transformation. Furthermore, differential invariants are applied to obtain exact solutions.

Citation: Rafael de la Rosa, María Santos Bruzón. Differential invariants of a generalized variable-coefficient Gardner equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 747-757. doi: 10.3934/dcdss.2018047
##### References:

show all references

##### References:
Solutions of equation (31), $\Delta=\beta^2-4\alpha \gamma$, $\epsilon=\pm 1$.
 $j$ $F_j(z)$ $1$ $\frac{2 \epsilon \alpha\,\mbox{sech}(\sqrt{\alpha}z)}{\sqrt{\Delta}- \epsilon \beta \mbox{sech}(\sqrt{\alpha}z)}, \, \alpha>0, \Delta>0$ $2$ $\frac{2 \epsilon \alpha\,\mbox{csch}(\sqrt{\alpha}z)}{\sqrt{-\Delta}-\epsilon \beta \mbox{csch}(\sqrt{\alpha}z)}, \, \alpha>0, \Delta <0$ $3$ $-\frac{\alpha}{\beta}\left[ 1\pm \mbox{tanh}(\frac{\sqrt{\alpha}}{2}z)\right], \, \alpha>0, \Delta =0$ $4$ $-\frac{\alpha}{\beta}\left[ 1\pm \mbox{coth}(\frac{\sqrt{\alpha}}{2}z)\right], \, \alpha>0, \Delta =0$
 $j$ $F_j(z)$ $1$ $\frac{2 \epsilon \alpha\,\mbox{sech}(\sqrt{\alpha}z)}{\sqrt{\Delta}- \epsilon \beta \mbox{sech}(\sqrt{\alpha}z)}, \, \alpha>0, \Delta>0$ $2$ $\frac{2 \epsilon \alpha\,\mbox{csch}(\sqrt{\alpha}z)}{\sqrt{-\Delta}-\epsilon \beta \mbox{csch}(\sqrt{\alpha}z)}, \, \alpha>0, \Delta <0$ $3$ $-\frac{\alpha}{\beta}\left[ 1\pm \mbox{tanh}(\frac{\sqrt{\alpha}}{2}z)\right], \, \alpha>0, \Delta =0$ $4$ $-\frac{\alpha}{\beta}\left[ 1\pm \mbox{coth}(\frac{\sqrt{\alpha}}{2}z)\right], \, \alpha>0, \Delta =0$
Solutions of equation (32), $\Delta=\beta^2-4\alpha \gamma$, $\epsilon=\pm 1$.
 $i$ $F_i(z)$ $1$ $\left[\frac{-\alpha\beta\,\mbox{sech}^2(\sqrt{\alpha}z)}{\beta^2-\alpha\gamma(1+\epsilon \mbox{tanh}(\sqrt{\alpha}z))^2}\right]^{1/2}, \, \alpha>0$ $2$ $\left[\frac{\alpha\beta\,\mbox{csch}^2(\sqrt{\alpha}z)}{\beta^2-\alpha\gamma(1+\epsilon \mbox{coth}(\sqrt{\alpha}z))^2}\right]^{1/2},\, \alpha>0$ $3$ $\left[\frac{2\alpha}{\epsilon \sqrt{\Delta} \mbox{cosh}(2\sqrt{\alpha}z)-\beta}\right]^{1/2}, \, \alpha>0, \Delta >0$ $4$ $\left[\frac{2\alpha}{\epsilon \sqrt{\Delta} \mbox{cos}(2\sqrt{\alpha}z)-\beta}\right]^{1/2}, \, \alpha <0, \Delta >0$ $5$ $\left[\frac{2\alpha}{\epsilon \sqrt{-\Delta} \mbox{sinh}(2\sqrt{\alpha}z)-\beta}\right]^{1/2}, \, \alpha>0, \Delta <0$ $6$ $\left[\frac{2\alpha}{\epsilon \sqrt{\Delta} \mbox{sin}(2\sqrt{-\alpha}z)-\beta}\right]^{1/2}, \, \alpha <0, \Delta >0$ $7$ $\left[\frac{-\alpha\,\mbox{sech}^2(\sqrt{\alpha}z)}{\beta+2\epsilon \sqrt{\alpha\gamma}\mbox{tanh}(\sqrt{\alpha}z)}\right]^{1/2}, \, \alpha>0, \gamma >0$ $8$ $\left[\frac{-\alpha\,\mbox{sec}^2(\sqrt{-\alpha}z)}{\beta+2\epsilon \sqrt{-\alpha\gamma} \mbox{tan}(\sqrt{-\alpha}z)}\right]^{1/2}, \alpha <0, \gamma>0$ $9$ $\left[\frac{\alpha\,\mbox{csch}^2(\sqrt{-\alpha}z)}{\beta+2\epsilon \sqrt{\alpha\gamma} \mbox{coth}(\sqrt{\alpha}z)}\right]^{1/2}, \alpha>0, \gamma>0$ $10$ $\left[\frac{-\alpha\,\mbox{csc}^2(\sqrt{-\alpha}z)}{\beta+2\epsilon \sqrt{-\alpha\gamma} \mbox{cot}(\sqrt{-\alpha}z)}\right]^{1/2}\, \alpha <0, \gamma>0$ $11$ $\left[-\frac{\alpha}{\beta} (1+\epsilon\mbox{tanh}(\frac{\sqrt{\alpha}}{2}z))\right]^{1/2}\, \alpha>0, \Delta=0$ $12$ $\left[-\frac{\alpha}{\beta} (1+\epsilon\mbox{coth}(\frac{\sqrt{\alpha}}{2}z))\right]^{1/2}\, \alpha>0, \Delta=0$ $13$ $4\left[\frac{\alpha e^{2\epsilon\sqrt{\alpha}z}}{(e^{2\epsilon\sqrt{\alpha}z}-4\beta)^2-64\alpha\gamma} \right]^{1/2}, \, \alpha>0$
 $i$ $F_i(z)$ $1$ $\left[\frac{-\alpha\beta\,\mbox{sech}^2(\sqrt{\alpha}z)}{\beta^2-\alpha\gamma(1+\epsilon \mbox{tanh}(\sqrt{\alpha}z))^2}\right]^{1/2}, \, \alpha>0$ $2$ $\left[\frac{\alpha\beta\,\mbox{csch}^2(\sqrt{\alpha}z)}{\beta^2-\alpha\gamma(1+\epsilon \mbox{coth}(\sqrt{\alpha}z))^2}\right]^{1/2},\, \alpha>0$ $3$ $\left[\frac{2\alpha}{\epsilon \sqrt{\Delta} \mbox{cosh}(2\sqrt{\alpha}z)-\beta}\right]^{1/2}, \, \alpha>0, \Delta >0$ $4$ $\left[\frac{2\alpha}{\epsilon \sqrt{\Delta} \mbox{cos}(2\sqrt{\alpha}z)-\beta}\right]^{1/2}, \, \alpha <0, \Delta >0$ $5$ $\left[\frac{2\alpha}{\epsilon \sqrt{-\Delta} \mbox{sinh}(2\sqrt{\alpha}z)-\beta}\right]^{1/2}, \, \alpha>0, \Delta <0$ $6$ $\left[\frac{2\alpha}{\epsilon \sqrt{\Delta} \mbox{sin}(2\sqrt{-\alpha}z)-\beta}\right]^{1/2}, \, \alpha <0, \Delta >0$ $7$ $\left[\frac{-\alpha\,\mbox{sech}^2(\sqrt{\alpha}z)}{\beta+2\epsilon \sqrt{\alpha\gamma}\mbox{tanh}(\sqrt{\alpha}z)}\right]^{1/2}, \, \alpha>0, \gamma >0$ $8$ $\left[\frac{-\alpha\,\mbox{sec}^2(\sqrt{-\alpha}z)}{\beta+2\epsilon \sqrt{-\alpha\gamma} \mbox{tan}(\sqrt{-\alpha}z)}\right]^{1/2}, \alpha <0, \gamma>0$ $9$ $\left[\frac{\alpha\,\mbox{csch}^2(\sqrt{-\alpha}z)}{\beta+2\epsilon \sqrt{\alpha\gamma} \mbox{coth}(\sqrt{\alpha}z)}\right]^{1/2}, \alpha>0, \gamma>0$ $10$ $\left[\frac{-\alpha\,\mbox{csc}^2(\sqrt{-\alpha}z)}{\beta+2\epsilon \sqrt{-\alpha\gamma} \mbox{cot}(\sqrt{-\alpha}z)}\right]^{1/2}\, \alpha <0, \gamma>0$ $11$ $\left[-\frac{\alpha}{\beta} (1+\epsilon\mbox{tanh}(\frac{\sqrt{\alpha}}{2}z))\right]^{1/2}\, \alpha>0, \Delta=0$ $12$ $\left[-\frac{\alpha}{\beta} (1+\epsilon\mbox{coth}(\frac{\sqrt{\alpha}}{2}z))\right]^{1/2}\, \alpha>0, \Delta=0$ $13$ $4\left[\frac{\alpha e^{2\epsilon\sqrt{\alpha}z}}{(e^{2\epsilon\sqrt{\alpha}z}-4\beta)^2-64\alpha\gamma} \right]^{1/2}, \, \alpha>0$
 [1] Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032 [2] Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053 [3] Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515 [4] Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167 [5] Hermann Brunner, Stefano Maset. Time transformations for delay differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 751-775. doi: 10.3934/dcds.2009.25.751 [6] Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031 [7] Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221 [8] Berat Karaagac. New exact solutions for some fractional order differential equations via improved sub-equation method. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 447-454. doi: 10.3934/dcdss.2019029 [9] Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481 [10] Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703 [11] Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345 [12] Hermann Brunner, Stefano Maset. Time transformations for state-dependent delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (1) : 23-45. doi: 10.3934/cpaa.2010.9.23 [13] Linghai Zhang. Long-time asymptotic behaviors of solutions of $N$-dimensional dissipative partial differential equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 1025-1042. doi: 10.3934/dcds.2002.8.1025 [14] Kai Liu. Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3915-3934. doi: 10.3934/dcdsb.2018117 [15] Mike Crampin, David Saunders. Homogeneity and projective equivalence of differential equation fields. Journal of Geometric Mechanics, 2012, 4 (1) : 27-47. doi: 10.3934/jgm.2012.4.27 [16] Yanqing Wang, Donghui Yang, Jiongmin Yong, Zhiyong Yu. Exact controllability of linear stochastic differential equations and related problems. Mathematical Control & Related Fields, 2017, 7 (2) : 305-345. doi: 10.3934/mcrf.2017011 [17] David Salas, Lionel Thibault, Emilio Vilches. On smoothness of solutions to projected differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2255-2283. doi: 10.3934/dcds.2019095 [18] Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231 [19] Enrique Zuazua. Controllability of partial differential equations and its semi-discrete approximations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 469-513. doi: 10.3934/dcds.2002.8.469 [20] Paul Bracken. Connections of zero curvature and applications to nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1165-1179. doi: 10.3934/dcdss.2014.7.1165

2018 Impact Factor: 0.545