# 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
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##### 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), $\Delta=\beta^2-4\alpha \gamma$, $\epsilon=\pm 1$.
 $i$ $F_i(z)$ $1$ $\left[\frac{-\alpha\beta\,\mbox{sech}^2(\sqrt{\alpha}z)}{\beta^2-\alpha\gamma(1+\epsilon \mbox{tanh}(\sqrt{\alpha}z))^2}\right]^{1/2}, \, \alpha>0$ $2$ $\left[\frac{\alpha\beta\,\mbox{csch}^2(\sqrt{\alpha}z)}{\beta^2-\alpha\gamma(1+\epsilon \mbox{coth}(\sqrt{\alpha}z))^2}\right]^{1/2},\, \alpha>0$ $3$ $\left[\frac{2\alpha}{\epsilon \sqrt{\Delta} \mbox{cosh}(2\sqrt{\alpha}z)-\beta}\right]^{1/2}, \, \alpha>0, \Delta >0$ $4$ $\left[\frac{2\alpha}{\epsilon \sqrt{\Delta} \mbox{cos}(2\sqrt{\alpha}z)-\beta}\right]^{1/2}, \, \alpha <0, \Delta >0$ $5$ $\left[\frac{2\alpha}{\epsilon \sqrt{-\Delta} \mbox{sinh}(2\sqrt{\alpha}z)-\beta}\right]^{1/2}, \, \alpha>0, \Delta <0$ $6$ $\left[\frac{2\alpha}{\epsilon \sqrt{\Delta} \mbox{sin}(2\sqrt{-\alpha}z)-\beta}\right]^{1/2}, \, \alpha <0, \Delta >0$ $7$ $\left[\frac{-\alpha\,\mbox{sech}^2(\sqrt{\alpha}z)}{\beta+2\epsilon \sqrt{\alpha\gamma}\mbox{tanh}(\sqrt{\alpha}z)}\right]^{1/2}, \, \alpha>0, \gamma >0$ $8$ $\left[\frac{-\alpha\,\mbox{sec}^2(\sqrt{-\alpha}z)}{\beta+2\epsilon \sqrt{-\alpha\gamma} \mbox{tan}(\sqrt{-\alpha}z)}\right]^{1/2}, \alpha <0, \gamma>0$ $9$ $\left[\frac{\alpha\,\mbox{csch}^2(\sqrt{-\alpha}z)}{\beta+2\epsilon \sqrt{\alpha\gamma} \mbox{coth}(\sqrt{\alpha}z)}\right]^{1/2}, \alpha>0, \gamma>0$ $10$ $\left[\frac{-\alpha\,\mbox{csc}^2(\sqrt{-\alpha}z)}{\beta+2\epsilon \sqrt{-\alpha\gamma} \mbox{cot}(\sqrt{-\alpha}z)}\right]^{1/2}\, \alpha <0, \gamma>0$ $11$ $\left[-\frac{\alpha}{\beta} (1+\epsilon\mbox{tanh}(\frac{\sqrt{\alpha}}{2}z))\right]^{1/2}\, \alpha>0, \Delta=0$ $12$ $\left[-\frac{\alpha}{\beta} (1+\epsilon\mbox{coth}(\frac{\sqrt{\alpha}}{2}z))\right]^{1/2}\, \alpha>0, \Delta=0$ $13$ $4\left[\frac{\alpha e^{2\epsilon\sqrt{\alpha}z}}{(e^{2\epsilon\sqrt{\alpha}z}-4\beta)^2-64\alpha\gamma} \right]^{1/2}, \, \alpha>0$
 $i$ $F_i(z)$ $1$ $\left[\frac{-\alpha\beta\,\mbox{sech}^2(\sqrt{\alpha}z)}{\beta^2-\alpha\gamma(1+\epsilon \mbox{tanh}(\sqrt{\alpha}z))^2}\right]^{1/2}, \, \alpha>0$ $2$ $\left[\frac{\alpha\beta\,\mbox{csch}^2(\sqrt{\alpha}z)}{\beta^2-\alpha\gamma(1+\epsilon \mbox{coth}(\sqrt{\alpha}z))^2}\right]^{1/2},\, \alpha>0$ $3$ $\left[\frac{2\alpha}{\epsilon \sqrt{\Delta} \mbox{cosh}(2\sqrt{\alpha}z)-\beta}\right]^{1/2}, \, \alpha>0, \Delta >0$ $4$ $\left[\frac{2\alpha}{\epsilon \sqrt{\Delta} \mbox{cos}(2\sqrt{\alpha}z)-\beta}\right]^{1/2}, \, \alpha <0, \Delta >0$ $5$ $\left[\frac{2\alpha}{\epsilon \sqrt{-\Delta} \mbox{sinh}(2\sqrt{\alpha}z)-\beta}\right]^{1/2}, \, \alpha>0, \Delta <0$ $6$ $\left[\frac{2\alpha}{\epsilon \sqrt{\Delta} \mbox{sin}(2\sqrt{-\alpha}z)-\beta}\right]^{1/2}, \, \alpha <0, \Delta >0$ $7$ $\left[\frac{-\alpha\,\mbox{sech}^2(\sqrt{\alpha}z)}{\beta+2\epsilon \sqrt{\alpha\gamma}\mbox{tanh}(\sqrt{\alpha}z)}\right]^{1/2}, \, \alpha>0, \gamma >0$ $8$ $\left[\frac{-\alpha\,\mbox{sec}^2(\sqrt{-\alpha}z)}{\beta+2\epsilon \sqrt{-\alpha\gamma} \mbox{tan}(\sqrt{-\alpha}z)}\right]^{1/2}, \alpha <0, \gamma>0$ $9$ $\left[\frac{\alpha\,\mbox{csch}^2(\sqrt{-\alpha}z)}{\beta+2\epsilon \sqrt{\alpha\gamma} \mbox{coth}(\sqrt{\alpha}z)}\right]^{1/2}, \alpha>0, \gamma>0$ $10$ $\left[\frac{-\alpha\,\mbox{csc}^2(\sqrt{-\alpha}z)}{\beta+2\epsilon \sqrt{-\alpha\gamma} \mbox{cot}(\sqrt{-\alpha}z)}\right]^{1/2}\, \alpha <0, \gamma>0$ $11$ $\left[-\frac{\alpha}{\beta} (1+\epsilon\mbox{tanh}(\frac{\sqrt{\alpha}}{2}z))\right]^{1/2}\, \alpha>0, \Delta=0$ $12$ $\left[-\frac{\alpha}{\beta} (1+\epsilon\mbox{coth}(\frac{\sqrt{\alpha}}{2}z))\right]^{1/2}\, \alpha>0, \Delta=0$ $13$ $4\left[\frac{\alpha e^{2\epsilon\sqrt{\alpha}z}}{(e^{2\epsilon\sqrt{\alpha}z}-4\beta)^2-64\alpha\gamma} \right]^{1/2}, \, \alpha>0$
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