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:
[1]

K. R. Adem and C. M. Khalique, Exact solutions and conservation laws of Zakharov-Kuznetsov modified equal width equation with power law nonlinearity, Nonlinear Anal. RWA, 13 (2012), 1692-1702. doi: 10.1016/j.nonrwa.2011.12.001.

[2]

Y. Y. Bagderina, Invariants of a family of third-order ordinary differential equations, J. Phys. A: Math. Theor. , 42 (2009), 085204, 21pp. doi: 10.1088/1751-8113/42/8/085204.

[3]

Y. Y. Bagderina, Symmetries and invariants of the systems of two linear second-order ordinary differential equations, Commun Nonlinear Sci Numer Simulat, 19 (2014), 3513-3522. doi: 10.1016/j.cnsns.2014.02.023.

[4]

M. S. BruzónM. L. Gandarias and M. Senthilvelan, On the nonlocal symmetries of certain nonlinear oscillators and their general solution, Physics Letters A, 375 (2011), 2985-2987. doi: 10.1016/j.physleta.2011.06.036.

[5]

R. de la RosaM. L. Gandarias and M. S. Bruzón, Symmetries and conservation laws of a fifth-order KdV equation with time-dependent coefficients and linear damping, Nonlinear Dyn, 84 (2016), 135-141. doi: 10.1007/s11071-015-2254-3.

[6]

R. de la RosaM. L. Gandarias and M. S. Bruzón, Equivalence transformations and conservation laws for a generalized variable-coefficient Gardner equation, Commun Nonlinear Sci Numer Simulat, 40 (2016), 71-79. doi: 10.1016/j.cnsns.2016.04.009.

[7]

R. de la Rosa and M. S. Bruzón, On the classical and nonclassical symmetries of a generalized Gardner equation, Applied Mathematics and Nonlinear Sciences, 1 (2016), 263-272.

[8]

M. L. Gandarias and N. H. Ibragimov, Equivalence group of a fourth-order evolution equation unifying various non-linear models, Commun Nonlinear Sci Numer Simulat, 13 (2008), 259-268. doi: 10.1016/j.cnsns.2005.12.011.

[9]

N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley & Sons, New York, 1999.

[10]

N. H. Ibragimov, Invariants of a remarkable family of nonlinear equations, Nonlinear Dyn, 30 (2002), 155-166. doi: 10.1023/A:1020406015011.

[11]

A. G. Johnpillai and C. M. Khalique, Gruop analysis of KdV equation with time dependent coefficients, Applied Mathematics and Computation, 216 (2010), 3761-3771. doi: 10.1016/j.amc.2010.05.043.

[12]

A. G. Johnpillai and C. M. Khalique, Conservation laws of KdV equation with time dependent coefficients, Commun Nonlinear Sci Numer Simulat, 16 (2011), 3081-3089. doi: 10.1016/j.cnsns.2010.10.031.

[13]

A. G. Johnpillai and C. M. Khalique, Lie group classification and invariant solutions of mKdV equation with time-dependent coefficients, Commun Nonlinear Sci Numer Simulat, 16 (2011), 1207-1215. doi: 10.1016/j.cnsns.2010.06.025.

[14]

N. A. Kudryashov and D. I. Sinelshchikov, A note on the Lie symmetry analysis and exact solutions for the extended mKdV equation, Acta. Appl. Math., 113 (2011), 41-44. doi: 10.1007/s10440-010-9582-6.

[15]

M. Lakshmanan and K. M. Tamizhmani, Lie-Bäcklund symmetries of certain nonlinear evolution equations under perturbation around their solutions, J. Math. Phys., 26 (1985), 1189-1200. doi: 10.1063/1.526524.

[16]

X. Li and M. Wang, A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with high-order nonlinear terms, Phys. Lett. A, 361 (2007), 115-118. doi: 10.1016/j.physleta.2006.09.022.

[17]

S. Lie, Über Differentialinvarianten, Math. Ann., 24 (1884), 537-578. doi: 10.1007/BF01447449.

[18]

S. Lie, Klassifikation und Integration von gewohnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten Ⅰ, Ⅱ, Math. Ann., 32 (1888), 213-281. doi: 10.1007/BF01444068.

[19]

H. Liu and J. Li, Lie symmetry analysis and exact solutions for the extended mKdV equation, Acta. Appl. Math., 109 (2010), 1107-1119. doi: 10.1007/s10440-008-9362-8.

[20]

R. M. MiuraC. S. Gardner and M. D. Kruskal, Korteweg-de Vries equation and generalizations. Ⅱ. Existence of conservation laws and constants of motion, J. Math. Phys., 9 (1968), 1204-1209. doi: 10.1063/1.1664701.

[21]

R. M. Miura, The Korteweg-de Vries equation: A survey of results, SIAM Review, 18 (1976), 412-459. doi: 10.1137/1018076.

[22]

M. Molati and M. P. Ramollo, Symmetry classification of the Gardner equation with time-dependent coefficients arising in stratified fluids, Commun Nonlinear Sci Numer Simulat, 17 (2012), 1542-1548. doi: 10.1016/j.cnsns.2011.09.002.

[23]

P. J. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511609565.

[24]

L. V. Ovsyannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982.

[25]

R. O. Popovych and N. M. Ivanova, New results on group classification of nonlinear diffusion-convection equations, J. Phys. A: Math. Gen., 37 (2004), 7547-7565. doi: 10.1088/0305-4470/37/30/011.

[26]

M. RosaM. S. Bruzón and M. L. Gandarias, Lie symmetries and conservation laws of a Fisher equation with nonlinear convection term, Discrete and Continuous Dynamical Systems -Series S, 8 (2015), 1331-1339. doi: 10.3934/dcdss.2015.8.1331.

[27]

M. SenthilvelanM. Torrisi and A. Valenti, Equivalence transformations and differential invariants of a generalized nonlinear Schrödinger equation, J. Phys. A: Math. Gen., 39 (2006), 3703-3713. doi: 10.1088/0305-4470/39/14/013.

[28]

C. Sophocleous and R. Traciná, Differential invariants for quasi-linear and semi-linear wave-type equations, Applied Mathematics and Computation, 202 (2008), 216-228. doi: 10.1016/j.amc.2008.01.033.

[29]

M. TorrisiR. Traciná and A. Valenti, On the linearization of semilinear wave equations, Nonlinear Dyn, 36 (2004), 97-106. doi: 10.1023/B:NODY.0000034649.74389.09.

[30]

R. Traciná, Invariants of a family of nonlinear wave equations, Commun Nonlinear Sci Numer Simulat, 9 (2004), 127-133. doi: 10.1016/S1007-5704(03)00021-2.

[31]

C. Tsaousi and C. Sophocleous, Differential invariants for systems of linear hyperbolic equations, J. Math. Anal. Appl., 363 (2010), 238-248. doi: 10.1016/j.jmaa.2009.08.023.

[32]

C. TsaousiR. Traciná and C. Sophocleous, Differential invariants for third-order evolution equations, Commun Nonlinear Sci Numer Simulat, 20 (2015), 352-359. doi: 10.1016/j.cnsns.2014.05.024.

[33]

O. VaneevaO. Popovych and C. Sophocleous, Equivalence transformations in the study of integrability, Phys. Scr., 89 (2014), 038003-038012. doi: 10.1088/0031-8949/89/03/038003.

[34]

O. VaneevaO. Kuriksha and C. Sophocleous, Enhanced group classification of Gardner equations with time-dependent coefficients, Commun Nonlinear Sci Numer Simulat, 22 (2015), 1243-1251. doi: 10.1016/j.cnsns.2014.09.016.

[35]

E. Yomba, The sub-ODE method for finding exact travelling wave solutions of generalized nonlinear Camassa-Holm, and generalized nonlinear Schrödinger equations, Phys. Lett. A, 372 (2008), 215-222. doi: 10.1016/j.physleta.2007.03.008.

show all references

References:
[1]

K. R. Adem and C. M. Khalique, Exact solutions and conservation laws of Zakharov-Kuznetsov modified equal width equation with power law nonlinearity, Nonlinear Anal. RWA, 13 (2012), 1692-1702. doi: 10.1016/j.nonrwa.2011.12.001.

[2]

Y. Y. Bagderina, Invariants of a family of third-order ordinary differential equations, J. Phys. A: Math. Theor. , 42 (2009), 085204, 21pp. doi: 10.1088/1751-8113/42/8/085204.

[3]

Y. Y. Bagderina, Symmetries and invariants of the systems of two linear second-order ordinary differential equations, Commun Nonlinear Sci Numer Simulat, 19 (2014), 3513-3522. doi: 10.1016/j.cnsns.2014.02.023.

[4]

M. S. BruzónM. L. Gandarias and M. Senthilvelan, On the nonlocal symmetries of certain nonlinear oscillators and their general solution, Physics Letters A, 375 (2011), 2985-2987. doi: 10.1016/j.physleta.2011.06.036.

[5]

R. de la RosaM. L. Gandarias and M. S. Bruzón, Symmetries and conservation laws of a fifth-order KdV equation with time-dependent coefficients and linear damping, Nonlinear Dyn, 84 (2016), 135-141. doi: 10.1007/s11071-015-2254-3.

[6]

R. de la RosaM. L. Gandarias and M. S. Bruzón, Equivalence transformations and conservation laws for a generalized variable-coefficient Gardner equation, Commun Nonlinear Sci Numer Simulat, 40 (2016), 71-79. doi: 10.1016/j.cnsns.2016.04.009.

[7]

R. de la Rosa and M. S. Bruzón, On the classical and nonclassical symmetries of a generalized Gardner equation, Applied Mathematics and Nonlinear Sciences, 1 (2016), 263-272.

[8]

M. L. Gandarias and N. H. Ibragimov, Equivalence group of a fourth-order evolution equation unifying various non-linear models, Commun Nonlinear Sci Numer Simulat, 13 (2008), 259-268. doi: 10.1016/j.cnsns.2005.12.011.

[9]

N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley & Sons, New York, 1999.

[10]

N. H. Ibragimov, Invariants of a remarkable family of nonlinear equations, Nonlinear Dyn, 30 (2002), 155-166. doi: 10.1023/A:1020406015011.

[11]

A. G. Johnpillai and C. M. Khalique, Gruop analysis of KdV equation with time dependent coefficients, Applied Mathematics and Computation, 216 (2010), 3761-3771. doi: 10.1016/j.amc.2010.05.043.

[12]

A. G. Johnpillai and C. M. Khalique, Conservation laws of KdV equation with time dependent coefficients, Commun Nonlinear Sci Numer Simulat, 16 (2011), 3081-3089. doi: 10.1016/j.cnsns.2010.10.031.

[13]

A. G. Johnpillai and C. M. Khalique, Lie group classification and invariant solutions of mKdV equation with time-dependent coefficients, Commun Nonlinear Sci Numer Simulat, 16 (2011), 1207-1215. doi: 10.1016/j.cnsns.2010.06.025.

[14]

N. A. Kudryashov and D. I. Sinelshchikov, A note on the Lie symmetry analysis and exact solutions for the extended mKdV equation, Acta. Appl. Math., 113 (2011), 41-44. doi: 10.1007/s10440-010-9582-6.

[15]

M. Lakshmanan and K. M. Tamizhmani, Lie-Bäcklund symmetries of certain nonlinear evolution equations under perturbation around their solutions, J. Math. Phys., 26 (1985), 1189-1200. doi: 10.1063/1.526524.

[16]

X. Li and M. Wang, A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with high-order nonlinear terms, Phys. Lett. A, 361 (2007), 115-118. doi: 10.1016/j.physleta.2006.09.022.

[17]

S. Lie, Über Differentialinvarianten, Math. Ann., 24 (1884), 537-578. doi: 10.1007/BF01447449.

[18]

S. Lie, Klassifikation und Integration von gewohnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten Ⅰ, Ⅱ, Math. Ann., 32 (1888), 213-281. doi: 10.1007/BF01444068.

[19]

H. Liu and J. Li, Lie symmetry analysis and exact solutions for the extended mKdV equation, Acta. Appl. Math., 109 (2010), 1107-1119. doi: 10.1007/s10440-008-9362-8.

[20]

R. M. MiuraC. S. Gardner and M. D. Kruskal, Korteweg-de Vries equation and generalizations. Ⅱ. Existence of conservation laws and constants of motion, J. Math. Phys., 9 (1968), 1204-1209. doi: 10.1063/1.1664701.

[21]

R. M. Miura, The Korteweg-de Vries equation: A survey of results, SIAM Review, 18 (1976), 412-459. doi: 10.1137/1018076.

[22]

M. Molati and M. P. Ramollo, Symmetry classification of the Gardner equation with time-dependent coefficients arising in stratified fluids, Commun Nonlinear Sci Numer Simulat, 17 (2012), 1542-1548. doi: 10.1016/j.cnsns.2011.09.002.

[23]

P. J. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511609565.

[24]

L. V. Ovsyannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982.

[25]

R. O. Popovych and N. M. Ivanova, New results on group classification of nonlinear diffusion-convection equations, J. Phys. A: Math. Gen., 37 (2004), 7547-7565. doi: 10.1088/0305-4470/37/30/011.

[26]

M. RosaM. S. Bruzón and M. L. Gandarias, Lie symmetries and conservation laws of a Fisher equation with nonlinear convection term, Discrete and Continuous Dynamical Systems -Series S, 8 (2015), 1331-1339. doi: 10.3934/dcdss.2015.8.1331.

[27]

M. SenthilvelanM. Torrisi and A. Valenti, Equivalence transformations and differential invariants of a generalized nonlinear Schrödinger equation, J. Phys. A: Math. Gen., 39 (2006), 3703-3713. doi: 10.1088/0305-4470/39/14/013.

[28]

C. Sophocleous and R. Traciná, Differential invariants for quasi-linear and semi-linear wave-type equations, Applied Mathematics and Computation, 202 (2008), 216-228. doi: 10.1016/j.amc.2008.01.033.

[29]

M. TorrisiR. Traciná and A. Valenti, On the linearization of semilinear wave equations, Nonlinear Dyn, 36 (2004), 97-106. doi: 10.1023/B:NODY.0000034649.74389.09.

[30]

R. Traciná, Invariants of a family of nonlinear wave equations, Commun Nonlinear Sci Numer Simulat, 9 (2004), 127-133. doi: 10.1016/S1007-5704(03)00021-2.

[31]

C. Tsaousi and C. Sophocleous, Differential invariants for systems of linear hyperbolic equations, J. Math. Anal. Appl., 363 (2010), 238-248. doi: 10.1016/j.jmaa.2009.08.023.

[32]

C. TsaousiR. Traciná and C. Sophocleous, Differential invariants for third-order evolution equations, Commun Nonlinear Sci Numer Simulat, 20 (2015), 352-359. doi: 10.1016/j.cnsns.2014.05.024.

[33]

O. VaneevaO. Popovych and C. Sophocleous, Equivalence transformations in the study of integrability, Phys. Scr., 89 (2014), 038003-038012. doi: 10.1088/0031-8949/89/03/038003.

[34]

O. VaneevaO. Kuriksha and C. Sophocleous, Enhanced group classification of Gardner equations with time-dependent coefficients, Commun Nonlinear Sci Numer Simulat, 22 (2015), 1243-1251. doi: 10.1016/j.cnsns.2014.09.016.

[35]

E. Yomba, The sub-ODE method for finding exact travelling wave solutions of generalized nonlinear Camassa-Holm, and generalized nonlinear Schrödinger equations, Phys. Lett. A, 372 (2008), 215-222. doi: 10.1016/j.physleta.2007.03.008.

Table 1.  Solutions of equation (31), $\Delta=\beta^2-4\alpha \gamma$, $\epsilon=\pm 1$.
$j$ $F_j(z)$
$1$ $\frac{2 \epsilon \alpha\,\mbox{sech}(\sqrt{\alpha}z)}{\sqrt{\Delta}- \epsilon \beta \mbox{sech}(\sqrt{\alpha}z)}, \, \alpha>0, \Delta>0$
$2$ $\frac{2 \epsilon \alpha\,\mbox{csch}(\sqrt{\alpha}z)}{\sqrt{-\Delta}-\epsilon \beta \mbox{csch}(\sqrt{\alpha}z)}, \, \alpha>0, \Delta <0$
$3$ $-\frac{\alpha}{\beta}\left[ 1\pm \mbox{tanh}(\frac{\sqrt{\alpha}}{2}z)\right], \, \alpha>0, \Delta =0$
$4$ $-\frac{\alpha}{\beta}\left[ 1\pm \mbox{coth}(\frac{\sqrt{\alpha}}{2}z)\right], \, \alpha>0, \Delta =0$
$j$ $F_j(z)$
$1$ $\frac{2 \epsilon \alpha\,\mbox{sech}(\sqrt{\alpha}z)}{\sqrt{\Delta}- \epsilon \beta \mbox{sech}(\sqrt{\alpha}z)}, \, \alpha>0, \Delta>0$
$2$ $\frac{2 \epsilon \alpha\,\mbox{csch}(\sqrt{\alpha}z)}{\sqrt{-\Delta}-\epsilon \beta \mbox{csch}(\sqrt{\alpha}z)}, \, \alpha>0, \Delta <0$
$3$ $-\frac{\alpha}{\beta}\left[ 1\pm \mbox{tanh}(\frac{\sqrt{\alpha}}{2}z)\right], \, \alpha>0, \Delta =0$
$4$ $-\frac{\alpha}{\beta}\left[ 1\pm \mbox{coth}(\frac{\sqrt{\alpha}}{2}z)\right], \, \alpha>0, \Delta =0$
Table 2.  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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