The auxiliary equation method proposed by Sirendaoreji is extended to construct new types of elliptic function solutions of nonlinear evolution equations. The effectiveness of the extended method is demonstrated by applications to the RKL model, the generalized derivative NLS equation and the Kundu-Eckhaus equation. Not only are the Jacobian elliptic function solutions are derived, but also the solitary wave solutions and trigonometric function solutions are obtained in a unified way.
References
M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, UK, 1991. MR1149378 M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, UK, 1991. MR1149378
C. Rogers and W. F. Shadwick, Bäcklund Transformations and Their Applications, vol. 161 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1982. MR658491 C. Rogers and W. F. Shadwick, Bäcklund Transformations and Their Applications, vol. 161 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1982. MR658491
V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer Series in Nonlinear Dynamics, Springer, Berlin, Germany, 1991. MR1146435 V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer Series in Nonlinear Dynamics, Springer, Berlin, Germany, 1991. MR1146435
R. Hirota, The Direct Method in Soliton Theory, vol. 155 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 2004. MR2085332 R. Hirota, The Direct Method in Soliton Theory, vol. 155 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 2004. MR2085332
J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” Journal of Mathematical Physics, vol. 24, no. 3, pp. 522–526, 1983. MR692140 0531.35069 10.1063/1.525721 J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” Journal of Mathematical Physics, vol. 24, no. 3, pp. 522–526, 1983. MR692140 0531.35069 10.1063/1.525721
S.-Y. Lou, “On the coherent structures of the Nizhnik-Novikov-Veselov equation,” Physics Letters A, vol. 277, no. 2, pp. 94–100, 2000. MR1803708 1273.35264 10.1016/S0375-9601(00)00699-X S.-Y. Lou, “On the coherent structures of the Nizhnik-Novikov-Veselov equation,” Physics Letters A, vol. 277, no. 2, pp. 94–100, 2000. MR1803708 1273.35264 10.1016/S0375-9601(00)00699-X
R. Radha, C. S. Kumar, M. Lakshmanan, and C. R. Gilson, “The collision of multimode dromions and a firewall in the two-component long-wave-short-wave resonance interaction equation,” Journal of Physics A, vol. 42, no. 10, Article ID 102002, 2009. MR2485853 1163.35037 10.1088/1751-8113/42/10/102002 R. Radha, C. S. Kumar, M. Lakshmanan, and C. R. Gilson, “The collision of multimode dromions and a firewall in the two-component long-wave-short-wave resonance interaction equation,” Journal of Physics A, vol. 42, no. 10, Article ID 102002, 2009. MR2485853 1163.35037 10.1088/1751-8113/42/10/102002
G.-Q. Xu and X.-Z. Huang, “New variable separation solutions for two nonlinear evolution equations in higher dimensions,” Chinese Physics Letters, vol. 30, no. 3, Article ID 130202, 2013. G.-Q. Xu and X.-Z. Huang, “New variable separation solutions for two nonlinear evolution equations in higher dimensions,” Chinese Physics Letters, vol. 30, no. 3, Article ID 130202, 2013.
W. Malfliet, “Solitary wave solutions of nonlinear wave equations,” American Journal of Physics, vol. 60, no. 7, pp. 650–654, 1992. MR1174588 1219.35246 10.1119/1.17120 W. Malfliet, “Solitary wave solutions of nonlinear wave equations,” American Journal of Physics, vol. 60, no. 7, pp. 650–654, 1992. MR1174588 1219.35246 10.1119/1.17120
R.-X. Yao and Z.-B. Li, “New exact solutions for three nonlinear evolution equations,” Physics Letters A, vol. 297, no. 3-4, pp. 196–204, 2002. MR1912462 0995.35003 10.1016/S0375-9601(02)00294-3 R.-X. Yao and Z.-B. Li, “New exact solutions for three nonlinear evolution equations,” Physics Letters A, vol. 297, no. 3-4, pp. 196–204, 2002. MR1912462 0995.35003 10.1016/S0375-9601(02)00294-3
Y.-P. Liu and Z.-B. Li, “A Maple package for finding exact solitary wave solutions of coupled nonlinear evolution equations,” Computer Physics Communications, vol. 155, no. 1, pp. 65–76, 2003. MR2001782 1196.37004 10.1016/S0010-4655(03)00347-3 Y.-P. Liu and Z.-B. Li, “A Maple package for finding exact solitary wave solutions of coupled nonlinear evolution equations,” Computer Physics Communications, vol. 155, no. 1, pp. 65–76, 2003. MR2001782 1196.37004 10.1016/S0010-4655(03)00347-3
M. A. Abdou and A. A. Soliman, “New applications of variational iteration method,” Physica D, vol. 211, no. 1-2, pp. 1–8, 2005. MR2200437 1084.35539 10.1016/j.physd.2005.08.002 M. A. Abdou and A. A. Soliman, “New applications of variational iteration method,” Physica D, vol. 211, no. 1-2, pp. 1–8, 2005. MR2200437 1084.35539 10.1016/j.physd.2005.08.002
J.-H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 695–700, 2005. 1072.35502 J.-H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 695–700, 2005. 1072.35502
S. Liu, Z. Fu, S. Liu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Physics Letters A, vol. 289, no. 1-2, pp. 69–74, 2001. MR1862082 0972.35062 10.1016/S0375-9601(01)00580-1 S. Liu, Z. Fu, S. Liu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Physics Letters A, vol. 289, no. 1-2, pp. 69–74, 2001. MR1862082 0972.35062 10.1016/S0375-9601(01)00580-1
G.-Q. Xu and Z.-B. Li, “On the Painlevé integrability, periodic wave solutions and soliton solutions of generalized coupled higher-order nonlinear Schrödinger equations,” Chaos, Solitons and Fractals, vol. 26, no. 5, pp. 1363–1375, 2005. MR2149321 1072.35586 10.1016/j.chaos.2005.04.007 G.-Q. Xu and Z.-B. Li, “On the Painlevé integrability, periodic wave solutions and soliton solutions of generalized coupled higher-order nonlinear Schrödinger equations,” Chaos, Solitons and Fractals, vol. 26, no. 5, pp. 1363–1375, 2005. MR2149321 1072.35586 10.1016/j.chaos.2005.04.007
G.-Q. Xu and Z.-B. Li, “Applications of Jacobi elliptic function expansion method for nonlinear differential-difference equations,” Communications in Theoretical Physics, vol. 43, no. 3, pp. 385–388, 2005. MR2185672 10.1088/0253-6102/43/3/001 G.-Q. Xu and Z.-B. Li, “Applications of Jacobi elliptic function expansion method for nonlinear differential-difference equations,” Communications in Theoretical Physics, vol. 43, no. 3, pp. 385–388, 2005. MR2185672 10.1088/0253-6102/43/3/001
G.-Q. Xu, “New types of exact solutions for the fourth-order dispersive cubic-quintic nonlinear Schrödinger equation,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5967–5971, 2011. MR2770216 1210.35241 10.1016/j.amc.2010.12.008 G.-Q. Xu, “New types of exact solutions for the fourth-order dispersive cubic-quintic nonlinear Schrödinger equation,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5967–5971, 2011. MR2770216 1210.35241 10.1016/j.amc.2010.12.008
Z. Yan, “Periodic, solitary and rational wave solutions of the 3D extended quantum Zakharov-Kuznetsov equation in dense quantum plasmas,” Physics Letters A, vol. 373, no. 29, pp. 2432–2437, 2009. 1231.76362 Z. Yan, “Periodic, solitary and rational wave solutions of the 3D extended quantum Zakharov-Kuznetsov equation in dense quantum plasmas,” Physics Letters A, vol. 373, no. 29, pp. 2432–2437, 2009. 1231.76362
B. Hong, “New exact Jacobi elliptic functions solutions for the generalized coupled Hirota-Satsuma KdV system,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 472–479, 2010. MR2678558 1200.35260 10.1016/j.amc.2010.05.079 B. Hong, “New exact Jacobi elliptic functions solutions for the generalized coupled Hirota-Satsuma KdV system,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 472–479, 2010. MR2678558 1200.35260 10.1016/j.amc.2010.05.079
B. Hong and D. Lu, “New Jacobi elliptic function-like solutions for the general KdV equation with variable coefficients,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1594–1600, 2012. MR2887542 1255.35193 10.1016/j.mcm.2011.10.057 B. Hong and D. Lu, “New Jacobi elliptic function-like solutions for the general KdV equation with variable coefficients,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1594–1600, 2012. MR2887542 1255.35193 10.1016/j.mcm.2011.10.057
Y. Zhou, M. Wang, and Y. Wang, “Periodic wave solutions to a coupled KdV equations with variable coefficients,” Physics Letters A, vol. 308, no. 1, pp. 31–36, 2003. MR1972296 1008.35061 10.1016/S0375-9601(02)01775-9 Y. Zhou, M. Wang, and Y. Wang, “Periodic wave solutions to a coupled KdV equations with variable coefficients,” Physics Letters A, vol. 308, no. 1, pp. 31–36, 2003. MR1972296 1008.35061 10.1016/S0375-9601(02)01775-9
E. Yomba, “The extended F-expansion method and its application for solving the nonlinear wave, CKGZ, GDS, DS and GZ equations,” Physics Letters A, vol. 340, no. 1–4, pp. 149–160, 2005. 1145.35455 E. Yomba, “The extended F-expansion method and its application for solving the nonlinear wave, CKGZ, GDS, DS and GZ equations,” Physics Letters A, vol. 340, no. 1–4, pp. 149–160, 2005. 1145.35455
E. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters A, vol. 277, no. 4-5, pp. 212–218, 2000. MR1827770 1167.35331 10.1016/S0375-9601(00)00725-8 E. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters A, vol. 277, no. 4-5, pp. 212–218, 2000. MR1827770 1167.35331 10.1016/S0375-9601(00)00725-8
E. Fan, “Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method,” Journal of Physics A, vol. 35, no. 32, pp. 6853–6872, 2002. MR1930895 1039.35029 10.1088/0305-4470/35/32/306 E. Fan, “Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method,” Journal of Physics A, vol. 35, no. 32, pp. 6853–6872, 2002. MR1930895 1039.35029 10.1088/0305-4470/35/32/306
G. Xu, “An elliptic equation method and its applications in nonlinear evolution equations,” Chaos, Solitons and Fractals, vol. 29, no. 4, pp. 942–947, 2006. MR2222144 1142.35587 10.1016/j.chaos.2005.08.058 G. Xu, “An elliptic equation method and its applications in nonlinear evolution equations,” Chaos, Solitons and Fractals, vol. 29, no. 4, pp. 942–947, 2006. MR2222144 1142.35587 10.1016/j.chaos.2005.08.058
Y. Chen and Z. Yan, “Weierstrass semi-rational expansion method and new doubly periodic solutions of the generalized Hirota-Satsuma coupled KdV system,” Applied Mathematics and Computation, vol. 177, no. 1, pp. 85–91, 2006. MR2234499 1094.65104 10.1016/j.amc.2005.10.037 Y. Chen and Z. Yan, “Weierstrass semi-rational expansion method and new doubly periodic solutions of the generalized Hirota-Satsuma coupled KdV system,” Applied Mathematics and Computation, vol. 177, no. 1, pp. 85–91, 2006. MR2234499 1094.65104 10.1016/j.amc.2005.10.037
D.-S. Wang and H.-B. Li, “Elliptic equation's new solutions and their applications to two nonlinear partial differential equations,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 762–771, 2007. MR2327751 1118.65379 10.1016/j.amc.2006.10.026 D.-S. Wang and H.-B. Li, “Elliptic equation's new solutions and their applications to two nonlinear partial differential equations,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 762–771, 2007. MR2327751 1118.65379 10.1016/j.amc.2006.10.026
O. P. Layeni, “A new rational auxiliary equation method and exact solutions of a generalized Zakharov system,” Applied Mathematics and Computation, vol. 215, no. 8, pp. 2901–2907, 2009. MR2563406 1180.35445 10.1016/j.amc.2009.09.034 O. P. Layeni, “A new rational auxiliary equation method and exact solutions of a generalized Zakharov system,” Applied Mathematics and Computation, vol. 215, no. 8, pp. 2901–2907, 2009. MR2563406 1180.35445 10.1016/j.amc.2009.09.034
H. Li, K. Wang, and J. Li, “Exact traveling wave solutions for the Benjamin-Bona-Mahony equation by improved Fan sub-equation method,” Applied Mathematical Modelling, vol. 37, no. 14-15, pp. 7644–7652, 2013. MR3089584 10.1016/j.apm.2013.03.027 H. Li, K. Wang, and J. Li, “Exact traveling wave solutions for the Benjamin-Bona-Mahony equation by improved Fan sub-equation method,” Applied Mathematical Modelling, vol. 37, no. 14-15, pp. 7644–7652, 2013. MR3089584 10.1016/j.apm.2013.03.027
S. Sirendaoreji, “A new auxiliary equation and exact travelling wave solutions of nonlinear equations,” Physics Letters A, vol. 356, no. 2, pp. 124–130, 2006. 1160.35527 S. Sirendaoreji, “A new auxiliary equation and exact travelling wave solutions of nonlinear equations,” Physics Letters A, vol. 356, no. 2, pp. 124–130, 2006. 1160.35527
S. Sirendaoreji, “Auxiliary equation method and new solutions of Klein-Gordon equations,” Chaos, Solitons and Fractals, vol. 31, no. 4, pp. 943–950, 2007. MR2262187 1143.35341 10.1016/j.chaos.2005.10.048 S. Sirendaoreji, “Auxiliary equation method and new solutions of Klein-Gordon equations,” Chaos, Solitons and Fractals, vol. 31, no. 4, pp. 943–950, 2007. MR2262187 1143.35341 10.1016/j.chaos.2005.10.048
S. Zhang and T.-C. Xia, “A generalized auxiliary equation method and its application to $(2+1)$-dimensional asymmetric Nizhnik-Novikov-Vesselov equations,” Journal of Physics A, vol. 40, no. 2, pp. 227–248, 2007. MR2303494 10.1088/1751-8113/40/2/003 S. Zhang and T.-C. Xia, “A generalized auxiliary equation method and its application to $(2+1)$-dimensional asymmetric Nizhnik-Novikov-Vesselov equations,” Journal of Physics A, vol. 40, no. 2, pp. 227–248, 2007. MR2303494 10.1088/1751-8113/40/2/003
D.-J. Huang, D.-S. Li, and H.-Q. Zhang, “Explicit and exact travelling wave solutions for the generalized derivative Schrödinger equation,” Chaos, Solitons and Fractals, vol. 31, no. 3, pp. 586–593, 2007. MR2262292 1139.35092 D.-J. Huang, D.-S. Li, and H.-Q. Zhang, “Explicit and exact travelling wave solutions for the generalized derivative Schrödinger equation,” Chaos, Solitons and Fractals, vol. 31, no. 3, pp. 586–593, 2007. MR2262292 1139.35092
E. Yomba, “The sub-ODE method for finding exact travelling wave solutions of generalized nonlinear Camassa-Holm, and generalized nonlinear Schrödinger equations,” Physics Letters A, vol. 372, no. 3, pp. 215–222, 2008. MR2380081 1217.81074 10.1016/j.physleta.2007.03.008 E. Yomba, “The sub-ODE method for finding exact travelling wave solutions of generalized nonlinear Camassa-Holm, and generalized nonlinear Schrödinger equations,” Physics Letters A, vol. 372, no. 3, pp. 215–222, 2008. MR2380081 1217.81074 10.1016/j.physleta.2007.03.008
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, NY, USA, 1992. MR1225604 M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, NY, USA, 1992. MR1225604
R. Radhakrishnan, A. Kundu, and M. Lakshmanan, “Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: integrability and soliton interaction in non-Kerr media,” Physical Review E, vol. 60, no. 3, pp. 3314–3323, 1999. R. Radhakrishnan, A. Kundu, and M. Lakshmanan, “Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: integrability and soliton interaction in non-Kerr media,” Physical Review E, vol. 60, no. 3, pp. 3314–3323, 1999.
W.-P. Hong, “Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic-quintic non-Kerr terms,” Optics Communications, vol. 194, no. 1–3, pp. 217–223, 2001. W.-P. Hong, “Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic-quintic non-Kerr terms,” Optics Communications, vol. 194, no. 1–3, pp. 217–223, 2001.
M. Wang, X. Li, and J. Zhang, “Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation,” Physics Letters A, vol. 363, no. 1-2, pp. 96–101, 2007. MR2287805 1197.81129 10.1016/j.physleta.2006.10.077 M. Wang, X. Li, and J. Zhang, “Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation,” Physics Letters A, vol. 363, no. 1-2, pp. 96–101, 2007. MR2287805 1197.81129 10.1016/j.physleta.2006.10.077
H. Triki and T. R. Taha, “Exact analytic solitary wave solutions for the RKL model,” Mathematics and Computers in Simulation, vol. 80, no. 4, pp. 849–854, 2009. MR2576474 1186.35210 10.1016/j.matcom.2009.08.031 H. Triki and T. R. Taha, “Exact analytic solitary wave solutions for the RKL model,” Mathematics and Computers in Simulation, vol. 80, no. 4, pp. 849–854, 2009. MR2576474 1186.35210 10.1016/j.matcom.2009.08.031
W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations,” Physica D, vol. 56, no. 4, pp. 303–367, 1992. MR1169610 0763.35088 10.1016/0167-2789(92)90175-M W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations,” Physica D, vol. 56, no. 4, pp. 303–367, 1992. MR1169610 0763.35088 10.1016/0167-2789(92)90175-M
A. Kundu, “Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger type equations,” Journal of Mathematical Physics, vol. 25, no. 12, pp. 3433–3438, 1984. MR767547 10.1063/1.526113 A. Kundu, “Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger type equations,” Journal of Mathematical Physics, vol. 25, no. 12, pp. 3433–3438, 1984. MR767547 10.1063/1.526113
A. Kundu, “Exact solutions to higher-order nonlinear equations through gauge transformation,” Physica D, vol. 25, no. 1–3, pp. 399–406, 1987. MR887472 10.1016/0167-2789(87)90113-8 A. Kundu, “Exact solutions to higher-order nonlinear equations through gauge transformation,” Physica D, vol. 25, no. 1–3, pp. 399–406, 1987. MR887472 10.1016/0167-2789(87)90113-8
W. Eckhaus, The Long-Time Behaviour for čommentComment on ref. [45a: We split this reference to ?,45b?]. Please check.Perturbed Wave-Equations and Related Problems}, Preprint no. 404, Department of Mathematics, University of Utrecht, 1985. W. Eckhaus, The Long-Time Behaviour for čommentComment on ref. [45a: We split this reference to ?,45b?]. Please check.Perturbed Wave-Equations and Related Problems}, Preprint no. 404, Department of Mathematics, University of Utrecht, 1985.
W. Eckhaus, “The long-time behaviour for perturbed wave-equations and related problems,” in Trends in Applications of Pure Mathematics to Mechanics, vol. 249 of Lecture Notes in Physics, pp. 168–194, Springer, Berlin, Germany, 1986. MR851805 0629.35085 10.1007/BFb0016391 W. Eckhaus, “The long-time behaviour for perturbed wave-equations and related problems,” in Trends in Applications of Pure Mathematics to Mechanics, vol. 249 of Lecture Notes in Physics, pp. 168–194, Springer, Berlin, Germany, 1986. MR851805 0629.35085 10.1007/BFb0016391
F. Calogero and W. Eckhaus, “Nonlinear evolution equations, rescalings, model PDEs and their integrability. I,” Inverse Problems, vol. 3, no. 2, pp. 229–262, 1987. MR913396 0645.35087 10.1088/0266-5611/3/2/008 F. Calogero and W. Eckhaus, “Nonlinear evolution equations, rescalings, model PDEs and their integrability. I,” Inverse Problems, vol. 3, no. 2, pp. 229–262, 1987. MR913396 0645.35087 10.1088/0266-5611/3/2/008
W.-G. Zhang and W.-X. Ma, “Explicit solitary-wave solutions to generalized Pochhammer-Chree equations,” Applied Mathematics and Mechanics, vol. 20, no. 6, pp. 625–632, 1999. MR1719171 0935.35132 W.-G. Zhang and W.-X. Ma, “Explicit solitary-wave solutions to generalized Pochhammer-Chree equations,” Applied Mathematics and Mechanics, vol. 20, no. 6, pp. 625–632, 1999. MR1719171 0935.35132
H. Sakaguchi and B. A. Malomed, “Stable localized pulses and zigzag stripes in a two-dimensional diffractive-diffusive Ginzburg-Landau equation,” Physica D, vol. 159, no. 1-2, pp. 91–100, 2001. MR1861651 0981.78008 10.1016/S0167-2789(01)00334-7 H. Sakaguchi and B. A. Malomed, “Stable localized pulses and zigzag stripes in a two-dimensional diffractive-diffusive Ginzburg-Landau equation,” Physica D, vol. 159, no. 1-2, pp. 91–100, 2001. MR1861651 0981.78008 10.1016/S0167-2789(01)00334-7
Y. Shang, “Explicit and exact solutions for a generalized long-short wave resonance equations with strong nonlinear term,” Chaos, Solitons and Fractals, vol. 26, no. 2, pp. 527–539, 2005. \endinput MR2143966 Y. Shang, “Explicit and exact solutions for a generalized long-short wave resonance equations with strong nonlinear term,” Chaos, Solitons and Fractals, vol. 26, no. 2, pp. 527–539, 2005. \endinput MR2143966
