# American Institute of Mathematical Sciences

August  2018, 11(4): 577-582. doi: 10.3934/dcdss.2018032

## Exact solutions of nonlinear partial differential equations

 1 Department of Electrical and Systems Engineering, Washington University, St. Louis, MO 63130, USA

* Corresponding author: Barbara Abraham-Shrauner

Received  November 2016 Revised  May 2017 Published  November 2017

Tests for determination of which nonlinear partial differential equations may have exact analytic nonlinear solutions of any of two types of hyperbolic functions or any of three types of Jacobian elliptic functions are presented. The Power Index Method is the principal method employed that extends the calculation of the power index for the most nonlinear terms to all terms in the nonlinear partial differential equations. An additional test is the identification of the net order of differentiation of each term in the nonlinear differential equations. The nonlinear differential equations considered are evolution equations. The tests extend the homogeneous balance condition that is necessary to conditions that may only be sufficient but are very simple to apply. Superposition of Jacobian elliptic functions is also presented with the introduction of a new basis that simplifies the calculations.

Citation: Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032
##### References:
 [1] B. Abraham-Shrauner, Comment on Comment on "Superposition of elliptic functions as solutions for a large number of nonlinear equations" J. Math. Phys., 56 (2015), 112101, 2 pages. doi: 10.1063/1.4936075. [2] W. F. Ames, Ad-hoc Exact Technique for Nonlinear Partial Differential Equations, in Nonlinear Partial Differential Equations, (ed. W.F. Ames) Academic Press, New York, (1967),68-72. [3] D. Baldwin, U. Göktas, W. Hereman, L. Hong, R. S. Martino and J. C. Miller, Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs, J. Symbolic Comp., 37 (2004), 669-705.  doi: 10.1016/j.jsc.2003.09.004. [4] H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, New York, 1962. [5] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000), 212-218.  doi: 10.1016/S0375-9601(00)00725-8. [6] W. Hereman, P. Banerjee, A. Korpel, G. Assanto, A. Van Immerzeele and A. Meerpoel, Exact solitary wave solutions of nonlinear evolution and wave equations using a new direct algebraic method, J. Phys. A: Math. Gen., 19 (1986), 607-628.  doi: 10.1088/0305-4470/19/5/016. [7] F. T. Hioe, Superposition solutions of coupled nonlinear equations, Phys. Lett. A, 234 (1997), 351-357.  doi: 10.1016/S0375-9601(97)00609-9. [8] L. Huibin and W. Kelin, Exact solutions for two nonlinear equations: I, J. Phys. A: Math. Gen., 23 (1990), 3923-3928.  doi: 10.1088/0305-4470/23/17/021. [9] A. Karczewska, P. Rozmej and E. Infeld, Shallow-water solitons dynamics beyond the Korteweg-de Vries equation, Phys. Rev. E, 90 (2014), 012907, 8 pages. [10] A. Khare and U. Sukhatme, Linear superposition in nonlinear equations Phys. Rev. Lett., 88 (2002), 244101, 4 pages. doi: 10.1103/PhysRevLett.88.244101. [11] A. Khare and A. Saxena, Linear superposition for a class of nonlinear equations, Phys. Lett. A, 377 (2013), 2761-2765.  doi: 10.1016/j.physleta.2013.08.015. [12] A. Khare and A. Saxena, Superposition of elliptic functions as solutions for a large number of nonlinear equations J. Math. Phys., 55 (2014), 032701, 25 pages. doi: 10.1063/1.4866781. [13] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys., 60 (1992), 650-654.  doi: 10.1119/1.17120. [14] F. Verheest, C. P. Olivier and W. A. Hereman, Modified Korteweg-de Vries solitons at supercritical densities in two electron temperature plasmas, J. Plasma Phys., 82 (2016), 905820208, 13 pages. doi: 10.1017/S0022377816000349. [15] M. Wang, Y. Zhou and Z. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A, 216 (1996), 67-75.  doi: 10.1016/0375-9601(96)00283-6. [16] Y. M. Zhao, New Jacobi elliptic function solutions for the Zakharov equations, J. Appl. Math., 2012 (2012), Art. ID 854619, 16 pp.

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##### References:
 [1] B. Abraham-Shrauner, Comment on Comment on "Superposition of elliptic functions as solutions for a large number of nonlinear equations" J. Math. Phys., 56 (2015), 112101, 2 pages. doi: 10.1063/1.4936075. [2] W. F. Ames, Ad-hoc Exact Technique for Nonlinear Partial Differential Equations, in Nonlinear Partial Differential Equations, (ed. W.F. Ames) Academic Press, New York, (1967),68-72. [3] D. Baldwin, U. Göktas, W. Hereman, L. Hong, R. S. Martino and J. C. Miller, Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs, J. Symbolic Comp., 37 (2004), 669-705.  doi: 10.1016/j.jsc.2003.09.004. [4] H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, New York, 1962. [5] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000), 212-218.  doi: 10.1016/S0375-9601(00)00725-8. [6] W. Hereman, P. Banerjee, A. Korpel, G. Assanto, A. Van Immerzeele and A. Meerpoel, Exact solitary wave solutions of nonlinear evolution and wave equations using a new direct algebraic method, J. Phys. A: Math. Gen., 19 (1986), 607-628.  doi: 10.1088/0305-4470/19/5/016. [7] F. T. Hioe, Superposition solutions of coupled nonlinear equations, Phys. Lett. A, 234 (1997), 351-357.  doi: 10.1016/S0375-9601(97)00609-9. [8] L. Huibin and W. Kelin, Exact solutions for two nonlinear equations: I, J. Phys. A: Math. Gen., 23 (1990), 3923-3928.  doi: 10.1088/0305-4470/23/17/021. [9] A. Karczewska, P. Rozmej and E. Infeld, Shallow-water solitons dynamics beyond the Korteweg-de Vries equation, Phys. Rev. E, 90 (2014), 012907, 8 pages. [10] A. Khare and U. Sukhatme, Linear superposition in nonlinear equations Phys. Rev. Lett., 88 (2002), 244101, 4 pages. doi: 10.1103/PhysRevLett.88.244101. [11] A. Khare and A. Saxena, Linear superposition for a class of nonlinear equations, Phys. Lett. A, 377 (2013), 2761-2765.  doi: 10.1016/j.physleta.2013.08.015. [12] A. Khare and A. Saxena, Superposition of elliptic functions as solutions for a large number of nonlinear equations J. Math. Phys., 55 (2014), 032701, 25 pages. doi: 10.1063/1.4866781. [13] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys., 60 (1992), 650-654.  doi: 10.1119/1.17120. [14] F. Verheest, C. P. Olivier and W. A. Hereman, Modified Korteweg-de Vries solitons at supercritical densities in two electron temperature plasmas, J. Plasma Phys., 82 (2016), 905820208, 13 pages. doi: 10.1017/S0022377816000349. [15] M. Wang, Y. Zhou and Z. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A, 216 (1996), 67-75.  doi: 10.1016/0375-9601(96)00283-6. [16] Y. M. Zhao, New Jacobi elliptic function solutions for the Zakharov equations, J. Appl. Math., 2012 (2012), Art. ID 854619, 16 pp.
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