# zbMATH — the first resource for mathematics

Exact solutions of the equation for surface waves in a convecting fluid. (English) Zbl 1428.35454
Summary: A method for finding exact solutions and the first integrals is presented. The basic idea of the method is to use the value of the Fuchs index that appears in the Painlevé test to construct the auxiliary equation for finding the first integrals and exact solutions of nonlinear differential equations. It allows us to obtain the first integrals and new exact solutions of some nonlinear ordinary differential equations. The main feature of the method is that we do not assign a solution function at the beginning, we find this function during calculations. This approach is conceptually equivalent to the third step of the Painlevé test and sometimes allows us to change this step. Our approach generalizes a number of other methods for finding exact solutions of nonlinear differential equations. We demonstrate a method for finding the traveling wave solutions and the first integrals of the well-known nonlinear evolution equation for description of surface waves in a convecting liquid. The general solution of this equation at some conditions on parameters and new traveling wave solutions of the fourth-order equation are found.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35C05 Solutions to PDEs in closed form 35C07 Traveling wave solutions
Full Text:
##### References:
  Kudryashov, N. A., Exact soliton solutions of the generalized evolution equation of wave dynamics, J. Appl. Math. Mech., 52, 3, 360-365 (1988)  Kudryashov, N. A., Exact solutions of the generalized Kuramoto-Sivashinsky equation, Phys. Lett. A., 147, 287-291 (1990)  Kudryashov, N. A., On types of nonlinear nonintegrable equations with exact solutions, Phys. Lett. A, 155, 269-275 (1991)  Parkes, E. J.; Duffy, B. R., An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. Phys. Commun., 98, 288-300 (1996) · Zbl 0948.76595  Malfliet, W.; Hereman, W., The tanh method: I exact solutions of nonlinear evolution and wave equations, Phys. Scripta, 54, 563-568 (1996) · Zbl 0942.35034  Fan, E., Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 227, 4-5, 212-218 (2000) · Zbl 1167.35331  Polyanin, A. D.; Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations (2003), Boca Ration, Chapman and Hall/CRC · Zbl 1015.34001  Fu, Z. T.; Liu, S. K.; Liu, S. D., New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. Lett. A, 290, 1-2, 72-76 (2001) · Zbl 0977.35094  Fu, Z. T.; Liu, S. K.; Liu, S. D., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289, 1-2, 69-74 (2001) · Zbl 0972.35062  Kudryashov, N. A., Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos Soliton Fractals, 24, 1217-1231 (2005) · Zbl 1069.35018  Kudryashov, N. A., Exact solitary waves of the fisher equations, Phys. Lett. A., 342, 99-106 (2005) · Zbl 1222.35054  Vitanov, N. R., Application of simplest equations of bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity, Commun. Nonlinear Sci. Numer. Simulat., 15, 2050-2060 (2010) · Zbl 1222.35062  Wang, M. L.; Li, X.; Zhang, J., The g′/g - expansion method and evolution equations in mathematical physics, Phys. Lett. A, 372, 417-421 (2008)  Zhang, J.; Wei, X.; Lu, Y., A generalized g′/g - expansion method and its applications, Phys. Lett. A, 372, 3653-3658 (2008) · Zbl 1220.37070  Kudryashov, N. A., A note on the g’/g-expansion method, Appl. Math. Comput., 217, 1755-1758 (2010) · Zbl 1203.35228  Kudryashov, N. A.; Loguinova, N. B., Extended simplest equation method for nonlinear differential equations, Appl. Math. Comput., 205, 396-402 (2008) · Zbl 1168.34003  Biswas, A., Solitary wave solution for the generalized Kawahara equation, Appl. Math. Lett., 22, 208-210 (2009) · Zbl 1163.35468  Kabir, M. M.; Khajeh, A.; Aghdam, E. A.; Koma, A. Y., Modified Kudryashov method for finding exact solitary wave solutions of higher-order nonlinear equations, Math. Methods Appl. Sci., 34, 2, 213-219 (2011) · Zbl 1206.35063  Ryabov, P. N.; Sinelshchikov, D. I.; Kochanov, M. B., Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations, Appl. Math. Comput., 218, 7, 3965-3972 (2011) · Zbl 1246.35015  Kudryashov, N. A., One method for finding exact solutions of nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simulat., 17, 2248-2253 (2012) · Zbl 1250.35055  Kudryashov, N. A.; Loguinova, N. B., Be careful with exp - function method, Commun. Nonlinear Sci. Numer. Simulat., 14, 1881-1890 (2009) · Zbl 1221.35344  Kudryashov, N. A., On “new traveling wave solutions” of the Kdv and the Kdv—burgers equations, Nonlinear Sci. Numer. Simulat., 14, 1891-1900 (2009) · Zbl 1221.35343  Kudryashov, N. A., Seven common errors in finding exact solutions of nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simulat., 14, 3507-3529 (2009) · Zbl 1221.35342  Kudryashov, N. A., Exact solutions and the first integral of the duffing-van der Pol equation, Regul. Chaot. Dyn., 23, 4 (2018)  Weiss, J., The Painleve property for partial differential equations. II: Backlund transformation, lax pairs, and the Schwarzian derivative, J. Math. Phys., 24, 1405-1413 (1983) · Zbl 0531.35069  Kudryashov, N. A., From singular manifold equations to integrable evolution equations, J. Phys. A Math. Gen., 27, 2457-2470 (1994) · Zbl 0839.35119  Nayfe, A. H.; Mook, D. T., Nonlinear Oscillations (1995), Wiley: Wiley New York  Kudryashov, N. A.; Zaharchenko, A. S., A note on solutions of the generalized fisher equation, Appl. Math. Lett., 32, 53-56 (2014) · Zbl 1327.35165  Polyanin, A. D.; Zaitsev, V. F., Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems (2018), CRC Press: CRC Press Boca RatonLondon · Zbl 1419.34001  Alfaro, C. M.; Depassier, M. C., Solitary waves in a shallow viscous fluid sustained by an adverse temperature gradient, Phys. Rev. Lett., 62, 22, 2597-2599 (1989)  Aspe, H.; Depassier, M. C., Evolution equation of surface waves in a convecting fluid, Phys. Rev. A, 22, 19, 4135-4142 (1990)  Garazo, A. N.; Velarde, M. G., Dissipative Korteweg-de vries description of Maragoni-Benard oscillatory convection, Phys. Fluids A Fluid Dyn., 3, 2295-2300 (1991) · Zbl 0745.76077  Kliakhander, I. L.; Porubov, A. V.; Velarde, M. G., Localized finite-amplitude disturbances and selection of solitary waves, Phys. Rev. E., 62, 4, 4959-4962 (2000)  Porubov, A. V., Exact travelling wave solutions of nonlinear evolution of surface waves in a convecting fluid, J. Phys. A. Math. Gen., 26, L797-L800 (1993) · Zbl 0844.76040  Porubov, A. V., Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid layer, Phys. Lett. A, 221, 391-394 (1996) · Zbl 0972.76546  Kudryashov, N. A., Painlevé analysis and exact solutions of the fourth-order equation for description of nonlinear waves, Commun. Nonlinear Sci. Numer. Simulat., 28, 1-9 (2015)  Cerverȯ, J. M.; Zurrȯn, O., Integrability of the perturbed Kdv equation for convecting fluids: Symmetry analysis and solutions, J. Nonlinear Math. Phys., 3, 1-2, 1-23 (1996) · Zbl 1044.35518  Abourabia, A. M.; Soliman, I. M., Verification of the physical validation of the solutions of the perturbed Kdv equation for connective fluids, Eur. Phys. J. Plus, 130, 130:159 (2015)  Conejo-Pérez, O.; Rosu, H. C., Solutions of the perturbed Kdv equation for convecting fluids by factorizations, Cent. Eur. J. Phys., 9, 4, 523-526 (2010)  Lou, S.-y.; Huang, G.-x.; Ruan, H.-y., Exact solitary waves in a convecting fluid, J. Phys. A. Math. Gen., 24, L587-L590 (1991) · Zbl 0735.76057  Lou, S.-y.; Zhu, Y. J.; Wang, L. Y., Exact solitary wave solutions of surface waves in a convecting fluids, Commun. Theor. Phys., 32, 563-566 (1999)  Kudryashov, N. A., Solitary and periodic solutions of the generalized Kuramoto-Sivashinsky equation, Regul. Chaot. Dyn., 13, 3, 234-238 (2008) · Zbl 1229.35229  Kudryashov, N. A., Fuchs indices and the first integrals of nonlinear differential equations, Chaos Solit. Fractals, 26, 2, 591-603 (2005) · Zbl 1076.34500  Kudryashov, N. A., Fourth-order analogies to the Painlevé equations, J. Phys. A. Math. Gen., 35, 4617-4632 (2002) · Zbl 1066.34086  Kudryashov, N. A., Asymptotic and exact solutions of the Fitzhugh-Nagumo model, Regul. Chaot. Dyn., 23, 152-160 (2018) · Zbl 1401.34004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.