×

New Weierstrass elliptic wave solutions of the Davey-Stewartson equation with power law nonlinearity. (English) Zbl 1455.35219

In this paper the author considers the (2+1)-dimensional Davey-Stewartson (DS) equations and obtains some previously known and new solutions through the Weierstrass elliptic function method. The paper is organized as follows. The first section is an introduction to the subject. Section 2 deals with the Weierstrass elliptic function method. In section 3, the author gives some particular travelling wave solutions of the (2+1)-dimensional (DS) equations and restates the main points in section 4. The paper is supported by an appendix containing some properties of Weierstrass elliptic functions. The paper is well documented.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35C07 Traveling wave solutions
33E05 Elliptic functions and integrals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Abramovitz and I. A. Stegun,Handbook of Mathematical Functions, 9th ed., Dover, New York, 1972. · Zbl 0543.33001
[2] N. I. Akhiezer,Elements of the Theory of Elliptic Functions, Transl. Math. Monogr. 79, Amer. Math. Soc., Providence, RI, 1990. · Zbl 0694.33001
[3] A. Bekirand A. C. Cevikel,New solitons and periodic solutions for nonlinear physical models in mathematical physics, Nonlinear Anal. Real World Appl. 11 (2010), 3275- 3285. · Zbl 1196.35178
[4] A. H. Bhrawy, M. A. Abdelkawy and A. Biswas,Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi’s elliptic function method, Comm. Nonlinear Sci. Numer. Simulation 18 (2013), 915-925. · Zbl 1261.35044
[5] M. Boiti, J. J.-P. Leon, L. Martina and F. Pempinelli,Scattering of localized solitons in the plane, Phys. Lett. A 132 (1988), 432-439.
[6] K. Chanderasekharam,Elliptic Functions, Springer, Berlin, 1985. · Zbl 0575.33001
[7] A. Davey and K. Stewartson,On three-dimensional packets of surfaces waves, Proc. Roy. Soc. London Ser. A 338 (1974), 101-110. · Zbl 0282.76008
[8] G. Ebadi and A. Biswas,TheGG0method and 1-soliton solution of the Davey-Stewartson equation, Math. Computer Modelling 53 (2011), 694-698. · Zbl 1217.35171
[9] A. El Achab,Constructing of exact solutions to the nonlinear Schrödinger equation (NLSE) with power-law nonlinearity by the Weierstrass elliptic function method, Optik 127 (2016), 1229-1232.
[10] A. El Achab,Elliptic travelling wave solutions to a generalized boussinesq equation, Abstr. Appl. Anal. 2014, art. 256019, 7 pp. · Zbl 1468.35168
[11] A. El Achab,On the integrability of the generalized Pochhammer-Chree (PC) equations, Phys. A 545 (2020), art. 123576, 9 pp.
[12] A. El Achab,Weierstrass elliptic solutions to a Zakharov equation in plasmas with power law nonlinearity, Appl. Math. (Warsaw) 42 (2015), 13-22. · Zbl 1327.35339
[13] A. El Achab and A. Amine,A construction of new exact periodic wave and solitary wave solutions for the 2D Ginzburg-Landau equation, Nonlinear Dynam. 91 (2018), 995-999.
[14] A. S. Fokas and P. M. Santini,Coherent structures in multidimensions, Phys. Rev. Lett. 63 (1989), 1329-1333.
[15] A. S. Fokas and P. M. Santini,Dromions and a boundary value problem for the Davey-Stewartson 1 equation, Phys. D 44 (1990), 99-130. · Zbl 0707.35144
[16] Y. Gurefe, E. Misirli, Y. Pandir, A. Sonmezoglu and M. Ekici,New exact solutions of the Davey-Stewartson equation with power-law nonlinearity, Bull. Malays. Math. Sci. Soc. 38 (2015), 1223-1234. · Zbl 1320.35145
[17] J. Hietarinta,One-dromion solutions for generic classes of equations, Phys. Lett. A 149 (1990), 113-118.
[18] J. Hietarinta and R. Hirota,Multidromion solutions to the Davey-Stewartson equation, Phys. Lett. A 145 (1990), 237-244.
[19] E. L. Ince,Ordinary Differential Equations, Dover, New York, 1956.
[20] C. G. J. Jacobi,Fundamenta Nova Theoriae Functionum Ellipticarum, Borntraeger, Königsberg, 1829.
[21] H. Jafari and M. Alipour,Solution of the Davey-Stewartson equation using homotopy analysis method, Nonlinear Anal. Model. Control 15 (2010), 423-433. · Zbl 1353.35265
[22] H. Jafari, A. Sooraki, Y. Talebi and A. Biswas,The first integral method and traveling wave solutions to Davey-Stewartson equation, Nonlinear Anal. Model. Control 17 (2012), 182-193. · Zbl 1311.35289
[23] H. Leblond,Spatiotemporal optical pulse control using microwaves, Phys. Rev. Lett. 95 (2005), no. 3, art. 033902, 4 pp.
[24] A. Lesfari,Introduction à la géométrie algébrique complexe, Hermann, Paris, 2015. · Zbl 1327.14001
[25] M. Mirzazadeh,Soliton solutions of Davey-Stewartson equation by trial equation method and ansatz approach, Nonlinear Dynam. 82 (2015), 1775-1780. · Zbl 1348.35049
[26] J. Nickel, V. S. Serov and H. W. Schurmann,Some elliptic travelling wave solutions for Novikov-Veselov equation, Progr. Electromagnetc Res. 61 (2006), 323-331.
[27] R. Radha and M. Lakshmanan,Localized coherent structures and integrability in a generalized (2 + 1)-dimensional nonlinear Schrödinger equation, Chaos Solitons Fractals 8 (1997), 17-25. · Zbl 0918.35129
[28] H. W. Schurmann, V. S. Serov and J. Nickel,Superposition in nonlinear wave and evolution equations, Int. J. Theoret. Phys. 45 (2006), 1057-1073. · Zbl 1101.81054
[29] J. Shi, J. Li and S. Li,Analytical travelling wave solutions and parameter analysis for the (2+1)-dimensional Davey-Stewartson-type equations, Pramana-J. Phys. 81 (2013), 747-762.
[30] K. Weierstrass,Mathematische werke V, Johnson, New York, 1915.
[31] E. T. Whittaker and G. Watson,A Course of Modern Analysis, Cambridge Univ. Press, Cambridge, 1988. · JFM 45.0433.02
[32] H. A. Zedan and S. J. Monaquel,The sine-cosine method for the Davey-Stewartson equations, Appl. Math. E-Notes 10 (2010), 103-111. · Zbl 1219.35216
[33] H. A. Zedan and S. Sh. Tantawy,Solution of Davey-Stewartson equations by homotopy perturbation method, Comput. Math. Math. Phys. 49 (2009), 1382-1388. · Zbl 1199.35359
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.