Exact soliton solutions of a \(D\)-dimensional nonlinear Schrödinger equation with damping and diffusive terms. (English) Zbl 1258.35181

Helal, M. A.; Seadawy, A. R.

Z. Angew. Math. Phys. 62, No. 5, 839-847 (2011).

Summary: In the present study, we apply function transformation methods to the \(D\)-dimensional nonlinear Schrödinger (NLS) equation with damping and diffusive terms. As special cases, this method applies to the sine-Gordon, sinh-Gordon, and other equations. Also, the results show that these equations depend on only one function that can be obtained analytically by solving an ordinary differential equation. Furthermore, certain exact solutions of these three equations are shown to lead to the exact soliton solutions of a \(D\)-dimensional NLS equation with damping and diffusive terms. Finally, our results imply that the planar solitons, \(N\) multiple solitons, propagational breathers, and quadric solitons are solutions to the sine-Gordon, sinh-Gordon, and \(D\)-dimensional NLS equations.

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35C08	Soliton solutions

Keywords:

soliton solutions; NLS equation; sine-Gordon equation; sinh-Gordon equation

References:

[1]	Helal M.A.: Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics. Chaos Solitons Fractals 13, 1917–1929 (2002) · Zbl 0997.35063
[2]	Dodd R.K., Eilbeck J.C., Gibbon J.D., Morris H.C.: Solitons and Nonlinear Wave Equations. Academic Press, London (1984) · Zbl 0496.35001
[3]	Zabusky N.J., Galvin C.J.: Shallow water waves, the KdV equation and solitons. J. Fluid Mech. 47, 811–824 (1971)
[4]	Zabusky N.J., Kruskal M.D.: Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett 15, 240–243 (1965) · Zbl 1201.35174
[5]	Wadati M.: Introduction to solitons. Pramana J. Phys 57, 841–847 (2001)
[6]	Lamb G.L.: Elements of Soliton Theory. Wiley, New York (1980) · Zbl 0445.35001
[7]	Ablowitz M.J., Clarkson P.A.: Solitons, Non-Linear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991) · Zbl 0762.35001
[8]	Novikov S., Manakov S., Pitaevskii L.B., Zakharov V.E.: Theory of Solitons. Plenum, New York (1984)
[9]	Freitas S.D., de Oliveira J.R.: A variational approach in the dissipative cubic-quintic nonlinear Schrödinger equation. Mod. Phys. Lett. B 16, 27–32 (2002) · Zbl 1079.78526
[10]	Fibich G.: Self-focusing in the damped nonlinear Schrödinger equation. SIAM J. Appl. Math. 61, 1680–1705 (2001) · Zbl 0987.78018
[11]	Agrawal G.P.: Nonlinear Fiber Optics. Academic, San Diego (1989)
[12]	Zakharov V.E., Shabat A.B.: Exact theory of two dimensional self- focusing and one dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69 (1972)
[13]	Hasegawa A., Tappert F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl. Phys. Lett. 23, 142–145 (1973)
[14]	Abdullaev F.: Theory of Solitons in Inhomogeneous Media. Wiley, New York (1994) · Zbl 0811.35043
[15]	Gardner C.S., Greene J.M., Kruskal M.D., Miura R.M.: Methods for solving the KdV equation. Phys. Rev. Lett 19, 1095–1097 (1967) · Zbl 1103.35360
[16]	Bullough R.K., Caudrey P.J.: Solitons. Springer, Heidelberg (1980)
[17]	Kaup D.J., Malomed B.A.: Variational principle for the Zakharov- Shabat equations. Physica D 84, 319–328 (1995) · Zbl 1194.35415
[18]	Tsurumi T., Wadati M.: Instability of the Bose-Einstein condensate under magnetic trap. J. Phys. Soc. Jpn. 66, 3035–3039 (1997) · Zbl 0967.76618
[19]	Tsurumi T., Wadati M.: Stability of the D-dimensional nonlinear Schrödinger equation under confined potential. J. Phys. Soc. Jpn. 68, 1531–1536 (1999) · Zbl 0974.35117
[20]	Zakharov V.E.: Collapse of langmuir waves. Sov. Phys. JETP 35, 908–912 (1972)
[21]	Zakharov V.E., Synakh V.S.: The nature of the self-focusing singularity. Sov. Phys. JETP 41, 465–470 (1975)
[22]	Derrick G.H.: Comments on Nonlinear Wave Equations as Models for Elementary Particles. J. Math. Phys 5, 1252–1254 (1964)
[23]	Gibbon J.D., Freeman N.C., Johnson R.S.: Correspondence between the classical {\(\lambda\)}{\(\phi\)} 4, double and single sine-gordon equations for three-dimensional solitons. Phys. Lett. A 65, 380–382 (1978)
[24]	Gibbon J.D., Freeman N.C., Davery A.: Three-dimensional multiple soliton-like solutions of non-linear Klein-Gordon equations. J. Phys. A: Math. Gen. L 11, L93–L96 (1978)
[25]	Wenhua H.: General soliton solutions of an n-dimensional nonlinear Schrödinger equation. J. Phys. A: Math. Gen. 25, L515–L520 (1992) · Zbl 0754.35157
[26]	Khater A.H., Callebaut D.K., El-Kallawy O.H.: Bäcklund transformations and exact soliton solutions for nonlinear Schrödinger-type equations. Nuovo Cimento B 113, 1121–1130 (1998)
[27]	Khater A.H., Helal M.A., Seadawy A.R.: General soliton solutions of a n-dimensional nonlinear Schrödinger equation. IL Nuovo Cimento B 115, 1303–1311 (2000)