# Soliton immersion for nonlinear Schrodinger equation with gravity

### Abstract

One of the developed directions of mathematics is studying of nonlinear differential equations in partial derivatives. Investigation in this area is topical, since the results get the theoretical and practical applications. There are some different approaches for solving of the equations. Methods of the theory of solitons allow to construct the solutions of the nonlinear differential equations in partial derivatives. One of the methods for solving of the equations is the inverse scattering method. The aim of the work is to construct a surface corresponding to a regular onesolitonic solution of the nonlinear Schrodinger equation with gravity in (1+1)-dimension. In this work the nonlinear Schrodinger equation with gravity in (1+1)-dimensions, as well as solitonic immersion in Fokas-Gelfand sense are considered. According to the approach the nonlinear differential equations in (1+1)-dimension are given in the form of zero curvature condition and are compatibility condition of the linear system equations, i.e. Lax representation. In this case there is a surface with immersion function. The surface defined by the immersion function is identified to the surface in three-dimensional space. Surface with coefficients of the first fundamental form corresponding to the regular onesolitonic solution of the nonlinear Schrodinger equation is found by soliton immersion.### References

[1] Ablowitz M.J., Clarkson P.A. Solitons, Non-linear Evolution Equations and Inverse Scattering. - Cambridge: Cambridge University Press, 1992. – 516 p.

[2] Myrzakulov R., Vijayalakshmi S. et all. A (2+1)-dimensional integrable spin model: Geometrical and gauge equivalent counterparts, solitons and localized coherent structures // J. Phys. Lett. A., - 1997. - No233A. - P. 391–396.

[3] Ceyhan O., Fokas A.S., Gurses M. Deformations of surfaces associated with integrable Gauss-Mainardi-Codazzi equations // J. Math. Phys., - 2000. - No4. - P. 2551–2270.

[4] Makhankov V.G., Myrzakuov R. Riemann Problem on a Plane and Nonlinear Schroedinger Equation // In book "Communication of the Joint Institute for Nuclear Research,"Р5-84-742, - Dubna, - 1984, - P. 6.

[5] Zhunussova Zh. Geometrical features of the soliton solution // Proceedings of the 9th ISAAC Congress, Springer, Series: Trends in Mathematics, ISBN 978-3-319-12576-3, - 2015. - P. 671-677.

[6] Zhunussova Zh. Reconstruction of surface corresponding to domain wall solution // Proceeding of the forth International conference "Modern problems of Applied mathematics and information technologies, Al-Khorezmiy 2014, Samarkand, - 2014. - P. 283.

[2] Myrzakulov R., Vijayalakshmi S. et all. A (2+1)-dimensional integrable spin model: Geometrical and gauge equivalent counterparts, solitons and localized coherent structures // J. Phys. Lett. A., - 1997. - No233A. - P. 391–396.

[3] Ceyhan O., Fokas A.S., Gurses M. Deformations of surfaces associated with integrable Gauss-Mainardi-Codazzi equations // J. Math. Phys., - 2000. - No4. - P. 2551–2270.

[4] Makhankov V.G., Myrzakuov R. Riemann Problem on a Plane and Nonlinear Schroedinger Equation // In book "Communication of the Joint Institute for Nuclear Research,"Р5-84-742, - Dubna, - 1984, - P. 6.

[5] Zhunussova Zh. Geometrical features of the soliton solution // Proceedings of the 9th ISAAC Congress, Springer, Series: Trends in Mathematics, ISBN 978-3-319-12576-3, - 2015. - P. 671-677.

[6] Zhunussova Zh. Reconstruction of surface corresponding to domain wall solution // Proceeding of the forth International conference "Modern problems of Applied mathematics and information technologies, Al-Khorezmiy 2014, Samarkand, - 2014. - P. 283.

Published

2018-07-18

How to Cite

ZHUNUSSOVA, Zh. Kh..
Soliton immersion for nonlinear Schrodinger equation with gravity.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 92, n. 4, p. 11-19, july 2018. ISSN 1563-0277. Available at: <http://bm.kaznu.kz/index.php/kaznu/article/view/449>. Date accessed: 10 dec. 2018.
Section

Mathematics

Keywords
nonlinear equation, immersion, surface, solitonic solution, fundamental form, zero curvature condition