Generalized Sub-Equation Method for the (1+1)-Dimensional Resonant Nonlinear Schrodinger’s Equation

Abstract

Interest in studying nonlinear models has been increasing in recent years. Dynamical systems, in which the state of the system changes continuously over time, have nonlinear interactions. The use of unique nonlinear differential equations is inescapable in the evaluation of such systems. In mathematical point of view, for obtaining analytical solutions of nonlinear differential equations, it must be fully integrable. Consequently, the importance of fully integrable nonlinear differential equations for nonlinear science has become indisputable. Among these equations, one of the most studied by physicists and mathematicians is the nonlinear Schrödinger equation. This equation has undergone many modifications to evaluate different phenomena. In this study, the resonant nonlinear Schrödinger equation, which is the most important of these physical equations in terms of explaining many physical phenomena, is solved analytically with the generalized sub-equation method.

Keywords

• Generalized Sub-Equation Method
• (1+1)-Dimensional Resonant Nonlinear Schrodinger’s Equation
• Exact Solution

(1+1)-Boyutlu Rezonant Doğrusal Olmayan Schrödinger Denklemi İçin Genelleştirilmiş Alt Denklem Yöntemi

Abstract

Doğrusal olmayan modelleri incelemeye olan ilgi son yıllarda artmaktadır. Sistemin durumunun zaman içinde sürekli olarak değiştiği dinamik sistemler doğrusal olmayan etkileşimlere sahiptir. Bu tür sistemlerin değerlendirilmesinde benzersiz doğrusal olmayan diferansiyel denklemlerin kullanılması kaçınılmazdır. Matematiksel bakış açısına göre, doğrusal olmayan diferansiyel denklemlerin analitik çözümlerini elde etmek için, tamamen integre edilebilir olmalıdır. Sonuç olarak, doğrusal olmayan bilim için tamamen integre edilebilir doğrusal olmayan diferansiyel denklemlerin önemi tartışılmaz hale gelmiştir. Bu denklemler arasında fizikçiler ve matematikçiler tarafından en çok çalışılanlardan biri doğrusal olmayan Schrödinger denklemidir. Bu denklem, farklı olayları değerlendirmek için birçok değişikliğe uğramıştır. Bu çalışmada birçok fiziksel olguyu açıklama açısından bu fiziksel denklemlerin en önemlisi olan rezonans doğrusal olmayan Schrödinger denklemi genelleştirilmiş alt denklem yöntemi ile analitik olarak çözülmüştür.

Keywords

• Genelleştirilmiş Alt Denklem Yöntemi
• (1+1) Boyutlu Resonant Doğrusal Olmayan Schrödinger Denklemi
• Tam Çözüm

References

1. Tahir, M., & Awan, A. U. (2019). The study of complexitons and periodic solitary-wave solutions with fifth-order KdV equation in (2+ 1) dimensions. Modern Physics Letters B, 33(33), 1950411.
2. Berti, A., & Berti, V. (2013). A thermodynamically consistent Ginzburg–Landau model for superfluid transition in liquid helium. Zeitschrift für angewandte Mathematik und Physik, 64(4), 1387-1397.
3. Kengne, E., Lakhssassi, A., Vaillancourt, R., & Liu, W. M. (2012). Exact solutions for generalized variable-coefficients Ginzburg-Landau equation: Application to Bose-Einstein condensates with multi-body interatomic interactions. Journal of mathematical physics, 53(12), 123703.
4. Rivers, R. J. (2001). Zurek-Kibble causality bounds in time-dependent Ginzburg-Landau theory and quantum field theory. Journal of low temperature physics, 124(1), 41-83.
5. Tasbozan, O., Kurt, A., & Tozar, A. (2019). New optical solutions of complex Ginzburg–Landau equation arising in semiconductor lasers. Applied Physics B, 125(6), 1-12.
6. Khamrakulov, K. P. (2019). Two-soliton molecule bouncing in a dipolar Bose–Einstein condensates under the effect of gravity. Modern Physics Letters B, 33(36), 1950452.
7. Seadawy, A. R., Iqbal, M., & Lu, D. (2019). Analytical methods via bright–dark solitons and solitary wave solutions of the higher-order nonlinear Schrödinger equation with fourth-order dispersion. Modern Physics Letters B, 33(35), 1950443.
8. Yan, X. W. (2020). Generalized (3+ 1)-dimensional Boussinesq equation: Breathers, rogue waves and their dynamics. Modern Physics Letters B, 34(01), 2050003.

9. Díaz, P., Laroze, D., Ávila, A., & Malomed, B. A. (2019). Two-dimensional composite solitons in a spin-orbit-coupled fermi gas in free space. Communications in Nonlinear Science and Numerical Simulation, 70, 372-383.
10. Tian, Z., & Du, J. (2019). Analogue Hawking radiation and quantum soliton evaporation in a superconducting circuit. The European Physical Journal C, 79(12), 1-7.
11. Arshad, M., Seadawy, A. R. & Lu, D. (2019). Study of bright–dark solitons of strain wave equation in micro-structured solids and its applications. Modern Physics Letters B, 33(33), 1950417.
12. Marsh, D. J., & Pop, A. R. (2015). Axion dark matter, solitons and the cusp–core problem. Monthly Notices of the Royal Astronomical Society, 451(3), 2479-2492.
13. Tala-Tebue, E., & Seadawy, A. R. (2018). Construction of dispersive optical solutions of the resonant nonlinear Schrödinger equation using two different methods. Modern Physics Letters B, 32(33), 1850407.
14. Zayed, E. M. E., & Alurrfi, K. A. E. (2016). Extended auxiliary equation method and its applications for finding the exact solutions for a class of nonlinear Schrödinger-type equations. Applied Mathematics and Computation, 289, 111-131.
15. Durur, H. (2020). Different types analytic solutions of the (1+ 1)-dimensional resonant nonlinear Schrödinger’s equation using -expansion method. Modern Physics Letters B, 34(03), 2050036.
16. Tasbozan, O., & Kurt, A. (2020). New Exact Solutions of the MkdV-Sine-Gordon Equation, Sohag Journal of Mathematics, 7, 1-4.
17. Williams, F., Tsitoura, F., Horikis, T. P., & Kevrekidis, P. G. (2020). Solitary waves in the resonant onlinear Schrödinger equation: Stability and dynamical properties. Physics Letters A, 384(22), 126441.
18. Lee, J. H., & Pashaev, O. K. (2007). Solitons of the resonant nonlinear Schrödinger equation with nontrivial boundary conditions: Hirota bilinear method. Theoretical and Mathematical Physics, 152(1), 991-1003.
19. Başhan, A., Uçar, Y., Yağmurlu, N. M., & Esen, A. (2018). A new perspective for quintic B-spline based Crank-Nicolson-differential quadrature method algorithm for numerical solutions of the nonlinear Schrödinger equation. The European Physical Journal Plus, 133(1), 1-15.
20. Başhan, A., & Esen, A. (2021). Single soliton and double soliton solutions of the quadratic‐nonlinear Korteweg‐de Vries equation for small and long‐times. Numerical Methods for Partial Differential Equations, 37(2), 1561-1582.
21. Başhan, A. (2019). A mixed methods approach to Schrödinger equation: Finite difference method and quartic B-spline based differential quadrature method. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 9(2), 223-235.
22. Başhan, A. (2018). An effective application of differential quadrature method based on modified cubic B-splines to numerical solutions of the KdV equation. Turkish Journal of Mathematics, 42(1), 373-394.
23. Başhan, A. (2021). Modification of quintic B-spline differential quadrature method to nonlinear Korteweg-de Vries equation and numerical experiments. Applied Numerical Mathematics, 167, 356-374.
24. Bashan, A., Yagmurlu, N. M., Ucar, Y., & Esen, A. (2017). An effective approach to numerical soliton solutions for the Schrödinger equation via modified cubic B-spline differential quadrature method. Chaos, Solitons & Fractals, 100, 45-56.