A Finite Difference Scheme for (2+1)D Cubic-Quintic Nonlinear Schrödinger Equations with Nonlinear Damping
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| Format: | Preprint |
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2024
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| _version_ | 1866916349232545792 |
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| author | Le, Anh Ha Huynh, Toan T. Nguyen, Quan M. |
| author_facet | Le, Anh Ha Huynh, Toan T. Nguyen, Quan M. |
| contents | Solitons of the purely cubic nonlinear Schrödinger equation in a space dimension of $n \geq 2$ suffer critical and supercritical collapses. These solitons can be stabilized in a cubic-quintic nonlinear medium. In this paper, we analyze the Crank-Nicolson finite difference scheme for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping. We show that both the discrete solution, in the discrete $L^2$-norm, and discrete energy are bounded. By using appropriate settings and estimations, the existence and the uniqueness of the numerical solution are proved. In addition, the error estimations are established in terms of second order for both space and time in discrete $L^2$-norm and $H^1$-norm. Numerical simulations for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping are conducted to validate the convergence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_12311 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Finite Difference Scheme for (2+1)D Cubic-Quintic Nonlinear Schrödinger Equations with Nonlinear Damping Le, Anh Ha Huynh, Toan T. Nguyen, Quan M. Numerical Analysis Solitons of the purely cubic nonlinear Schrödinger equation in a space dimension of $n \geq 2$ suffer critical and supercritical collapses. These solitons can be stabilized in a cubic-quintic nonlinear medium. In this paper, we analyze the Crank-Nicolson finite difference scheme for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping. We show that both the discrete solution, in the discrete $L^2$-norm, and discrete energy are bounded. By using appropriate settings and estimations, the existence and the uniqueness of the numerical solution are proved. In addition, the error estimations are established in terms of second order for both space and time in discrete $L^2$-norm and $H^1$-norm. Numerical simulations for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping are conducted to validate the convergence. |
| title | A Finite Difference Scheme for (2+1)D Cubic-Quintic Nonlinear Schrödinger Equations with Nonlinear Damping |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2407.12311 |