Dynamics of Plane Waves in the Fractional Nonlinear Schrödinger Equation with Long-Range Dispersion
We analytically and numerically investigate the stability and dynamics of the plane wave solutions of the fractional nonlinear Schrödinger (NLS) equation, where the long-range dispersion is described by the fractional Laplacian <inline-formula><math xmlns="http://www.w3.org/1998/Math/M...
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MDPI AG
2021-08-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/8/1394 |
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| author | Siwei Duo Taras I. Lakoba Yanzhi Zhang |
| author_facet | Siwei Duo Taras I. Lakoba Yanzhi Zhang |
| author_sort | Siwei Duo |
| collection | DOAJ |
| description | We analytically and numerically investigate the stability and dynamics of the plane wave solutions of the fractional nonlinear Schrödinger (NLS) equation, where the long-range dispersion is described by the fractional Laplacian <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo></mrow><mrow><mi>α</mi><mo>/</mo><mn>2</mn></mrow></msup></semantics></math></inline-formula>. The linear stability analysis shows that plane wave solutions in the defocusing NLS are always stable if the power <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></semantics></math></inline-formula> but unstable for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. In the focusing case, they can be linearly unstable for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></semantics></math></inline-formula>. We then apply the split-step Fourier spectral (SSFS) method to simulate the nonlinear stage of the plane waves dynamics. In agreement with earlier studies of solitary wave solutions of the fractional focusing NLS, we find that as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></semantics></math></inline-formula> decreases, the solution evolves towards an increasingly localized pulse existing on the background of a “sea” of small-amplitude dispersive waves. Such a highly localized pulse has a broad spectrum, most of whose modes are excited in the nonlinear stage of the pulse evolution and are not predicted by the linear stability analysis. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>, we always find the solution to undergo collapse. We also show, for the first time to our knowledge, that for initial conditions with nonzero group velocities (traveling plane waves), an onset of collapse is delayed compared to that for a standing plane wave initial condition. For defocusing fractional NLS, even though we find traveling plane waves to be linearly unstable for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, we have never observed collapse. As a by-product of our numerical studies, we derive a stability condition on the time step of the SSFS to guarantee that this method is free from numerical instabilities. |
| first_indexed | 2024-03-10T08:20:27Z |
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| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T08:20:27Z |
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| spelling | doaj.art-601b932dd1364d1fb06fdc143cf84ac62023-11-22T10:00:44ZengMDPI AGSymmetry2073-89942021-08-01138139410.3390/sym13081394Dynamics of Plane Waves in the Fractional Nonlinear Schrödinger Equation with Long-Range DispersionSiwei Duo0Taras I. Lakoba1Yanzhi Zhang2Department of Mathematics, University of South Carolina, Columbia, SC 29208, USADepartment of Mathematics and Statistics, University of Vermont, Burlington, VT 05405, USADepartment of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USAWe analytically and numerically investigate the stability and dynamics of the plane wave solutions of the fractional nonlinear Schrödinger (NLS) equation, where the long-range dispersion is described by the fractional Laplacian <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo></mrow><mrow><mi>α</mi><mo>/</mo><mn>2</mn></mrow></msup></semantics></math></inline-formula>. The linear stability analysis shows that plane wave solutions in the defocusing NLS are always stable if the power <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></semantics></math></inline-formula> but unstable for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. In the focusing case, they can be linearly unstable for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></semantics></math></inline-formula>. We then apply the split-step Fourier spectral (SSFS) method to simulate the nonlinear stage of the plane waves dynamics. In agreement with earlier studies of solitary wave solutions of the fractional focusing NLS, we find that as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></semantics></math></inline-formula> decreases, the solution evolves towards an increasingly localized pulse existing on the background of a “sea” of small-amplitude dispersive waves. Such a highly localized pulse has a broad spectrum, most of whose modes are excited in the nonlinear stage of the pulse evolution and are not predicted by the linear stability analysis. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>, we always find the solution to undergo collapse. We also show, for the first time to our knowledge, that for initial conditions with nonzero group velocities (traveling plane waves), an onset of collapse is delayed compared to that for a standing plane wave initial condition. For defocusing fractional NLS, even though we find traveling plane waves to be linearly unstable for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, we have never observed collapse. As a by-product of our numerical studies, we derive a stability condition on the time step of the SSFS to guarantee that this method is free from numerical instabilities.https://www.mdpi.com/2073-8994/13/8/1394fractional nonlinear Schrödinger equationfractional Laplacianplane wave solutionmodulation instabilitysplit-step methodnumerical stability |
| spellingShingle | Siwei Duo Taras I. Lakoba Yanzhi Zhang Dynamics of Plane Waves in the Fractional Nonlinear Schrödinger Equation with Long-Range Dispersion Symmetry fractional nonlinear Schrödinger equation fractional Laplacian plane wave solution modulation instability split-step method numerical stability |
| title | Dynamics of Plane Waves in the Fractional Nonlinear Schrödinger Equation with Long-Range Dispersion |
| title_full | Dynamics of Plane Waves in the Fractional Nonlinear Schrödinger Equation with Long-Range Dispersion |
| title_fullStr | Dynamics of Plane Waves in the Fractional Nonlinear Schrödinger Equation with Long-Range Dispersion |
| title_full_unstemmed | Dynamics of Plane Waves in the Fractional Nonlinear Schrödinger Equation with Long-Range Dispersion |
| title_short | Dynamics of Plane Waves in the Fractional Nonlinear Schrödinger Equation with Long-Range Dispersion |
| title_sort | dynamics of plane waves in the fractional nonlinear schrodinger equation with long range dispersion |
| topic | fractional nonlinear Schrödinger equation fractional Laplacian plane wave solution modulation instability split-step method numerical stability |
| url | https://www.mdpi.com/2073-8994/13/8/1394 |
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