Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation
Fractal and fractional calculus have important theoretical and practical value. In this paper, analytical solutions, including the <i>N</i>-fractal-soliton solution with fractal characteristics in time and soliton characteristics in space as well as the long-time asymptotic solution of a...
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doaj-87a5e8bcd2b947b6b4804c3d22ff48cb2021-09-26T01:30:50ZengMDPI AGSymmetry2073-89942021-08-01131593159310.3390/sym13091593Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type EquationBo Xu0Sheng Zhang1School of Mathematics, China University of Mining and Technology, Xuzhou 221116, ChinaSchool of Mathematical Sciences, Bohai University, Jinzhou 121013, ChinaFractal and fractional calculus have important theoretical and practical value. In this paper, analytical solutions, including the <i>N</i>-fractal-soliton solution with fractal characteristics in time and soliton characteristics in space as well as the long-time asymptotic solution of a local time-fractional nonlinear Schrödinger (NLS)-type equation, are obtained by extending the Riemann–Hilbert (RH) approach together with the symmetries of the associated spectral function, jump matrix, and solution of the related RH problem. In addition, infinitely many conservation laws determined by an expression, one end of which is the partial derivative of local fractional-order in time, and the other end is the partial derivative of integral order in space of the local time-fractional NLS-type equation are also obtained. Constraining the time variable to the Cantor set, the obtained one-fractal-soliton solution is simulated, which shows the solution possesses continuous and non-differentiable characteristics in the time direction but keeps the soliton continuous and differentiable in the space direction. The essence of the fractal-soliton feature is that the time and space variables are set into two different dimensions of 0.631 and 1, respectively. This is also a concrete example of the same object showing different geometric characteristics on two scales.https://www.mdpi.com/2073-8994/13/9/1593<i>N</i>-fractal-soliton solutionlong-time asymptotic solutionconservation lawRiemann–Hilbert approachlocal time-fractional NLS-type equationtwo-scale |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Bo Xu Sheng Zhang |
spellingShingle |
Bo Xu Sheng Zhang Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation Symmetry <i>N</i>-fractal-soliton solution long-time asymptotic solution conservation law Riemann–Hilbert approach local time-fractional NLS-type equation two-scale |
author_facet |
Bo Xu Sheng Zhang |
author_sort |
Bo Xu |
title |
Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation |
title_short |
Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation |
title_full |
Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation |
title_fullStr |
Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation |
title_full_unstemmed |
Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation |
title_sort |
riemann–hilbert approach for constructing analytical solutions and conservation laws of a local time-fractional nonlinear schrödinger type equation |
publisher |
MDPI AG |
series |
Symmetry |
issn |
2073-8994 |
publishDate |
2021-08-01 |
description |
Fractal and fractional calculus have important theoretical and practical value. In this paper, analytical solutions, including the <i>N</i>-fractal-soliton solution with fractal characteristics in time and soliton characteristics in space as well as the long-time asymptotic solution of a local time-fractional nonlinear Schrödinger (NLS)-type equation, are obtained by extending the Riemann–Hilbert (RH) approach together with the symmetries of the associated spectral function, jump matrix, and solution of the related RH problem. In addition, infinitely many conservation laws determined by an expression, one end of which is the partial derivative of local fractional-order in time, and the other end is the partial derivative of integral order in space of the local time-fractional NLS-type equation are also obtained. Constraining the time variable to the Cantor set, the obtained one-fractal-soliton solution is simulated, which shows the solution possesses continuous and non-differentiable characteristics in the time direction but keeps the soliton continuous and differentiable in the space direction. The essence of the fractal-soliton feature is that the time and space variables are set into two different dimensions of 0.631 and 1, respectively. This is also a concrete example of the same object showing different geometric characteristics on two scales. |
topic |
<i>N</i>-fractal-soliton solution long-time asymptotic solution conservation law Riemann–Hilbert approach local time-fractional NLS-type equation two-scale |
url |
https://www.mdpi.com/2073-8994/13/9/1593 |
work_keys_str_mv |
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