研究目的
Investigating the dynamical behavior resulting from an initial discontinuity in focusing media using the focusing nonlinear Schr?dinger equation.
研究成果
The study demonstrates how a relatively broad class of Riemann problems can be effectively studied using Whitham modulation theory for the focusing nonlinear Schr?dinger equation, despite the elliptic nature of the Whitham equations. The space-time plane divides into expanding domains where the solution is described by a slow modulation of genus-0, genus-1, or genus-2 solutions, depending on the specifics of the initial datum.
研究不足
The Whitham equations for the focusing NLS equation are elliptic, making them unsuitable for studying initial value problems in general. The study is limited to a relatively broad class of Riemann problems that can still be effectively studied using Whitham modulation theory.
1：Experimental Design and Method Selection:
The study uses numerical simulations and Whitham modulation theory to analyze the behavior of solutions to the focusing nonlinear Schr?dinger equation with initial discontinuities.
2：Sample Selection and Data Sources:
Initial conditions with a jump in either or both the amplitude and the local wave number are considered.
3：List of Experimental Equipment and Materials:
Numerical simulations were performed using an eighth-order Fourier split-step code with 4096 spatial grid points, spatial domain [?200, 200], and integration time step 10?
4：Experimental Procedures and Operational Workflow:
The space-time plane is divided into expanding domains where the solution is described by a slow modulation of genus-0, genus-1, or genus-2 solutions.
5：Data Analysis Methods:
Analytical and numerical methods are used to show the arrangement of solutions depending on the specifics of the initial datum.
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