33,767 research outputs found
The semi-discrete AKNS system: Conservation laws, reductions and continuum limits
In this paper, the semi-discrete Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy
is shown in spirit composed by the Ablowitz-Ladik flows under certain
combinations. Furthermore, we derive its explicit Lax pairs and infinitely many
conservation laws, which are non-trivial in light of continuum limit.
Reductions of the semi-discrete AKNS hierarchy are investigated to include the
semi-discrete Korteweg-de Vries (KdV), the semi-discrete modified KdV, and the
semi-discrete nonlinear Schr\"odinger hierarchies as its special cases.
Finally, under the uniform continuum limit we introduce in the paper, the above
results of the semi-discrete AKNS hierarchy, including Lax pairs, infinitely
many conservation laws and reductions, recover their counterparts of the
continuous AKNS hierarchy
Solutions to the complex Korteweg-de Vries equation: Blow-up solutions and non-singular solutions
In the paper two kinds of solutions are derived for the complex Korteweg-de
Vries equation, including blow-up solutions and non-singular solutions. We
derive blow-up solutions from known 1-soliton solution and a double-pole
solution. There is a complex Miura transformation between the complex
Korteweg-de Vries equation and a modified Korteweg-de Vries equation. Using the
transformation, solitons, breathers and rational solutions to the complex
Korteweg-de Vries equation are obtained from those of the modified Korteweg-de
Vries equation. Dynamics of the obtained solutions are illustrated.Comment: 12 figure
Landau-Zener-Stuckelberg interference in a multi-anticrossing system
We propose a universal analytical method to study the dynamics of a
multi-anticrossing system subject to driving by one single large-amplitude
triangle pulse, within its time scales smaller than the dephasing time. Our
approach can explain the main features of the Landau-Zener-Stuckelberg
interference patterns recently observed in a tripartite system [Nature
Communications 1:51 (2010)]. In particular, we focus on the effects of the size
of anticrossings on interference and compare the calculated interference
patterns with numerical simulations. In addition, Fourier transform of the
patterns can extract information on the energy level spectrum.Comment: 6 pages, 5 figure
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