616 research outputs found
-Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schr\"odinger Equation
In this paper, a general bright-dark soliton solution in the form of Pfaffian
is constructed for an integrable semi-discrete vector NLS equation via Hirota's
bilinear method. One- and two-bright-dark soliton solutions are explicitly
presented for two-component semi-discrete NLS equation; two-bright-one-dark,
and one-bright-two-dark soliton solutions are also given explicitly for
three-component semi-discrete NLS equation. The asymptotic behavior is analysed
for two-soliton solutions
N-Dark-Dark Solitons in the Generally Coupled Nonlinear Schroedinger Equations
N-dark-dark solitons in the generally coupled integrable NLS equations are
derived by the KP-hierarchy reduction method. These solitons exist when
nonlinearities are all defocusing, or both focusing and defocusing
nonlinearities are mixed. When these solitons collide with each other, energies
in both components of the solitons completely transmit through. This behavior
contrasts collisions of bright-bright solitons in similar systems, where
polarization rotation and soliton reflection can take place. It is also shown
that in the mixed-nonlinearity case, two dark-dark solitons can form a
stationary bound state.Comment: 26 pages, 3 figure
Self-adaptive moving mesh schemes for short pulse type equations and their Lax pairs
Integrable self-adaptive moving mesh schemes for short pulse type equations
(the short pulse equation, the coupled short pulse equation, and the complex
short pulse equation) are investigated. Two systematic methods, one is based on
bilinear equations and another is based on Lax pairs, are shown. Self-adaptive
moving mesh schemes consist of two semi-discrete equations in which the time is
continuous and the space is discrete. In self-adaptive moving mesh schemes, one
of two equations is an evolution equation of mesh intervals which is deeply
related to a discrete analogue of a reciprocal (hodograph) transformation. An
evolution equations of mesh intervals is a discrete analogue of a conservation
law of an original equation, and a set of mesh intervals corresponds to a
conserved density which play an important role in generation of adaptive moving
mesh. Lax pairs of self-adaptive moving mesh schemes for short pulse type
equations are obtained by discretization of Lax pairs of short pulse type
equations, thus the existence of Lax pairs guarantees the integrability of
self-adaptive moving mesh schemes for short pulse type equations. It is also
shown that self-adaptive moving mesh schemes for short pulse type equations
provide good numerical results by using standard time-marching methods such as
the improved Euler's method.Comment: 13 pages, 6 figures, To be appeared in Journal of Math-for-Industr
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