1,906 research outputs found
Numerical study of blow-up and stability of line solitons for the Novikov-Veselov equation
We study numerically the evolution of perturbed Korteweg-de Vries solitons
and of well localized initial data by the Novikov-Veselov (NV) equation at
different levels of the "energy" parameter . We show that as , NV behaves, as expected, similarly to its formal limit, the
Kadomtsev-Petviashvili equation. However at intermediate regimes, i.e. when is not very large, more varied scenarios are possible, in particular,
blow-ups are observed. The mechanism of the blow-up is studied
A new extended matrix KP hierarchy and its solutions
With the square eigenfunctions symmetry constraint, we introduce a new
extended matrix KP hierarchy and its Lax representation from the matrix KP
hierarchy by adding a new flow. The extended KP hierarchy contains two
time series and and eigenfunctions and adjoint
eigenfunctions as components. The extended matrix KP hierarchy and its
-reduction and reduction include two types of matrix KP hierarchy
with self-consistent sources and two types of (1+1)-dimensional reduced matrix
KP hierarchy with self-consistent sources. In particular, the first type and
second type of the 2+1 AKNS equation and the Davey-Stewartson equation with
self-consistent sources are deduced from the extended matrix KP hierarchy. The
generalized dressing approach for solving the extended matrix KP hierarchy is
proposed and some solutions are presented. The soliton solutions of two types
of 2+1-dimensional AKNS equation with self-consistent sources and two types of
Davey-Stewartson equation with self-consistent sources are studied.Comment: 17 page
Generalized Darboux transformations for the KP equation with self-consistent sources
The KP equation with self-consistent sources (KPESCS) is treated in the
framework of the constrained KP equation. This offers a natural way to obtain
the Lax representation for the KPESCS. Based on the conjugate Lax pairs, we
construct the generalized binary Darboux transformation with arbitrary
functions in time for the KPESCS which, in contrast with the binary Darboux
transformation of the KP equation, provides a non-auto-B\"{a}cklund
transformation between two KPESCSs with different degrees. The formula for
N-times repeated generalized binary Darboux transformation is proposed and
enables us to find the N-soliton solution and lump solution as well as some
other solutions of the KPESCS.Comment: 20 pages, no figure
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