53 research outputs found
On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations
Considered herein are the generalized Camassa-Holm and Degasperis-Procesi
equations in the spatially periodic setting. The precise blow-up scenarios of
strong solutions are derived for both of equations. Several conditions on the
initial data guaranteeing the development of singularities in finite time for
strong solutions of these two equations are established. The exact blow-up
rates are also determined. Finally, geometric descriptions of these two
integrable equations from non-stretching invariant curve flows in
centro-equiaffine geometries, pseudo-spherical surfaces and affine surfaces are
given.Comment: 26 page
Massive Thirring Model: Inverse Scattering and Soliton Resolution
In this paper the long-time dynamics of the massive Thirring model is
investigated. Firstly the nonlinear steepest descent method for Riemann-Hilbert
problem is explored to obtain the soliton resolution of the solutions to the
massive Thirring model whose initial data belong to some weighted-Sobolev
spaces. Secondly, the asymptotic stability of multi-solitons follow as a
corollary. The main difficulty in studying the massive Thirring model through
inverse scattering is that the corresponding Lax pair has singularities at the
origin and infinity. We overcome this difficulty by making use of two
transforms that separate the singularities.Comment: arXiv admin note: text overlap with arXiv:2009.04260,
arXiv:1907.0711
A new integrable two-component system with cubic nonlinearity
In this paper, a new integrable two-component system, mt=[m(uxvx−uv+uvx−uxv)]x,nt=[n(uxvx−uv+uvx−uxv)]x, where m=u−uxx and n=v−vxx, is proposed. Our system is a generalized version of the integrable system mt=[m(u2x−u2)]x, which was shown having cusped solution (cuspon) and W/M-shape soliton solutions by Qiao [J. Math. Phys. 47, 112701 (2006). The new system is proven integrable not only in the sense of Lax-pair but also in the sense of geometry, namely, it describes pseudospherical surfaces. Accordingly, infinitely many conservation laws are derived through recursion relations. Furthermore, exact solutions such as cuspons and W/M-shape solitons are also obtained
Solutions to the SU() self-dual Yang-Mills equation
In this paper we aim to derive solutions for the SU() self-dual
Yang-Mills (SDYM) equation with arbitrary . A set of
noncommutative relations are introduced to construct a matrix equation that can
be reduced to the SDYM equation. It is shown that these relations can be
generated from two different Sylvester equations, which correspond to the two
Cauchy matrix schemes for the (matrix) Kadomtsev-Petviashvili hierarchy and the
(matrix) Ablowitz-Kaup-Newell-Segur hierarchy, respectively. In each Cauchy
matrix scheme we investigate the possible reductions that can lead to the
SU SDYM equation and also analyze the physical significance of
some solutions, i.e. being Hermitian, positive-definite and of determinant
being one.Comment: 26 page
Talbot effect for the Manakov System on the torus
In this paper, the Talbot effect for the multi-component linear and nonlinear
systems of the dispersive evolution equations on a bounded interval subject to
periodic boundary conditions and discontinuous initial profiles is
investigated. Firstly, for a class of two-component linear systems satisfying
the dispersive quantization conditions, we discuss the fractal solutions at
irrational times. Next, the investigation to nonlinear regime is extended, we
prove that, for the concrete example of the Manakov system, the solutions of
the corresponding periodic initial-boundary value problem subject to initial
data of bounded variation are continuous but nowhere differentiable
fractal-like curve with Minkowski dimension at irrational times. Finally,
numerical experiments for the periodic initial-boundary value problem of the
Manakov system, are used to justify how such effects persist into the
multi-component nonlinear regime. Furthermore, it is shown in the nonlinear
multi-component regime that the interplay of different components may induce
subtle different qualitative profile between the jump discontinuities,
especially in the case that two nonlinearly coupled components start with
different initial profile
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