Long time asymptotics for the KPII equation
The long-time asymptotics of small Kadomtsev-Petviashvili II (KPII) solutions is derived using the inverse scattering theory and the stationary phase method.
math.AP↗
arXiv subjects
Derchyi Wu
Publications and source records attributed to Derchyi Wu.
Stability of Kadomtsev-Petviashvili multi-line solitons
We prove the long-standing inverse scattering theory (IST) of perturbed Kadomtsev Petviashvili multi-line solitons. Our work is the first rigorous IST of a multi-dimensional integrable system when both continuous and discrete scattering data are present, and the support of continuous scattering data does not degenerate into contours in the complex plane. As an application, an $L^\infty$-stability theorem of the Kadomtsev Petviashvili multi-line solitons is justified.
math.AP↗
The inverse scattering theory of Kadomtsev-Petviashvili II equations
An overview of the inverse scattering theory of the Kadomtsev Petviashvili II equation with an emphasis on the inverse problem for perturbed KP multi line solitons is provided. It is shown that, despite additional algebraic or analytic techniques are introduced due to new singular structures, there exists a consistency of the inverse scattering theories for different backgrounds such as the vacuum, $1$-line solitons, and multi line solitons.
nlin.SI↗
Stability of KdV solitons
We prove an orbital stability theorem of KdV $n$-solitons with explicit phase shifts in the soliton region with cones around the $x$-axis and lines determined by bound states of the KdV $n$-solitons removed.
math.AP↗
The direct scattering problem for perturbed Kadomtsev-Petviashvili multi line solitons
Regular Kadomtsev-Petviashvili II (KPII) line solitons have been investigated and classified successfully by the Grassmannians. The inverse scattering method provides a promising and powerful approach to study the stability properties of $\textrm{Gr}(N, M)_{> 0}$ KP solitons. In this paper, we complete rigorous analysis for the direct scattering problem of perturbed $\textrm{Gr}(N, M)_{> 0}$ KP solitons}.
nlin.SI↗
The direct scattering problem for the perturbed $\textrm{Gr}(1, 2)_{\ge 0}$ Kadomtsev-Petviashvili solitons
Regular Kadomtsev-Petviashvili (KP) solitons have been investigated and classified successfully by the Grassmannian. We provide rigorous analysis for the direct scattering problem of perturbed $\textrm{Gr}(1, 2)_{\ge 0}$ KP solitons.
nlin.SI↗
The direct problem for the perturbed Kadomtsev-Petviashvili II one line solitons
We provide rigorous analysis for the direct scattering theory of perturbed Kadomtsev-Petviashvili II one line solitons. Namely, for generic small initial data, the existence of the eigenfunction is proved by establishing uniform estimates of the Green function and the Cauchy integral equation for the eigenfunction is justified by analysing the spectral transform.
nlin.SI↗
The Cauchy problem for the Pavlov equation with large data
The Pavlov equation is one of the simplest integrable systems of vector fields arising from various problems of mathematical physics and differential geometry which are intensively studied in recent literature. In this report, solving a nonlinear Riemann-Hilbert problem via a Newtonian iteration scheme, we complete the inverse scattering theory and prove a short time unique solvability of the Cauchy problem of the Pavlov equation with large initial data.
nlin.SI↗
A twisted integrable hierarchy with $\mathbb D_2$ symmetry
A loop algebra approach to the Gerdjikov-Mikhailov-Valchev (GMV) equation is provided to exploit the associated twisted integrable structure and a new twisted integrable hierarchy is discovered. Using the twisted loop algebra structure, we obtain a transparent treatment of the associated scattering and inverse scattering theory and solve the initial value problem for the GMV equation.
nlin.SI↗
Twisted hierarchies associated with the generalized sine-Gordon equation
Twisted $U$- and twisted $U/K$-hierarchies are soliton hierarchies introduced by Terng to find higher flows of the generalized sine-Gordon equation. Twisted $\frac {O(J,J)}{O(J)\times O(J)}$-hierarchies are among the most important classes of twisted hierarchies. In this paper, interesting first and higher flows of twisted $\frac {O(J,J)}{O(J)\times O(J)}$-hierarchies are explicitly derived, the associated submanifold geometry is investigated and a unified treatment of the inverse scattering theory is provided.
nlin.SI↗
The Cauchy problem of the Ward equation with mixed scattering data
We solve the Cauchy problem of the Ward equation with both continuous and discrete scattering data.
nlin.SI↗
The Cauchy Problem of the Ward equation
We generalize the results of Villarroel, Fokas and Ioannidou, Dai, Terng and Uhlenbeck to study the inverse scattering problem of the Ward equation with non-small data and solve the Cauchy problem of the Ward equation with a non-small purely continuous scattering data.
math.AP↗