22 research outputs found
Initial-boundary value problem and long-time asymptotics for the Kundu--Eckhaus equation on the half-line
The initial-boundary value problem for the Kundu--Eckhaus equation on the
half-line is considered in this paper by using the Fokas method. We will show
that the solution can be expressed in terms of the solution of a
matrix Riemann--Hilbert problem formulated in the complex -plane.
Furthermore, based on a nonlinear steepest descent analysis of the associated
Riemann--Hilbert problem, we can give the precise asymptotic formulas for the
solution of the Kundu--Eckhaus equation on the half-line.Comment: arXiv admin note: text overlap with arXiv:1702.02084,
arXiv:0808.1534, arXiv:0812.1335 by other author
Long-time asymptotics for the Hirota equation on the half-line
We consider the Hirota equation on the quarter plane with the initial and
boundary values belonging to the Schwartz space. The goal of this paper is to
study the long-time behavior of the solution of this initial-boundary value
problem based on the asymptotic analysis of an associated matrix
Riemann--Hilbert problem.Comment: arXiv admin note: text overlap with arXiv:1712.0382
The Riemann-Hilbert approach to focusing Kundu-Eckhaus equation with nonzero boundary conditions
In this article, we focus on investigating the focusing Kundu-Eckhaus
equation with nonzero boundary condition. A appropriate two-sheeted Riemann
surface is introduced to map the spectral parameter into a single-valued
parameter . Starting from the Lax pair of Kundu-Eckhaus equation,two kind of
Jost solutions are construed. Further their asymptotic, analyticity, symmetries
as well as spectral matrix are detailed analyzed. It is shown that the solution
of Kundu-Eckhaus equation with nonzero boundary condition can characterized
with a matrix Riemann-Hilbert problem. Then a formula of -soliton solutions
is derived by solving Riemann-Hilbert problem. As applications, the first-order
explicit soliton solution is obtained.Comment: 26 page
Long time asymptotics behavior of the focusing nonlinear Kundu-Eckhaus equation
We study the Cauchy problem for the focusing nonlinear Kundu-Eckhaus equation
and construct long time asymptotic expansion of its solution in fixed
space-time cone with
. By using the inverse scattering transform,
Riemann-Hilbert approach and steepest descent method we
obtain the lone time asymptotic behavior of the solution, at the same time we
obtain the solitons in the cone compare with the all N-soliton the residual
error up to order .Comment: 33 pages. arXiv admin note: text overlap with arXiv:1604.07436 by
other author
Long-time asymptotics for the Nonlocal mKdV equation
In this paper, we study the Cauchy problem with decaying initial data for the
nonlocal modified Korteweg-de Vries equation (nonlocal mKdV)
which can be viewed as a generalization of the local classical mKdV equation.
We first formulate the Riemann-Hilbert problem associated with the Cauchy
problem of the nonlocal mKdV equation. Then we apply the Deift-Zhou nonlinear
steepest-descent method to analyze the long-time asymptotics for the solution
of the nonlocal mKdV equation. In contrast with the classical mKdV equation, we
find some new and different results on long-time asymptotics for the nonlocal
mKdV equation and some additional assumptions about the scattering data are
made in our main results.Comment: 27 page
Long-time asymptotics in the modified Landau-Lifshitz equation with nonzero boundary conditions
In this work, we consider the long-time asymptotics of the modified
Landau-Lifshitz equation with nonzero boundary conditions (NZBCs) at infinity.
The critical technique is the deformations of the corresponding matrix
Riemann-Hilbert problem via the nonlinear steepest descent method, as well as
we employ the -function mechanism to eliminate the exponential growths of
the jump matrices. The results indicate that the solution of the modified
Landau-Lifshitz equation with nonzero boundary conditions admits two different
asymptotic behavior corresponding to two types of regions in the -plane.
They are called the plane wave region with , and the modulated elliptic wave region with
, respectively.Comment: 34 pages, 10 figure
Inverse scattering transform for the Kundu-Eckhaus Equation with nonzero boundary condition
In this paper, we consider the initial value problem for both of the
defocusing and focusing Kundu-Eckhaus (KE) equation with non-zero boundary
conditions (NZBCs) at infinity by inverse scattering transform method. The
solutions of the KE equation with NZBCs can be reconstructed in terms of the
solution of an associated matrix Riemann-Hilbert problem (RHP). In
our formulation, both the direct and the inverse problems are posed in terms of
a suitable uniformization variable which allows us to develop the IST on the
standard complex plane instead of a two-sheeted Riemann surface or the cut
plane with discontinuities along the cuts. Furthermore, on the one hand, we
obtain the N-soliton solutions with simple pole of the defocusing and focusing
KE equation with the NZBCs, especially, the explicit one-soliton solutions are
given in details. And we prove that the scattering data of the
defocusing KE equation can only have simple zeros. On the other hand, we also
obtain the soliton solutions with double pole of the focusing KE equation with
NZBCs. And we show that the double pole solutions can be viewed as some proper
limit of the two simple pole soliton solutions. Some dynamical behaviors and
typical collisions of the soliton solutions of both of the defocusing and
focusing KE equation are shown graphically.Comment: 45 pages, 13 figure
Soliton resolution for a coupled generalized nonlinear Schr\"{o}dinger equations with weighted Sobolev initial data
In this work, we employ the steepest descent method in order
to study the Cauchy problem of the cgNLS equations with initial conditions in
weighted Sobolev space . The large time asymptotic behavior of the solution
and are derived in a fixed space-time cone
. Based on the resulting
asymptotic behavior, we prove the solution resolution conjecture of the cgNLS
equations which contains the soliton term confirmed by
-soliton on discrete spectrum and the
order term on continuous spectrum with residual error up to
.Comment: 37 page
Long-time asymptotics for the Sasa-Satsuma equation via nonlinear steepest descent method
We formulate a Riemann-Hilbert problem to solve the Cauchy problem
for the Sasa-Satsuma equation on the line, which allows us to give a
representation for the solution of Sasa-Satsuma equation. We then apply the
method of nonlinear steepest descent to compute the long-time asymptotics of
the Sasa-Satsuma equation.Comment: arXiv admin note: text overlap with arXiv:1712.07002,
arXiv:1712.0382
The Gerdjikov-Ivanov type derivative nonlinear Schr\"odinger equation: Long-time dynamics of nonzero boundary conditions
We consider the Gerdjikov--Ivanov type derivative nonlinear Schr\"odinger
equation \berr \ii q_{t}+q_{xx}-\ii q^2\bar{q}_{x}+\frac{1}{2}(|q|^4-q_0^4)q=0
\eerr on the line. The initial value is given and satisfies the
symmetric, nonzero boundary conditions at infinity, that is, as , and . The goal of this paper
is to study the asymptotic behavior of the solution of this initial-value
problem as . The main tool is the asymptotic analysis of an
associated matrix Riemann--Hilbert problem by using the steepest descent method
and the so-called -function mechanism. We show that the solution of
this initial value problem has a different asymptotic behavior in different
regions of the -plane. In the regions and
, the solution takes the form of a plane wave. In the region
, the solution takes the form of a
modulated elliptic wave