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On long-time asymptotics of solution to the non-local Lakshmanan–Porsezian–Daniel equation with step-like initial data
Wen-Yu Zhou,
Shou-Fu Tian,
Xiao-Fan Zhang School of Mathematics, China University of Mining and Technology, Xuzhou, P. R. China
Abstract:
The non-linear steepest descent method is employed to study the long-time asymptotics of solution to the non-local Lakshmanan–Porsezian–Daniel
equation with step-like initial data
$$
q(x,0)=q_0(x)\to\begin{cases}
0, &x\to-\infty,
\\
A, &x\to+\infty,
\end{cases}
$$
where
$A$ is an arbitrary positive constant. We first construct the basic Riemann–Hilbert (RH) problem. After that, to eliminate the influence of singularities, we use the Blaschke–Potapov factor to deform the original RH problem into
a regular RH problem which can be clearly solved. Then different asymptotic behaviors on the whole
$(x,t)$-plane are analyzed in detail. In the region
$(x/t)^2<1/(27\gamma)$ with
$\gamma>0$, there are three real saddle points
due to which the asymptotic behaviors have
a more complicated error term. We prove that the asymptotic solution constructed by the leading and error terms
depends on the values of
$\operatorname{Im}v(-\lambda_j)$,
$j=1,2,3$, where $v(\lambda_j) =-(1/(2\pi))\ln|1+r_1(\lambda_j)r_2(\lambda_j)|-(i/(2\pi))\Delta(\lambda_j)$, $\Delta(\lambda_j)=\int_{-\infty}^{\lambda_j}d \arg(1+r_1(\zeta)r_2(\zeta))$,
$r_i(\xi)$,
$i=1,2$, are the reflection coefficients and
$\lambda_j$ are the saddle points of the
phase function
$\theta(\xi,\mu)$. Besides, the leading term is characterized by parabolic cylinder functions and satisfies boundary conditions. In the region
$(x/t)^2>1/(27\gamma)$ with
$\gamma>0$, there are one real and two conjugate complex saddle points. Based on the positions of these points, we improve the extension forms of the jump contours and successfully obtain the large-time asymptotic results of
the solution in this case.
Keywords:
non-local Lakshmanan–Porsezian–Daniel equation, step-like initial data, Riemann–Hilbert problem, non-linear steepest descent method, long-time asymptotics.
UDC:
517.95
MSC: 35Q51,
35Q15,
35C20,
37K40 Received: 15.06.2024
Revised: 21.10.2024
Language: English
DOI:
10.4213/im9617
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