arXiv subjects

Binlu Feng

Publications and source records attributed to Binlu Feng.

4 recordsLinked to original sources

arXiv2024

Asymptotic behavior for a new higher-order nonlinear Schr\"{o}dinger equation

We investigate the Cauchy problem of a new higher-order nonlinear Schr\"{o}dinger equation (NHNSE) with weighted Sobolev initial data which is derived by ourselves. By applying $\bar{\partial}$-steepest descent method, we derive the long-time asymptotics of the NHNSE. Explicit steps are as follows: first of all, based on the spectral analysis of a Lax pair and scattering matrice, the solution of the NHNSE is exhibted through solving the corresponding Riemann-Hilbert problem. Secondly, by applying some properties of the Riemann-Hilbert problem, we obtain the long-time asymptotics of the solution to the NHNSE. As we know that the properties of the NHNSE presented in the paper have not been found in any scholar journals.

math.AP↗

arXiv2023

The general solutions for a non-isospectral integrable TD hierarchy via the inverse scattering transform

A non-isospectral Lax pair is first introduced from which a kind of non-isospectral integrable TD hierarchy is derived, whose reduction is an integrable system called the non-isospectral integrable TD system. Then by using the inverse scattering transform (IST) method, new general soliton solutions for the non-isospectral integrable TD hierarchy are obtained. Because we investigate soliton solutions of non-isospectral integrable systems by the IST method, a new Gel'fand-Levitan-Marchenko (GLM) equation needs to be constructed. Finally, we explicitly obtain the exact solutions of the non-isospectral integrable TD system. The method presented in the paper can be extensively applied to other integrable equations.

nlin.SI↗

arXiv2022

Long-time asymptotics for a complex cubic Camassa-Holm equation

In this paper, we investigate the Cauchy problem of the following complex cubic Camassa-Holm (ccCH) equation $$m_{t}=b u_{x}+\frac{1}{2}\left[m\left(|u|^{2}-\left|u_{x}\right|^{2}\right)\right]_{x}-\frac{1}{2} m\left(u \bar{u}_{x}-u_{x} \bar{u}\right), \quad m=u-u_{x x},$$ where $b>0$ is an arbitrary positive real constant. Long-time asymptotics of the equation is obtained through the $\bar{\partial}$-steepest descent method. Firstly, based on the spectral analysis of the Lax pair and scattering matrix, the solution of the equation is able to be constructed via solving the corresponding Riemann-Hilbert (RH) problem. Then, we present different long time asymptotic expansions of the solution $u(y,t)$ in different space-time solitonic regions of $\xi=y/t$. The half-plane ${(y,t):-\infty 0}$ is divided into four asymptotic regions: $\xi \in(-\infty,-1)$, $\xi \in (-1,0)$, $\xi \in (0,\frac{1}{8})$ and $\xi \in (\frac{1}{8},+\infty)$. When $\xi$ falls in $(-\infty,-1)\cup (\frac{1}{8},+\infty)$, no stationary phase point of the phase function $\theta(z)$ exists on the jump profile in the space-time region. In this case, corresponding asymptotic approximations can be characterized with an $N(\Lambda)$-solitons with diverse residual error order $O(t^{-1+2\varepsilon})$. There are four stationary phase points and eight stationary phase points on the jump curve as $\xi \in (-1,0)$ and $\xi \in (0,\frac{1}{8})$, respectively. The corresponding asymptotic form is accompanied by a residual error order $O(t^{-\frac{3}{4}})$.

math.AP↗

arXiv2022

A class of extended high-dimensional nonisospectral KdV hierarchies and symmetry

We construct a new class of N-dimensional Lie algebras and apply them to integrable systems. In this paper, we obtain a nonisospectral KdV integrable hierarchy by introducing a nonisospectral spectral problem. Then, a coupled nonisospectral KdV hierarchy is deduced by means of the corresponding higher-dimensional loop algebra. It follows that the K symmetries, {\tau} symmetries and their Lie algebra of the coupled nonisospectral KdV hierarchy are investigated. The bi-Hamiltonian structures of the both resulting hierarchies are derived by using the trace identity. Finally, we derive a multi-component nonisospectral KdV hierarchy related to the N-dimensional loop algebra, which generalizes the coupled results to an arbitrary number of components.

math-ph↗