Eugenio Montefusco, Benedetta Pellacci, Marco Squassina
The soliton dynamics in the semiclassical limit for a weakly coupled nonlinear focusing Schrödinger systems in presence of a nonconstant potential is studied by taking as initial data some rescaled ground state solutions of an associate elliptic system.
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Asymptotic Stability of Multi-Solitons for Coupled Nonlinear Schrödinger Equations via the $\bar{\partial}$-Method
本文基于$\bar{\partial}$-最速下降法，建立了聚焦耦合非线性薛定谔方程在加权索伯列夫空间中的长时间渐近行为，将误差估计改进至$t^{-3/4}$阶，并证明了多孤子解的渐近稳定性及碰撞规律。
Stability analysis of breathers for coupled nonlinear Schrodinger equations
本文研究了耦合非线性 Schrodinger 方程中非退化矢量孤子的谱稳定性和呼吸子的非线性稳定性，发现孤子在负 Krein 符号特征值下仍谱稳定，并利用 Lyapunov 方法证明了呼吸子的非线性稳定性。
Nonlinear stability of vector multi-solitons in coupled NLS and modified KdV equations
该研究证明耦合NLS和修正KdV方程的多孤子（含呼吸子）在特定Sobolev空间中非线性稳定，并给出Lyapunov泛函负特征值个数，方法基于可积性，同时推广至经典方程及李代数框架。
Enhanced PINNs for data-driven solitons and parameter discovery for (2+ 1)-dimensional coupled nonlinear Schrödinger systems
针对描述受扰光纤横向效应的（2+1）维C-CNLSEs，提出增强型PINN算法（含局部自适应激活函数、区域特异性加权损失函数），实现暗/反暗孤子结构恢复与常数/变系数参数发现，提升收敛速度与精度，为高维复杂孤子动力学研究提供新框架。
Dynamics of the combined nonlinear Schrödinger equation with inverse-square potential
本文研究 d=3,4,5 时带反平方势的组合能量临界 NLS，在阈值以下及阈值能量处分别用集中紧性、变分分析和调制分析刻画散射、爆破及临界动力学，并推广标准组合 NLS 的结果。
We consider the cubic Szego equation perturbed by a small Toeplitz potential (natural generalization of a multiplicative potential) and having as initial condition a soliton of the unperturbed equation. We show that the solution preserves the soliton shape for a long time, and we determine the effective ODEs satisfied by the modulation parameters of the soliton.
We show that a soliton scattered by an external delta potential splits into two solitons and a radiation term. Theoretical analysis gives the amplitudes and phases of the reflected and transmitted solitons with errors going to zero as the velocity of the incoming soliton tends to infinity. Numerical analysis shows that this asymptotic relation is valid for all but very slow solitons. We also show that the total transmitted mass, that is the square of the $L^2$ norm of the solution restricted on the transmitted side of the delta potential is in good agreement with the quantum transmission rate …
Modeling soliton-potential interactions for the 1-D cubic NLS via an invariant-preserving finite-difference method
该研究针对带光滑快速衰减空间依赖势的一维三次NLS，采用保质量和能量不变量的有限差分法，对不同入射速度下的孤子-势相互作用进行建模。
Multi-solitons for the nonlinear Schrödinger equation with repulsive Dirac delta potential
本文证明了带排斥狄拉克δ势和超临界聚焦非线性项的NLS方程多孤子解的存在性，主要创新在于处理静止基态部分，其线性化算子有两个不稳定特征值。
Consider the nonlinear Schrödinger equation (NLS) with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and, if the nonlinearity is focusing, then also solitons with positive large energy, which are unstable. In this paper we classify the global dynamics below the second lowest energy of solitons under small mass and radial symmetry constraints.
Transmission of fast solitons for the NLS with an external potential
该研究探讨含外势的三次NLS方程中加速孤子的动力学，证明足够大速度下孤子可有效传输，拓展了前人关于δ势两类情形的相关研究成果。
In this work we mainly consider the dynamics and scattering of a narrow soliton of NLS equation with a potential in $\mathbb{R}^3$, where the asymptotic state of the system can be far from the initial state in parameter space. Specifically, if we let a narrow soliton state with initial velocity $\upsilon_{0}$ to interact with an extra potential $V(x)$, then the velocity $\upsilon_{+}$ of outgoing solitary wave in infinite time will in general be very different from $\upsilon_{0}$. In contrast to our present work, previous works proved that the soliton is asymptotically stable under the assumpt…
In this paper, a class of coupled systems of nonlinear Schrodinger equations with sign-changing potential, including the linearly coupled case, is considered. The existence of non-trivial bound state solutions via linking methods for cones in Banach spaces is proved.
Two-solitons with logarithmic separation for 1D NLS with repulsive delta potential
研究一维带排斥δ势的聚焦非线性薛定谔方程，证明弱势下可构造对数分离双孤子解，强势下则不存在此类解。
Global dynamics below excited solitons for the non-radial NLS with potential
该研究针对带外势的三维三次非线性薛定谔方程，将Nakanishi关于径向情形下小质量激发孤子以下解的全局动力学结果，推广至非径向情形，证明此类解要么散射到基态，要么H¹范数随时间增长。
We investigate the validity of a soliton dynamics behavior in the semi-relativistic limit for the nonlinear Schrödinger equation in $\R^{N}, N\ge 3$, in presence of a singular external potential.
We show that the action potential signals generated inside axons are reaction-diffusion solitons or reaction-diffusion waves, refuting the Hodgkin and Huxley hypothesis that action potentials propagate along axons with an elastic wave mechanism. Reaction-diffusion action potential wavefronts and solitons annihilate at collision and boundaries of axons, in contrast with elastic waves, where amplitudes add up and reflect at boundaries. We numerically calculate the values of the speed of the action potential spikes and the dispersion relations. These findings suggest several experiments as valida…
We study the Gross-Pitaevskii equation with a slowly varying smooth potential, $V(x) = W(hx)$. We show that up to time $\log(1/h)/h $ and errors of size $h^2$ in $H^1$, the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian, $ (ξ^2 + \sech^2 * V (x))/2 $. This provides an improvement ($ h \to h^2 $) compared to previous works, and is strikingly confirmed by numerical simulations.
We consider asymptotic stability of a small solitary wave to supercritical 2-dimensional nonlinear Schrödinger equations $$ iu_t+Δu=Vu\pm |u|^{p-1}u \quad\text{for $(x,t)\in\mathbb{R}^2\times\mathbb{R}$,}$$ in the energy class.
The integrable nonlocal nonlinear Schrodinger (NNLS) equation with the self-induced parity-time-symmetric potential [Phys. Rev. Lett. 110 (2013) 064105] is investigated, which is an integrable extension of the standard NLS equation. Its novel higher-order rational solitons are found using the nonlocal version of the generalized perturbation (1, N-1)-fold Darboux transformation. These rational solitons illustrate abundant wave structures for the distinct choices of parameters (e.g., the strong and weak interactions of bright and dark rational solitons). Moreover, we also explore the dynamical b…
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