We study the collision of two fast solitons for the nonlinear Schrödinger equation in the presence of a spatially adiabatic external potential. For a high initial relative speed $\|v\|$ of the solitons, we show that, up to times of order $\log\|v\|$ after the collision, the solitons preserve their shape (in $L^2$-norm), and the dynamics of the centers of mass of the solitons is approximately determined by the external potential, plus error terms due to radiation damping and the extended nature of the solitons. We remark on how to obtain longer time scales under stronger assumptions on the initial condition and the external potential.
We give a complete characterisation of the reflectionless Schrödinger operators on the line with integrable potentials, solve the inverse scattering problem of reconstructing such potentials from the eigenvalues and norming constants, and derive the corresponding generalized soliton solutions of the Korteweg--de Vries equation
In this paper we give a simple and short proof of asymptotic stability of soliton for discrete nonlinear Schrödinger equation near anti-continuous limit. Our novel insight is that the analysis of linearized operator, usually non-symmetric, can be reduced to a study of simple self-adjoint operator almost like the free discrete Laplacian restricted on odd functions.
In this paper, we study the bifurcation problem from a line soliton for a stationary nonlinear Schrödinger equation on the product space $\mathbb{R} \times \mathbb{T}$. We extend earlier results to a larger class of the nonlinearity in the equation. The salient point of our analysis relies on a lower bound of solution to the ``auxiliary equation'' and then on the application of the Crandall-Rabinowitz argument
A generalized inhomogeneous higher-order nonlinear Schrodinger (GIHNLS) equation for the Heisenberg ferromagnetic spin chain system in (1+1)-dimensions under zero boundary condition at infinity is taken into account. The spectral analysis is first performed to generate a related matrix Riemann-Hilbert problem on the real axis. Then, through solving the resulting matrix Riemann-Hilbert problem by taking the jump matrix to be the identity matrix, the general bright multi-soliton solutions to the GIHNLS equation are attained. Furthermore, the one- and two-soliton solutions are written out and ana…
Exact analytical soliton solutions play an important role in soliton fields. Soliton solutions were obtained with some special constraints on the nonlinear parameters in nonlinear coupled systems, but they usually do not holds in real physical systems. We successfully release all usual constrain conditions on nonlinear parameters for exact analytical vector soliton solutions in $N$-component coupled nonlinear Schrödinger equations. The exact soliton solutions and their existence condition are given explicitly. Applications of these results are discussed in several present experimental paramete…
We have considered cubic nonlinear Schrödinger equation along with supersymmetric $\mathcal{PT}$ like potential and obtained exact stationary solutions in terms of bright and brigh-dark interacting solitons. The $\mathcal{PT}$ broken and $\mathcal{PT}$ unbroken regions are demonstrated also depicted. Connected and disconnected stable regions of bright and dark solitons are examined incorporating linear stability analysis validated by direct numerical simulations. Moreover, the strength of stability has been illustrated through excitations of bright and dark solitons.
In this work, inverse scattering transform for the sixth-order nonlinear Schrödinger equation with both zero and nonzero boundary conditions at infinity is given, respectively. For the case of zero nonzero boundary conditions, in terms of the Laurent's series and generalization of the residue theorem, the bound-state soliton is derived. For nonzero boundary conditions, using the robust inverse scattering transform, we present a matrix Riemann-Hilbert problem of the sixth-order nonlinear Schrödinger equation. Then based on the obtained Riemann-Hilbert problem, the rogue wave and breather wave s…
We explore the phenomena of absorption/emission of solitons by an integrable boundary for the nonlinear Schrödinger equation on the half-line. This is based on the investigation of time-dependent reflection matrices which satisfy the boundary zero curvature equation. In particular, this leads to absorption/emission processes at the boundary that can take place for solitons and higher-order solitons. As a consequence, the usual charges on the half-line are no longer conserved but we show explicitly how to restore an infinite set of conserved quantities by taking the boundary into account. The H…
The Cauchy problem of the modified nonlinear Schrödinger (mNLS) equation with the finite density type initial data is investigated via $\overline{\partial}$ steepest descent method. In the soliton region of space-time $x/t\in(5,7)$, the long-time asymptotic behavior of the mNLS equation is derived for large times. Furthermore, for general initial data in a non-vanishing background, the soliton resolution conjecture for the mNLS equation is verified, which means that the asymptotic expansion of the solution can be characterized by finite number of soliton solutions as the time $t$ tends to infi…
Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrodinger equation with the self-induced parity-time-symmetric potential
研究了具有自诱导宇称时间对称势的可积非局域非线性薛定谔方程[Phys. Rev. Lett. 110（2013）064105]，这是标准NLS方程的可积扩展。利用非局域形式的广义微扰（1，N-1）重达布变换，得到了新的高阶有理孤子.这些有理孤子说明了对于不同参数选择的丰富的波结构（例如，明、暗有理孤子的强相互作用和弱相互作用）。此外，我们还在数值模拟的基础上研究了这些高阶有理孤子在小噪声作用下的动力学行为。
Inverse scattering for the focusing nonlinear Schrödinger equation with elliptic background and full soliton gas
Scuola Internazionale Superiore di Studi Avanzati (SISSA)（高等国际研究学院）；University of Central Florida（中佛罗里达大学）；China University of Mining and Technology（中国矿业大学）
