正则短脉冲的孤立波与Ostrovsky方程

Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction

Coupled Boussinesq equations describe long weakly-nonlinear longitudinal strain waves in a bi-layer with a soft bonding between the layers (e.g. a soft adhesive). From the mathematical viewpoint, a particularly difficult case appears when the linear long-wave speeds in the layers are significantly different (high-contrast case). The traditional derivation of the uni-directional models leads to four uncoupled Ostrovsky equations, for the right- and left-propagating waves in each layer. However, the models impose a ``zero-mass constraint'' i.e. the initial conditions should necessarily have zero…

Traveling waves of the regularized short pulse equation

In the present work, we revisit the so-called regularized short pulse equation (RSPE) and, in particular, explore the traveling wave solutions of this model. We theoretically analyze and numerically evolve two sets of such solutions. First, using a fixed point iteration scheme, we numerically integrate the equation to find solitary waves. It is found that these solutions are well approximated by a truncated series of hyperbolic secants. The dependence of the soliton's parameters (height, width, etc) to the parameters of the equation is also investigated. Second, by developing a multiple scale …

非线性介质中孤立波相互作用的数值分析

Numerical analysis of solitary waves interaction in nonlinear medium

利用数值模拟研究了正则化长波方程孤立波（孤子）之间的相互作用。对于非线性介质的稳定模型的接收，使用线性预测和逐步逼近的方法。通过建模确定，取决于孤子的速度的比率和最高导数平衡的形式是可能的介质非平衡状态的自组织作为冲击波的形成和稳定的孤波的形式，作为全部或部分相互穿透的孤子的结果而创建。第三波中孤子的聚集也是可能的。冲击波可以作为振幅稳定的波阵面进入其它可能的共振态状态，并先于奇异性负阵面发展。

正则化长波方程的多次高阶摄动分析

Multiple-Time Higher-Order Perturbation Analysis of the Regularized Long-Wavelength Equation

通过考虑正则长波方程的长波极限，研究了其多时间高阶发展方程。作为第一个结果，方程的Korteweg-de Vries层次的孤立波解的特定情况下，在提供一个世俗的自由微扰理论中发挥了至关重要的作用。然后，作为一个结果，我们证明了相关的微扰级数可以求和，并给出了RLW方程的精确孤波解。最后，对N-孤子解以及多尺度方法在获得均匀微扰级数时的适用性局限性作了一些评论和考虑。

柱坐标中孤立波长波近似的论证

On the long-wave approximation of solitary waves in cylindrical coordinates

本文论证了柱坐标KdV方程作为径向对称波长波近似的有效性，证明正则化Boussinesq方程与柱KdV近似在L2空间中的误差估计，并指出文献中孤立波解类不包含L2解，从而揭示该方程在径向孤立波建模中的适用性障碍。

正则Boussinesq方程的Whitham调制方程（具有三次非线性）

Whitham modulation equations for the regularized Boussinesq equation with cubic nonlinearity

研究正则Boussinesq方程作为Fermi-Pasta-Ulam晶格的色散长波（准连续）近似，分类了显式周期行波解，并推导了描述周期行波解慢调制的Whitham调制方程，分析了其凸性及非线性特性，揭示了周期行波的调制不稳定性。

On the orbital stability of Gaussian solitary waves in the log-KdV equation

We consider the logarithmic Korteweg-de Vries (log-KdV) equation, which models solitary waves in anharmonic chains with Hertzian interaction forces. By using an approximating sequence of global solutions of the regularized generalized KdV equation in $H^1(\mathbb{R})$ with conserved $L^2$ norm and energy, we construct a weak global solution of the log-KdV equation in a subset of $H^1(\mathbb{R})$. This construction yields conditional orbital stability of Gaussian solitary waves of the log-KdV equation, provided uniqueness and continuous dependence of the constructed solution holds. Furthermore…

Localization in optical systems with an intensity-dependent dispersion

We address the nonlinear Schrodinger equation with intensity-dependent dispersion which was recently proposed in the context of nonlinear optical systems. Contrary to the previous findings, we prove that no solitary wave solutions exist if the sign of the intensity-dependent dispersion coincides with the sign of the constant dispersion, whereas a continuous family of such solutions exists in the case of the opposite signs. The family includes two particular solutions, namely cusped and bell-shaped solitons, where the former represents the lowest energy state in the family and the latter is a l…

Nonlinear dispersive regularization of inviscid gas dynamics

Ideal gas dynamics can develop shock-like singularities with discontinuous density. Viscosity typically regularizes such singularities and leads to a shock structure. On the other hand, in 1d, singularities in the Hopf equation can be non-dissipatively smoothed via KdV dispersion. Here, we develop a minimal conservative regularization of 3d ideal adiabatic flow of a gas with polytropic exponent $γ$. It is achieved by augmenting the Hamiltonian by a capillarity energy $β(ρ) (\nabla ρ)^2$. The simplest capillarity coefficient leading to local conservation laws for mass, momentum, energy and entr…

具有长程、非凸色散介质中的亮传播呼吸子

Bright traveling breathers in media with long-range, nonconvex dispersion

本文数值研究了非局域色散介质中亮传播呼吸子的存在与性质，通过求解Benjamin-Bona-Mahony方程和conduit方程，揭示了非凸色散导致的振荡背景及拓扑相位跳跃现象，为海洋内波和光脉冲等系统提供了理论见解。