Asymptotic Stability of Lattice Solitons in the Energy Space

Orbital and asymptotic stability for 1-soliton solutions of the Toda lattice equations as well as for small solitary waves of the FPU lattice equations are established in the energy space. Unlike analogous Hamiltonian PDEs, the lattice equations do not conserve the adjoint momentum. In fact, the Toda lattice equation is a bidirectional model that does not fit in with the existing theory for the Hamiltonian systems by Grillakis, Shatah and Strauss. To prove stability of 1-soliton solutions, we split a solution around a 1-soliton into a small solution that moves more slowly than the main solitary wave and an exponentially localized part. We apply a decay estimate for solutions to a linearized Toda equation which has been recently proved by Mizumachi and Pego to estimate the localized part. We improve the asymptotic stability results for FPU lattices in a weighted space obtained by Friesecke and Pego.     

Decaying periodic-wavetrain and solitary-vortex exact solutions for two-dimensional planetary viscous flows

Summary We give Taylor-Kovasznay solutions for two-dimensional motions in a layer of viscous fluid on a rotating globe in the β-plane approximation. These solutions represent travelling-wave-type periodic wavetrains and solitary vortices which decay away in time due to viscous effects. Due to the relevance of its scientific content, this paper has been given priority by the Journal Direction.     

A universal separatrix map for weak interactions of solitary waves in generalized nonlinear Schrödinger equations

It is known that weak interactions of two solitary waves in generalized nonlinear Schrödinger (NLS) equations exhibit fractal dependence on initial conditions, and the dynamics of these interactions is governed by a universal two-degree-of-freedom ODE system [Y. Zhu J. Yang, Universal fractal structures in the weak interaction of solitary waves in generalized nonlinear Schrödinger equations, Phys. Rev. E 75 (2007) 036605]. In this paper, this ODE system is analyzed comprehensively. Using asymptotic methods along separatrix orbits, a simple second-order map is derived. This map does not have any free parameters after variable rescalings, and thus is universal for all weak interactions of solitary waves in generalized NLS equations. Comparison between this map’s predictions and direct simulations of the ODE system shows that the map can capture the fractal-scattering phenomenon of the ODE system very well both qualitatively and quantitatively.     

A mathematical model for wave propagation in elastic tubes with inhomogeneities: Application to blood waves propagation

We study the propagation of waves in an elastic tube filled with an inviscid fluid. We consider the case of inhomogeneity whose mechanical and geometrical properties vary in space. We deduce a system of equations of the Boussinesq type as describing the wave propagation in the tube. Numerical simulations of these equations show that inhomogeneities prevent separation of right-going from left-going waves.Then reflected and transmitted coefficients are obtained in the case of localized constriction and localized rigidity. Next we focus on wavetrains incident on various types of anomalous regions. We show that the existence of anomalous regions modifies the wavetrain patterns.     

求解非线性差分方程孤立波解的直接代数法

推广了求解非线性差分方程孤立波解的直接代数法.用此方法研究了Hybrid晶格方程，借助于符号计算Maple,得到它的新孤波解.这种方法也可用于求解其他的差分方程. 关键词： 微分-差分方程 Hybrid晶格方程 行波解 孤     

Stability Analysis of Solitary Wave Solutions for Coupled and(2+1)-Dimensional Cubic Klein-Gordon Equations and Their Applications

The searching exact solutions in the solitary wave form of non-linear partial differential equations(PDEs play a significant role to understand the internal mechanism of complex physical phenomena. In this paper, we employ the proposed modified extended mapping method for constructing the exact solitary wave and soliton solutions of coupled Klein-Gordon equations and the(2+1)-dimensional cubic Klein-Gordon(K-G) equation. The Klein-Gordon equation are relativistic version of Schr¨odinger equations, which describe the relation of relativistic energy-momentum in the form of quantized version. We productively achieve exact solutions involving parameters such as dark and bright solitary waves, Kink solitary wave, anti-Kink solitary wave, periodic solitary waves, and hyperbolic functions in which severa solutions are novel. We plot the three-dimensional surface of some obtained solutions in this study. It is recognized that the modified mapping technique presents a more prestigious mathematical tool for acquiring analytical solutions o PDEs arise in mathematical physics.     

Improved Conditions for Global Asymptotic Stability of Cohen--Grossberg Neural Networks with Time-Varying Delays

The global asymptotic stability of delayed Cohen-Grossberg neural networks with impulses is investigated. Based on the new suitable Lyapunov functions and the Jacobsthal inequality, a set of novel sufficient criteria are derived for the global asymptotic stability of Cohen-Grossberg neural networks with time-varying delays and impulses. An illustrative example with its numerical simulations is given to demonstrate the effectiveness of the obtained results.     

Coupled Klein–Gordon equations and energy exchange in two-component systems

A system of coupled Klein–Gordon equations is proposed as a model for one-dimensional nonlinear wave processes in two-component media (e.g., long longitudinal waves in elastic bi-layers, where nonlinearity comes only from the bonding material). We discuss general properties of the model (Lie group classification, conservation laws, invariant solutions) and special solutions exhibiting an energy exchange between the two physical components of the system. To study the latter, we consider the dynamics of weakly nonlinear multi-phase wavetrains within the framework of two pairs of counter-propagating waves in a system of two coupled Sine–Gordon equations, and obtain a hierarchy of asymptotically exact coupled evolution equations describing the amplitudes of the waves. We then discuss modulational instability of these weakly nonlinear solutions and its effect on the energy exchange.     

用VOF法数值模拟孤立波迎撞及在后台阶上的演化

通过数值离散求解二维Navier-Stokes方程和利用VOF界面跟踪技术,分别对两个界面孤立波之间的迎撞问题和一个孤立波在后台阶地形上的演化问题进行数值模拟。     

求sine-Gordon 型方程孤波解的一种统一方法

借助于一个辅助常微分方程的解，提出了一种求sine-Gordon 型方程孤波解的统一方法，并用该法简洁地求得了三个著名的sine-Gordon 型方程，即单sine-Gordon 方程、双sine-Gordon 方程和mKdV-sine-Gordon方程的精确孤波解。     

非线性演化方程孤立波的同伦分析法求解

利用同伦分析法求解了Burgers方程，得到了其扭结形孤立波的近似解析解，该解非常接近于相应的精确解.结果表明，同伦分析法可用来求解非线性演化方程的孤立波解.同时，也对所用方法进行了一定扩展，得到了Kadomtsev-Petviashvili(KP)方程的钟形孤立子解.经过扩展后的方法能够更方便地用于求解更多非线性演化方程的高精度近似解析解. 关键词： Burgers方程 同伦分析法 KP方程 孤立波解     

修正cKdV方程组的孤立波结构及其稳定性

利用函数展开法求解修正耦合KdV(Coupled KdV,cKdV)方程组,得到几类孤立波解,包括扭结型-钟型、双扭结型、双钟型以及双扭结-双钟型结构的单孤子解.在不同的极限情况下,这些解分别退化为修正cKdV方程的扭结状或钟状孤波解.对孤立波的稳定性进行了数值研究,结果表明:修正cKdV方程既存在稳定的孤立波解,也存在不稳定的孤立波解.     

Periodic wavetrains for systems of coupled nonlinear Schrödinger equations

Exact, periodic wavetrains for systems of coupled nonlinear Schrödinger equations are obtained by the Hirota bilinear method and theta functions identities. Both the bright and dark soliton regimes are treated, and the solutions involve products of elliptic functions. The validity of these solutions is verified independently by a computer algebra software. The long wave limit is studied. Physical implications will be assessed.     

2N+1阶KdV型方程的孤波解

获得了２Ｎ＋１阶ＫｄＶ型方程的显式精确孤波解．作为特例，讨论了高阶广义ＫｄＶ型方程、高阶广义ＭＫｄＶ型方程和高阶广义Ｓｃｈａｍｅｌ的ＭＫｄＶ型方程．还研究了２Ｎ＋１阶ＫＰ型方程 关键词：     

Combined optical solitary waves of the Fokas—Lenells equation

In this work, we investigate the Fokas–Lenells equation describing the propagation of ultrashort pulses in optical fibers when certain terms of the next asymptotic order beyond those necessary for the nonlinear Schrö dinger equation are retained. In addition to group velocity dispersion and Kerr nonlinearity, the model involves both spatio-temporal dispersion and self-steepening terms. A class of exact combined solitary wave solutions of this equation is constructed for the first time, by adopting the complex envelope function ansatz. The influences of spatio-temporal dispersion on the characteristics of combined solitary waves is also discussed.     

Jacobi椭圆函数展开法及其在求解非线性波动方程中的应用

给出了Jacobi椭圆函数展开法,且应用该方法获得了几种非线性波方程的准确周期解.该方法包含了双曲函数展开法,应用该方法得到的周期解包含了冲击波解和孤波解. 关键词： Jacobi椭圆函数 非线性方程 周期解 孤波解     

On the generalized Kadomtsev-Petviashvili equation with generalized evolution and variable coefficients

In this Letter, the existence of the solitary wave solution of the Kadomtsev-Petviashvili equation with generalized evolution and time-dependent coefficients will be studied. We use the solitary wave ansätze-method to derive these solutions. A couple of conserved quantities are also computed. Moreover, some figures are plotted to see the effects of the coefficient functions on the propagation and asymptotic characteristics of the solitary waves.     

The modulation instability revisited

The modulational instability (or “Benjamin-Feir instability”) has been a fundamental principle of nonlinear wave propagation in systems without dissipation ever since it was discovered in the 1960s. It is often identified as a mechanism by which energy spreads from one dominant Fourier mode to neighboring modes. In recent work, we have explored how damping affects this instability, both mathematically and experimentally. Mathematically, the modulational instability changes fundamentally in the presence of damping: for waves of small or moderate amplitude, damping (of the right kind) stabilizes the instability. Experimentally, we observe wavetrains of small or moderate amplitude that are stable within the lengths of our wavetanks, and we find that the damped theory predicts the evolution of these wavetrains much more accurately than earlier theories. For waves of larger amplitude, neither the standard (undamped) theory nor the damped theory is accurate, because frequency downshifting affects the evolution in ways that are still poorly understood.     

Interface Shape and Concentration Distribution in Crystallization from Solution under Microgravity

The usual formally variable separation approach is valid only for completely integrable models. In this paper, we extend the method to a nonintegrable generalized Hirota-Satsuma equations. Some new exact solitary wave solutions and periodic wave solutions of the equations are also obtained.