Weak and Strong Interactions between Internal Solitary Waves

The interaction of weakly nonlinear long internal gravity waves is studied. Weak interactions occur when the wave phase speeds are unequal; this case includes that of a head-on collision. It is shown that each wave satisfies a Korteweg-de Vries equation, and the main effect of the interaction is described by a phase shift. Strong interactions occur when the wave phase speeds are nearly equal although the waves belong to different modes. This case is described by a pair of coupled Korteweg-de Vries equations, for which some preliminary numerical results are presented.     

Oblique Interactions between Internal Solitary Waves

In this paper, we study the oblique interaction of weakly, nonlinear, long internal gravity waves in both shallow and deep fluids. The interaction is classified as weak when where Δ₁=|c_m/c_n?cosδ|, Δ₂=|c_n/c_m?cosδ|,c_m,n, are the linear, long wave speeds for waves with mode numbers m, n, δ is the angle between the respective propagation directions, and α measures the wave amplitude. In this case, each wave is governed by its own Kortweg-de Vries (KdV) equation for a shallow fluid, or intermediate long-wave (ILW) equation for a deep fluid, and the main effect of the interaction is an 0(α) phase shift. A strong interaction (I) occurs when Δ_1,2 are 0(α), and this case is governed by two coupled Kadomtsev-Petviashvili (KP) equations for a shallow fluid, or two coupled two-dimensional ILW equations for deep fluids. A strong interaction (II) occurs when Δ₁ is 0(α), and (or vice versa), and in this case, each wave is governed by its own KdV equation for a shallow fluid, or ILW equation for a deep fluid. The main effect of the interaction is that the phase shift associated with Δ₁ leads to a local distortion of the wave speed of the mode n. When the interacting waves belong to the same mode (i.e., m = n) the general results simplify and we show that for a weak interaction the phase shift for obliquely interacting waves is always negative (positive) for (1/2+cosδ)>0(<0), while the interaction term always has the same polarity as the interacting waves.     

Strong Interactions between Solitary Waves Belonging to Different Wave Modes

Strong interactions between weakly nonlinear long waves are studied. Strong interactions occur when the linear long wave phase speeds are nearly equal although the waves belong to different modes. Specifically we study this situation in the context of internal wave modes propagating in a density stratified fluid. The interaction is described by two coupled Korteweg-deVries equations, which possess both dispersive and nonlinear coupling terms. It is shown that the coupled equations possess an exact analytical solution involving the characteristic “sech²” profile of the Korteweg-deVries equation. It is also shown that when the coefficients satisfy some special conditions, the coupled equations possess an n-solition solution analogous to the Korteweg-deVries n-solition solution. In general though the coupled equations are found not to be amenable to solution by the inverse scattering transform technique, and thus a numerical method has been employed in order to find solutions. This method is described in detail in Appendix A. Several numerical solutions of the coupled equations are presented.     

The Evolution of Finite Amplitude Solitary Rossby Waves on a Weak Shear

The evolution equation is derived for finite amplitude, long Rossby waves on a weak shear generalizing an earlier version given by Benney [1].     

相同模式内弧立波的强斜相互作用*

水文讨论分层流体中相同模式向孤立波的强斜相互作用,包括浅流体情形和深流体情形．采用Lazrange描述方法,发现在浅流体情形相互作用由KP方程描述;在深流体情形相互作用由二维的中等长波方程描述;在无限深情形相互作用由二维的BO方程描述．     

Solitary Waves for Nonconvex FPU Lattices

Solitary waves in a one-dimensional chain of atoms are investigated. The potential energy is required to be monotone and grow super-quadratically. The existence of solitary waves with a prescribed asymptotic strain is shown under certain assumptions on the asymptotic strain and the wave speed. It is demonstrated that the invariance of the equations allows one to transform a system with nonconvex potential energy density to the situation under consideration.     

微结构固体中孤立波的演变及非光滑孤立波

给出了包含宏观应变和微形变的全部二次项以及宏观应变三次项的一种新的自由能函数.利用新自由能函数并根据Mindlin微结构理论，建立了描述微结构固体中纵波传播的一种新模型.利用近来发展的奇行波系统的动力系统理论，分析了系统的所有相图分支，并给出了周期波解、孤立波解、准孤立尖波解、孤立尖波解以及紧孤立波解.孤立尖波解和紧孤立波解的得到，有效地证明了在一定条件下，微结构固体中可以形成和存在孤立尖波和紧孤立波等非光滑孤立波.此结果进一步推广了微结构固体中只存在光滑孤立波的已有结论.     

Envelope Solitary Waves and Periodic Waves in the AB Equations

This article deals with the envelope solitary waves and periodic waves in the AB equations that serve as model equations describing marginally unstable baroclinic wave packets in geophysical fluids and also ultra‐short pulses in nonlinear optics. An envelope solitary wave has a width proportional to its velocity and inversely proportional to its amplitude. The velocity of the envelope solitary wave is partially dependent on its amplitude in the sense that the amplitude determines the upper or lower limit of the velocity. When two envelope solitary waves collide, they survive the collision and retain their identities except for a shift in the positions of both the envelopes and the carrier waves. The periodic wave solutions in sine wave form may be stable or unstable depending upon the wave parameters. When the sine wave is destabilized by small perturbations, its long‐time evolution shows a Fermi–Pasta–Ulam‐type oscillation.     

Nonlinear Wave Equations for Oceanic Internal Solitary Waves

In the coastal ocean, the interaction of barotropic tidal currents with topographic features such as the continental shelf, sills in narrow straits, and bottom ridges are often observed to generate large amplitude, horizontally propagating internal solitary waves. These are long nonlinear waves and hence can be modeled by equations of the Korteweg–de Vries type. Typically they occur in regions of variable bottom topography, with the consequence that the appropriate nonlinear evolution equation has variable coefficients. Further, as these waves can be long‐lived it is necessary to take account of the effects of the Earth's background rotation. We review this family of model evolution equations and some of their pertinent solutions, obtained both asymptotically and numerically.     

Perturbations of Solitons and Solitary Waves

A direct perturbation method is developed to investigate the evolution of solitary waves in the presence of small perturbations. A uniformly valid first order solution is constructed. The method is applied to several nonlinear evolution equations which support solitons or solitary waves. Finally, the method is compared with other approaches in the literature.     

耦合BBM方程组孤立波解的轨道稳定性(英文)

本文研究具有Hamilton形式的耦合BBM方程组孤立波解的轨道稳定性.首先找到两族显式孤立波解.然后通过详细的谱分析证明出孤立波解的轨道稳定性.     

Convective Linear Stability of Solitary Waves for Boussinesq Equations

Boussinesq was the first to explain the existence of Scott Russell's solitary wave mathematically. He employed a variety of asymptotically equivalent equations to describe water waves in the small-amplitude, long-wave regime. We study the linearized stability of solitary waves for three linearly well-posed Boussinesq models. These are problems for which well-developed Lyapunov methods of stability analysis appear to fail. However, we are able to analyze the eigenvalue problem for small-amplitude solitary waves, by comparison to the equation that Boussinesq himself used to describe the solitary wave, which is now called the Korteweg–de Vries equation. With respect to a weighted norm designed to diminish as perturbations convect away from the wave profile, we prove that nonzero eigenvalues are absent in a half-plane of the form R λ>− b for some b >0, for all three Boussinesq models. This result is used to prove the decay of solutions of the evolution equations linearized about the solitary wave, in two of the models. This "convective linear stability" property has played a central role in the proof of nonlinear asymptotic stability of solitary-wave-like solutions in other systems.     

Orbital Stability of Solitary Waves for Generalized Zakharov System

In the interaction of laser-plasma the system of Zakharov equation plays an important role.This system attracted many scientists' wide interest and attention.And the formation, evolution and interaction of the Langmuir solutions differ from solutions of the KDV equation. Here we consider the following generalized Zakharov system     

Instability of Solitary Waves in Nonlinear Composite Media

In this paper,we investigate a class of Hamiltonian systems arising in nonlinear composite media.By detailed analysis and computation we obtain a decaying estimates on the semigroup and prove the orbitalinstability of two families of explicit solitary wave solutions (slow family in anisotropic case and solitary wavesin isotropic case),which theoretically verify the related guess and numerical results.     

Two-Wave Interactions for Weakly Nonlinear Hyperbolic Waves

This paper is devoted to studying the weakly nonlinear interaction of two waves whose propagation is governed by n × n hyperbolic systems of conservation laws. Our method of approach involves introducing two nonlinear phase variables and carrying out a perturbation analysis. This extended version of our previous single-wave-mode theory [5] is then applied to the equations of gas dynamics to study interacting sound waves. Numerical results for the wave-wave interaction are presented graphically in a set of figures.     

Evolution of Higher-Order Gray Hirota Solitary Waves

The defocusing Hirota equation has dark and gray soliton solutions which are stable on a background of periodic waves of constant amplitude. In this paper, gray solitary wave evolution for a higher-order defocusing Hirota equation is examined. A direct analysis is used to identify families of higher-order gray Hirota solitary waves, which are embedded for certain parameter values. Soliton perturbation theory is used to determine the detailed behavior of an evolving higher-order gray Hirota solitary wave. An integral expression for the first-order correction to the wave is found and analytical expressions for the steady-state and transient components of the solitary wave tail are derived. A subtle and complex picture of the development of solitary wave tails emerges. It is found that solitary wave tails develop for two reasons, one is decay of the solitary wave caused by resonance, the second is corrections at first-order to the background wave. Strong agreement is found between the theoretical predictions of the perturbation theory and numerical solutions of the governing equations.     

Accelerated Imaginary-time Evolution Methods for the Computation of Solitary Waves

Two accelerated imaginary-time evolution methods are proposed for the computation of solitary waves in arbitrary spatial dimensions. For the first method (with traditional power normalization), the convergence conditions as well as conditions for optimal accelerations are derived. In addition, it is shown that for nodeless solitary waves, this method converges if and only if the solitary wave is linearly stable. The second method is similar to the first method except that it uses a novel amplitude normalization. The performance of these methods is illustrated on various examples. It is found that while the first method is competitive with the Petviashvili method, the second method delivers much better performance than the first method and the Petviashvili method.     

Large Amplitude Solitary Waves in Unbounded Stratified Fluids

Solitary waves of arbitrary amplitude are found to exist for a class of unbounded stratified flow configurations. In special cases the solutions are similar to those obtained from the Benjamin-Ono equation.     

有限变形弹性圆杆中的孤波

在同时引入横向惯性和横向剪切应变的情况下,导出了有限变形弹性圆杆的非线性纵向波动方程,方程中包含了二次和三次的非线性项以及由横向剪切与横向惯性导致的两种几何弥散效应．借助Mathematica软件,利用双曲正割函数的有限展开法,对该方程和对应的截断的非线性方程进行求解,得到了非线性波动方程的孤波解,同时给出了这些解存在的必要条件．