Interaction of a Solitary Wave with an External Force Moving with Variable Speed

The interaction of a solitary wave with an external force moving with constant acceleration is studied within the forced Korteweg-de Vries equation. For the case of a weak isolated force an asymptotic model based on equations for the amplitude and position of the solitary wave is developed. Phase portraits for this asymptotic system are obtained analytically and numerically. Analysis has shown that an accelerated force of either sign can capture a solitary wave if the acceleration is less than a certain critical value, depending on the forcing amplitude (for the case of a constant force speed only a positive force can capture a solitary wave). Direct numerical simulation of the forced Korteweg-de Vries equation has confirmed the predictions of the asymptotic model. Also, it is shown numerically that the accelerated force can capture more than one solitary wave.     

Do peaked solitary water waves indeed exist?

Many models of shallow water waves, such as the famous Camassa–Holm equation, admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa–Holm equation. Besides, it is proved that Kelvin’s theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, unlike the traditional smooth waves that are dispersive with wave height, the phase speed of the peaked solitary waves has nothing to do with wave height, but depends (for a fixed wave height) on its decay length, i.e., the actual wavelength: in fact, the peaked solitary waves are dispersive with the actual wavelength when wave height is fixed. In addition, unlike traditional smooth waves whose kinetic energy decays exponentially from free surface to bottom, the kinetic energy of the peaked solitary waves either increases or almost keeps the same. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.     

Rossby Elevation Waves in the Presence of a Critical Layer

In a previous paper, we investigated the solitary-wave-like development of small-amplitude Rossby waves propagating in a zonal shear current, for the particular case when the Rossby wave speed equals the mean-flow velocity at a certain latitude in the β-plane. We presented a general theory for the nonlinear critical-layer theory, and illustrated it by explicitly describing the motion of a depression solitary wave (D-wave). Here, we report a continuation of that study and consider the more complex case of an elevation solitary wave (E-wave). The method involves matched asymptotic expansions between the outer flow away from the critical layer and the inner flow inside the latter, both these flows having different scalings. We showed previously that the critical-layer flow expansion diverged in the case of the E-wave on the separatrices bounding the open and closed streamlines, which led us to defer a detailed E-wave study. Thus, in this paper, we examine the motion in the additional layer located along the separatrices where this singularity is removed by using a third scaling and find that the previous undesirable distortions are discarded. The evolution equation is derived and is a Korteveg-de-Vries type-equation modified by new nonlinear terms generated by the nonlinear interactions occuring in the critical layer. This equation supports a family of E-waves provided that the mean flow obeys certain conditions. The energy exchange that occurs between the mean flow and the D or E-wave during the critical-layer formation is evaluated in the quasi-steady régime assumption.     

Stability of travelling wave solutions to a semilinear hyperbolic system with relaxation

We study a semilinear hyperbolic system with relaxation and investigate the asymptotic stability of travelling wave solutions with shock profile. It is shown that the travelling wave solution is asymptotically stable, provided the initial disturbance is suitably small. Moreover, we show that the time convergence rate is polynomially (resp. exponentially) fast as t→∞ if the initial disturbance decays polynomially (resp. exponentially) for x→∞. Our proofs are based on the space–time weighted energy method. Copyright © 2008 John Wiley & Sons, Ltd.     

Rossby Solitary Waves in the Presence of a Critical Layer

This study considers the evolution of weakly nonlinear long Rossby waves in a horizontally sheared zonal current. We consider a stable flow so that the nonlinear time scale is long. These assumptions enable the flow to organize itself into a large‐scale coherent structure in the régime where a competition sets in between weak nonlinearity and weak dispersion. This balance is often described by a Korteweg‐de‐Vries equation. The traditional assumption of a weak amplitude breaks down when the wave speed equals the mean flow velocity at a certain latitude, due to the appearance of a singularity in the leading‐order equation, which strongly modifies the flow in a critical layer. Here, nonlinear effects are invoked to resolve this singularity, because the relevant geophysical flows have high Reynolds numbers. Viscosity is introduced in order to render the nonlinear‐critical‐layer solution unique, but the inviscid limit is eventually taken. By the method of matched asymptotic expansions, this inner flow is matched at the edges of the critical layer with the outer flow. We will show that the critical‐layer–induced flow leads to a strong rearrangement of the related streamlines and consequently of the potential‐vorticity contours, particularly in the neighborhood of the separatrices between the open and closed streamlines. The symmetry of the critical layer vis‐à‐vis the critical level is also broken. This theory is relevant for the phenomenon of Rossby wave breaking and eventual saturation into a nonlinear wave. Spatially localized solutions are described by a Korteweg‐de‐Vries equation, modified by new nonlinear terms; depending on the critical‐layer shape, this leads to depression or elevation waves. The additional terms are made necessary at a certain order of the asymptotic expansion while matching the inner flow on the dividing streamlines. The new evolution equation supports a family of solitary waves. In this paper we describe in detail the case of a depression wave, and postpone for further discussion the more complex case of an elevation wave.     

Effects of Quadratic Singular Curves in Integrable Equations

In this paper, the effects of quadratic singular curves in integrable wave equations are studied by using the bifurcation theory of dynamical system. Some new singular solitary waves (pseudo‐cuspons) and periodic waves are found more weak than regular singular traveling waves such as peaked soliton (peakon), cusp soliton (cuspon), cusp periodic wave, etc. We show that while the first‐order derivatives of the new singular solitary wave and periodic waves exist, their second‐order derivatives are discontinuous at finite number of points for the solitary waves or at infinitely countable points for the periodic wave. Moreover, an intrinsic connection is constructed between the singular traveling waves and quadratic singular curves in the phase plane of traveling wave system. The new singular periodic waves, pseudo‐cuspons, and compactons emerge if corresponding periodic orbits or homoclinic orbits are tangent to a hyperbola, ellipse, and parabola. In particular, pseudo‐cuspon is proposed for the first time. Finally, we study the qualitative behavior of the new singular solitary wave and periodic wave solutions through theoretical analysis and numerical simulation.     

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.     

Transverse linear instability of solitary waves for coupled long-wave–short-wave interaction equations

In this paper, we investigate the transverse linear instability of one-dimensional solitary wave solutions of the coupled system of two-dimensional long-wave–short-wave interaction equations. We show that the one-dimensional solitary waves are linearly unstable to perturbations in the transverse direction if the coefficient of the term associated with transverse effects is negative. This transverse instability condition coincides with the non-existence condition identified in the literature for two-dimensional localized solitary wave solutions of the coupled system.     

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.     

Solitary Wave Transformation Due to a Change in Polarity

Solitary wave transformation in a zone with sign-variable coefficient for the quadratic nonlinear term is studied for the variable-coefficient Korteweg–de Vries equation. Such a change of sign implies a change in polarity for the solitary wave solutions of this equation. This situation can be realized for internal waves in a stratified ocean, when the pycnocline lies halfway between the seabed and the sea surface. The width of the transition zone of the variable nonlinear coefficient is allowed to vary over a wide range. In the case of a short transition zone it is shown using asymptotic theory that there is no solitary wave generation after passage through the turning point, where the coefficient of the quadratic nonlinear term goes to zero. In the case of a very wide transition zone it is shown that one or more solitary waves of the opposite polarity are generated after passage through the turning point. Here, asymptotic methods are effective only for the first (adiabatic) stage when the solitary wave is approaching the turning point. The results from the asymptotic theories are confirmed by direct numerical simulation. The hypothesis that the pedestal behind the solitary wave approaching the turning point has a significant role on the generation of the terminal solitary wave after the transition zone is examined. It is shown that the pedestal is not the sole contributor to the amplitude of the terminal solitary wave. A negative disturbance at the turning point due to the transformation in the zone of the variable nonlinear coefficient contributes as much to the process of the generation of the terminal solitary waves.     

Backlund Transformation and Exact Solution to Kadomtsev-Petviashvili Equation with Variable Coefficients

By using the homogeneous balance principle, we derive a Backlund transformation (BT) to (3+1)-dimensionaI Kadomtsev-Petviashvili (K-P) equation with variable coefficients if the variable coefficients are linearly dependent. Based on the BT, the exact solution of the (3+1)-dimensional K-P equation is given. By the same method, we derive a BT and the solution to (2+1)-dimensional K-P equation. The variable coefficients can change the amplitude of solitary wave, but cannot change the form of solitary wave.     

Approximate damped oscillatory solutions for compound KdV-Burgers equation and their error estimates

In this paper,we focus on studying approximate solutions of damped oscillatory solutions of the compound KdV-Burgers equation and their error estimates.We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdV-Burgers equation.We obtain some global phase portraits under different parameter conditions as well as the existence of bounded traveling wave solutions.Furthermore,we investigate the relations between the behavior of bounded traveling wave solutions and the dissipation coefficient r of the equation.We obtain two critical values of r,and find that a bounded traveling wave appears as a kink profile solitary wave if |r| is greater than or equal to some critical value,while it appears as a damped oscillatory wave if |r| is less than some critical value.By means of analysis and the undetermined coefficients method,we find that the compound KdV-Burgers equation only has three kinds of bell profile solitary wave solutions without dissipation.Based on the above discussions and according to the evolution relations of orbits in the global phase portraits,we obtain all approximate damped oscillatory solutions by using the undetermined coefficients method.Finally,using the homogenization principle,we establish the integral equations reflecting the relations between exact solutions and approximate solutions of damped oscillatory solutions.Moreover,we also give the error estimates for these approximate solutions.     

Baecklund Transformation and Exact Solution to Kadomtsev-Petviashvili Equation with Variable Coefficients

By using homogeneous balance principle,we derive a Baecklund trans-formation(BT) to (3 1)-dimensional Kadomtsev-Petviashvili( K-P) equation with variable coefficients if the variable coefficients are linearly dependent.Based on the BT,the exact solution of the (3 1)-dimensional K-P equation is given.By the same method,we derive a BT and the solution to (2 1)-dimensional K-P equation,The variable coefficients can change the amplitude of solitary wave,but cannot change the form of solitary wave.     

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.     

New periodic solitary-wave solutions for the Benjiamin Ono equation

In this paper, the new periodic solitary wave and doubly periodic solutions for (1 + 1)-dimensional Benjiamin Ono equation are obtained, using the bilinear method and extended homoclinic test approach. These results demonstrate that the integrable system has richly dynamical behavior even if it is (1 + 1)-dimensional.     

Three-dimensional gravity waves in a channel of variable depth

We consider existence of three-dimensional gravity waves traveling along a channel of variable depth. It is well known that the long-wave small-amplitude expansion for such waves results in the stationary Korteweg–de Vries equation, coefficients of which depend on the transverse topography of the channel. This equation has a single-humped solitary wave localized in the direction of the wave propagation. We show, however, that there exists an infinite set of resonant Fourier modes that travel at the same speed as the solitary wave does. This fact suggests that the solitary wave confined in a channel of variable depth is always surrounded by small-amplitude oscillatory disturbances in the far-field profile.     

有限深两层流中内孤立波的高阶解*

本文研究有限水深两层流中孤立波的三阶近似理论,并考虑了自由表面对孤立波的影响,运用坐标变形方法得到了三阶内孤立波的发展方程,求得波速的解析表达式。对方程进行了数值计算,得到了几种参数下三阶解曲线,指出自由表面对波型和波速的影响是二阶的。计算表明三阶解对一阶、二阶解有明显的改进,使其更加接近试验结果。     

Exact solitary wave solutions of nonlinear wave equations

The hyperbolic function method for nonlinear wave equations is presented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Gr?bner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.     

The asymptotic average-shadowing property and transitivity for flows

The asymptotic average-shadowing property is introduced for flows and the relationships between this property and transitivity for flows are investigated. It is shown that a flow on a compact metric space is chain transitive if it has positively (or negatively) asymptotic average-shadowing property and a positively (resp. negatively) Lyapunov stable flow is positively (resp. negatively) topologically transitive provided it has positively (resp. negatively) asymptotic average-shadowing property. Furthermore, two conditions for which a flow is a minimal flow are obtained.