[SIAM NEWS] Nonlinear Ocean-Wave Interactions on Flat Beaches

By M.J. Ablowitz and D.E. Baldwin

June 03, 2013

[M.J. Ablowitz is a professor of applied mathematics at the University of Colorado, Boulder, from which D.E. Baldwin recently received a PhD.]

People have always been fascinated by waves, particularly water and ocean waves. The mathematical study of water waves goes back to the origins of differential equations. While linear equations are often good models for small-amplitude waves, nonlinear equations are needed for larger amplitudes. We have seen and photographed interacting nonlinear waves that occur daily at two relatively flat beaches; a well-known nonlinear wave equation has solutions that are remarkably similar to what we observed

人们总是会被波吸引，尤其是水波和海波。水波的数学研究可以追溯到微分方程的起源。线性方程通常是小振幅波的良好模型，但更大振幅的波需要非线性方程。我们已经看到并拍摄了每天出现在两个相对平坦的海滩上相互作用的非线性波；一个著名的非线性波动方程具有与我们的观察非常相似的解。

Even Newton (1642-1727) was interested in providing a mathematical description of water waves, but many years would pass before this was feasible. In 1757 Euler derived the inviscid equations of fluid dynamics. Soon afterward, Laplace and Lagrange found linear approximations to the water-wave equations. In 1816 Cauchy’s study of the linear initial-value problem of water waves won a prize from the French Academy of Sciences. This work, an early application of Fourier analysis, was not well understood at the time. But in general, water-wave dynamics satisfy nonlinear equations because the wave amplitudes are not sufficiently small.

甚至牛顿（1642-1727）也对给出水波的数学描述非常感兴趣，但许多年过后，这才成为可能。1757年欧拉推导出了流体动力学的无粘性方程。不久之后，Laplace和Lagrange发现了水波方程的线性近似。1816年，Cauchy对水波线性初值问题的研究获得了法国科学院颁发的奖项。这项工作是傅立叶分析的一个早期应用，但在当时并没有被人们完全理解。但一般情况下，由于波振幅不是特别小，所以水波动力学满足非线性方程。

Figure 1. Short-stem X-type interaction; see also [2]. (a) Contour plot of an analytical line-soliton interaction solution of the KP equation (here e^Φ ≈ 2:3)). (b) Photograph taken in Mexico, December 31, 2011. (c) 3D plot of the solution shown in (a).

In 1847, Stokes derived the correct nonlinear boundary conditions on the water’s free surface and used it to show that the speed of a traveling wave in deep water depends on its amplitude. In the 1870s, understanding that the nonlinear water-wave equations are simplified when the water is shallow or the waves are long, Boussinesq derived (1+1)-dimensional equations (one space and one time dimension); he found a solitary wave solution that is localized and nonperiodic. In 1895 Korteweg and his student de Vries followed Boussinesq’s pioneering path and derived a unidirectional (1+1)-dimensional nonlinear equation for shallow water, usually called the Korteweg–de Vries (KdV) equation. They also found special periodic solutions, which they called cnoidal waves, that can be written in terms of Jacobian elliptic functions. The cnoidal wave, in a special limit, becomes a solitary wave. A solitary wave had been observed in 1834 by Russell, a naval engineer; he found that the wave’s speed depends on its amplitude, which agrees with the KdV equation’s solitary wave.

1847年，Stokes推导出了正确的水自由面的非线性边界条件，并用它证明了深水中行波速度依赖于振幅。19世纪70年代，在了解到浅水或长波情况下非线性水波方程可以被简化后，Boussinesq推导出了（1 + 1）维方程（一维空间和一维时间）；他发现了局部和非周期的孤立波解。1895年Korteweg和他的学生de Vries沿着Boussinesq的开创道路，导出了单向（1 + 1）维浅水非线性方程，这个方程通常被称为Korteweg–de Vries（KdV）方程。他们还发现了特殊的周期解，他们称之为椭圆余弦波（cnoidal waves），该解可以用雅可比椭圆函数的形式来表示。在一个特殊极限下，椭圆余弦波将变成一个孤立波。1834年，一位海军工程师Russell观察到了一个孤立波，他发现波速依赖于振幅，这与KdV方程的孤立波是一致的。

Between 1895 and 1960, most applications of the KdV equation involved water waves. But in the 1960s mathematicians found that the KdV equation is universal: It arises in wave problems with weak dispersion and weak quadratic nonlinearity. Besides water waves, the KdV equation arises in stratified fluids, plasma physics, elasticity, and lattice dynamics, among other settings. It was lattice dynamics that motivated Kruskal and Zabusky in 1965 to carry out a numerical study of the KdV equation. They discovered that these KdV solitary waves have special interaction properties: Their amplitudes before and after interaction are preserved, but there is a phase shift. They called these special solitary waves solitons. Soon afterward, in 1967, Gardner, Greene, Kruskal, and Miura developed a method—later named the inverse scattering transform (IST) method—for finding the solution. They also found a spectral interpretation for solitons. Their work spurred great interest, and many researchers made important contributions.

在1895年到1960年间，大多数KdV方程的应用都涉及水波。但在20世纪60年代，数学家们发现KdV方程是通用的：它出现在具有弱色散和弱二次非线性的波问题中。除了水波外，KdV方程还出现在分层流体、等离子物理、弹性学和晶体动力学等研究中。1965年，晶体动力学促动Kruskal和Zabusky开展了KdV方程的数值研究。他们发现，这些KdV孤立波具有特殊的相互作用性质：它们相互作用前后振幅保持不变，只是出现了相移。他们称这些特殊的孤立波为孤子。在不久后的1967年，Gardner，Greene，Kruskal和Miura开发了一种后来被称为反散射变换（inverse scattering transform，IST）的方法用于发现这个解。他们还发现了孤子的谱解释。他们的工作引起了极大的兴趣，许多研究者在上面作出了重要贡献。

Equations solvable with the IST method, like the KdV equation, are often called integrable. In 1970 Kadomtsev and Petviashvili (KP) found a multidimensional (two-space, one-time) generalization of the KdV equation; it is also integrable and can be derived from the water-wave equations in shallow water with surface tension included. Like the KdV equation, it has soliton solutions that can be written explicitly. The simplest is a plane-wave solution, which is essentially one-dimensional and satisfies the KdV equation.

KdV等可以用IST方法求解的方程，通常被称为可积方程。1970年，Kadomtsev和Petviashvili（KP）发现了KdV方程的一个多维（两维空间，一维时间）推广；它也是可积的，并且可以从含有表面张力的浅水波方程中推导出来。像KdV方程一样，它也有可以显式表达的孤子解。其中最简单的是平面波解，它本质上是一维的，并且满足KdV方程。

Figure 2. Contour plot (a) and photographs (b), (c) of a Y-type interaction (e^Φ = 0); see also [2]. (b) Taken in Mexico, January 6, 2010. (c) Taken in California, May 3, 2012.

The well-known two-soliton solutions, first found in the 1970s, are more interesting; surprisingly, similar interactions are visible on a daily basis on relatively flat beaches. It is useful to write the two-soliton solution of the KP equation, with small surface tension, with a phase-shift parameter that we label e^Φ. We concentrate on four cases: e^Φ order one, e^Φ large, e^Φ zero, and e^Φ small. Remarkably, we have seen each of these types at the beach; we call them short-stem X-, long-stem X-, Y-, and H-type interactions, respectively.

著名的两孤子解更有趣，它们最早在20世纪70年代被发现；令人惊讶的是，每天在相对平坦的海滩上都能看到类似的相互作用。写出表面张力很小、并且带有一个相移参数（我们记为e^Φ）的KP方程两孤子解是非常有用的。我们专注于下面四种情况：e^Φ一阶，e^Φ很大，e^Φ为0以及e^Φ很小等四种情况。很明显，我们已经在海滩上看到了上面的每一种类型；我们分别称之为短茎X，长茎X，Y和H型相互作用。

Before our recent observations, there was only one known photograph—of a long-stem X-type interaction, taken on the Oregon coast in the 1970s (see [1], page 291). MJA saw and photographed short- and long-stem X-type and Y-type interactions in Nuevo Vallarta, Mexico; he also occasionally saw and photographed more complex multisoliton interactions. Motivated by this and the KP equation’s analytic solutions, DEB traveled to Venice Beach, California, where he saw and photographed not only interactions of the types seen by MJA, but also H-type interactions. We observed these soliton interactions daily on relatively flat beaches, in shallow water, within about two hours of low tide. Being near a jetty helps the development of cross-waves but is not necessary if there is a good crosswind. We have seen mainly two-soliton interactions but occasionally have spotted more complex soliton interactions as well. Additional details and photos can be found in our paper [2], and we have also posted photos and videos on our websites.

在我们最近的观察之前，只有唯一一张已知的长茎X型相互作用的照片，它于上世纪70年代俄勒冈海岸拍得（见[1]，291页）。M.J. Ablowitz在Nuevo Vallarta, Mexico看到并拍下了短茎X型、长茎X型以及Y型相互作用；他也偶尔看到并拍下了更多复杂的多孤子相互作用。受此和KP方程解析解的促动，D.E. Baldwin前往Venice Beach, California旅行，在那里他看到和拍下了不仅由M.J. Ablowitz看到的相互作用类型，而且还有H型相互作用。我们每天在相对平坦的海滩上，大约在两小时左右的低潮期，观察着这些浅水中的孤子相互作用。码头附近有助于形成水波交叉，但如果有一个好的侧风那也不需要码头。我们主要看到了两孤子相互作用，但偶尔也会看见更复杂的孤子相互作用。额外的细节和照片可以在我们的论文[2]中找到，同时我们也将照片和视频放在了我们的网站上。

Along with the phase shift, some of these distinctive nonlinear interactions are explained in part by the stem height: Not just the sum of the wave heights away from the interaction, it can be considerably higher. This can be important in descriptions of tsunami propagation, which in certain cases can be modeled with the KP equation. Indeed, satellite images reveal local X- and Y-type interactions for the 2011 Japanese earthquake-induced tsunami. This made the effects of the tsunami even worse. Because the Japanese tsunami was close to shore, nonlinearity did not have time to amplify the stem height; other tsunamis might occur well away from shore, in which case nonlinear effects could become important. In such cases X- and Y-type tsunami interactions could be extremely destructive.

除了相移外，这些独特的非线性相互作用中有一些可以由茎高来部分解释：不单单是远离相互作用的波高之和，它可以高得多。这在海啸传播的描述中很重要，在某些情况下海啸传播可以用KP方程来描述。事实上，卫星图像展示了2011年日本大地震引起的海啸中局部X型和Y型相互作用。这使得海啸的影响更加恶劣。由于日本海啸非常接近海滨地区，非线性没有足够时间来放大茎高；其他海啸可能也会在远离海滨地区发生，此时非线性可能变得非常重要。在这样的情况下，X型和Y型海啸相互作用可能变得极具破坏力。

References

[1] M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.

[2] M.J. Ablowitz and D.E. Baldwin, Nonlinear shallow ocean-wave soliton interactions on flat beaches, Phys. Rev. E, 86 (2012), 036305.

Written on February 1st , 2015 by 李军