Introduction.

Tidal bores occur worldwide wherever a large high tide occurs in conjunction with a funnel shaped estuary. When conditions are right this results in a wave of elevation propagating many kilometers upriver. Some sites at which bores regularly occur include the Severn river in England, the Dordogne in France, and the Qiantang in China, the last of which has been a tourist destination for at least a thousand years and continues to fascinate today [4]. Figure 1 depicts a bore on the Qiantang river in September of 2024. Note the train of oscillations behind the propagating front. Figure 2 is a video depicting the arrival of a bore on the Severn.

Figure 1. The Qiantang bore - the Silver Dragon - photographed at Yanguan scenic area. The wave is propagating left to right. Photo courtesy of Michael Berry.

Figure 2. Surfers and kayakers riding the Severn river bore. For the higher resolution source file, see Movie1.mp4 [19.6 MB]. Video courtesy of Kenneth T.-R. McLaughlin and Richard M. McLaughlin.

The Korteweg-de Vries-Burgers (KdVB) equation is one of the simplest PDEs that incorporates nonlinear, dissipative, and dispersive effects. The KdVB equation with front-type initial data has been proposed [19, 26, 29] as a mathematical model for undular (non-turbulent) bore propagation. In particular we consider

where \(u:\mathbb{R}^2\to \mathbb{R}\) and \(\nu \in \mathbb{R}\). Traveling wave solutions of (1.1) exist and are unique up to translation [5]. The parameter \(\nu\) represents a dimensionless dispersion parameter: the traveling wave profile is monotone [5] if and only if \(|\nu|\leq \tfrac14\). As is evident from the previous figures traveling bores observed in nature have oscillatory tails corresponding to a non-monotone traveling front.

It was shown in [24] that monotone profiles of (1.1) are stable with respect to small perturbations of the initial front profile. Other related results include [1, 15, 17, 18, 28, 30]. Using a perturbative approach from the monotone fronts, researchers have established stability for some weakly non-monotone front solutions of (1.1); see [20, 21, 22, 23]. Numerically, we observe that the stability condition in [21] holds for \(|\nu|\lesssim 0.3\) for (1.1). Numerical evidence in [10] suggests that all traveling wave solutions of (1.1) are stable, though rigorously showing stability in this generality has remained an open problem. Notably, it has been shown that some traveling front solutions of the KdVB equation with a higher-order power-law nonlinearity are known to be unstable [16, 25].

In [3], we establish asymptotic stability of traveling waves in \(L_p\) for all \(p>2\) provided the following spectral condition is met: the operator

acting on \(L_2(\mathbb{R})\), where \(\phi\) solves (1.1), has exactly one bound state for some \(\epsilon>0\). Notably this condition does not involve the initial data, and the result holds for arbitrarily large perturbations of the traveling front solution.

We rigorously establish that traveling wave solutions of (1.1) with \(|\nu|\in[0,\frac14]\cup[0.2533, 3.9]\), a parameter region which includes oscillatory fronts, are stable with respect to perturbations of any size. In the monotone case, \(|\nu|\leq\frac14\), this is shown using classical analysis. More interesting for readers of DSWeb, perhaps, is the non-monotone case where we prove the spectral condition using techniques of validated numerics and computer assisted proof.

From Sturm theory showing that a one-dimensional Schrödinger operator has one bound state is equivalent to showing the solution to a certain ordinary differential equation has a single zero on the real line. We consider the coupled system

This gives a boundary value problem for \((\phi,\psi)\) coupled to an initial value problem for \((w,z)\): one must construct the heteroclinic connection, considering the first two components, between the stable manifold of the fixed point \((\phi,\psi) = (-1,0)\) and the unstable manifold of the fixed point \((\phi,\psi) = (1,0)\) and from that find \((w,z)\). The number of roots of \(w(z)\) is exactly equal to the number of bound states of (1.2).

Visualizing the stability condition.

In Figure 3(a) and (b), we plot the \(\phi\) and \(w\) components respectively of the solution of (1.3), with the desired asymptotic boundary conditions, against \(\nu\) and \(x\). We see in Figure 3(b) that there is a single bound state for \(\nu\) small, but that near \(\nu = 4\), \(w\) transitions from moving further away from the origin as \(x\to +\infty\) to moving toward it. Thus, for larger \(\nu\), the hypothesis of the Theorem given in [3] is not satisfied, and so no conclusions about stability can be made.

Figure 3. Plot of solutions of equation (1.3) as \(\nu\) varies between 0.26 and 4.5. (a) Plot of \(\phi\) against \(\nu\) and \(x\). (b) Plot of \(w\) against \(\nu\) and \(x\).

Rigorous results.

We rigorously verify (see [3]) that there is a single bound state, that is that \(w=0\) exactly once for each \(\nu\) as depicted in Figure 3(b), for \(\nu\in[0.2533, 3.9]\). We accomplish this by using computer assisted proof, aided by interval arithmetic. At a high level, we represent the stable manifold of the fixed point \((-1,0,1,0)\) of (1.3) as a series solution with basis elements chosen to capture the exponential decay of the solution to the fixed point (known as the parameterization method [7, 8, 9, 12, 27]), we represent the unstable manifold associated with the fixed point \((1,0)\) of (1.3) restricted to the first two components, and we use series solutions with the standard polynomial basis to represent the solution of (1.3) on finite intervals between the two far field regions. We then use the Newton-Kantorovich Theorem ([2, 6, 11, 13, 14]) to show that there is a nearby global solution to these individual pieces of solutions. We also carry out analysis to guarantee no bound states occur for \(x\ll0\). In Figure 4 we depict the rigorous enclosure of the solution of (1.3) with dashed blue lines. The actual error bound is too small to be seen in the figure. For details about the rigorous computations, see [3].

Figure 4. Plot of the solution \((\phi,\psi,w,z)\) of (1.3) against \(x\), for \(\nu = 2\), with solid black lines, and the rigorous enclosure of the numerical solution with dashed blue lines.

The Profiles.

We have two animations depicting the change in the front profile \(\phi(x;\nu)\) and the Sturm function \(w(x;\nu)\) as a function of the non-dimensional dispersion \(\nu\). Figure 5 shows the front profile \(\phi(x,\nu)\) as the dispersion parameter \(\nu\) increases from \(\nu=0.26\) to \(\nu = 4.5\). Note that monotonicity is lost at \(\nu=\frac14\). It is apparent that as the dispersion increases one sees more prominent oscillations in the front profile. Each oscillation in \(\phi\) corresponds to a potential well in \(\frac12 \phi_x\), leading to the possibility that the operator

will support multiple bound states.

Figure 6 shows \(w(x,\nu)\), the solution to

as \(\nu\) varies over the same range \([0.26,4.5].\) By the Sturm oscillation theorem the index of the operator, the number of bound state eigenvalues, is equal to the number of roots of \(w(x,\nu)\). There is always one root, and thus one bound state. Since \(w(x,\nu)\) tends asymptotically to a linear function it is clear that once the function "turns around" there will necessarily be a root far out on the negative axis. A parameter value of roughly \(\nu=4.1\) marks the first time where the function and its derivative are both negative. In this case it can be shown that there is necessarily a second zero of \(w(x,\nu)\). As \(\nu\) increases this zero moves closer to the origin. The birth of this second eigenvalue at roughly \(\nu = \pm 4.1\) implies that the stability condition fails to hold, but does not necessarily imply that an instability occurs.

Figure 5. The traveling wave profile as a function of the non-dimensional dispersion parameter \(\nu\). As the dispersion parameter grows the characteristic dispersive oscillations become more pronounced. For the higher resolution source file, see Movie2.mp4 [253 KB].

Figure 6. The solution to the zero-energy Schrodinger equation as a function of \(\nu\). There is a single root until roughly \(\nu=4.1\), where the slope of the solution at \(x=0\) first becomes negative. At this point a root is created for large, negative \(x\). This root moves towards the origin as \(\nu\) increases. For the higher resolution source file, see Movie3.mp4 [249 KB].

Acknowledgements.

The authors would like to thank Michael Berry, Kenneth T.-R. McLaughlin, and Richard M. McLaughlin for generously sharing their photos and videos. JCB would like to thank the organizers of the Ningbo International Conference on Water Waves and Bores, especially Professor Changzheng Qu, for their hospitality during the conference.

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