We are dealing with robustness of singular solution behaviour for continuum PDEs under stochastic perturbations. This is a well-known subject for SPDEs in fluid dynamics which form shocks by producing a vertical derivative in their solution profile in finite time. This is also known as wave breaking. For example, a well-studied topic in this field is the robustness of shock formation by the 1D Burgers equation () by the following stochastic perturbation
for the evolution in time of fluid momentum density on the real line under stochastic perturbations . In equation (1.1), the time differential is short notation for a stochastic integral; the spatial partial derivative acts rightward on all products; the symbol denotes a Stratonovich stochastic process; is a Brownian motion with spatially modulated amplitude ; and one interprets the effect of the stochasticity as a noisy perturbation of the transport velocity.
See  for more discussion.
Such stochastic perturbations are of current interest in the modelling of geophysical fluids, where they can express the uncertainty in the effects of the unresolved processes upon the resolved flow. In this context, a new class of stochastic fluid equations was introduced by , which crucially preserve many of the circulation properties of the respective deterministic equations. For instance in the stochastic quasi-geostrophic equations studied by  and , the potential vorticity is still preserved by material transport. These equations are derived from a stochastically constrained variational principle , where the action is given by
where is the velocity advecting quantity and is the Lagrangian of the deterministic fluid. The Lagrange multiplier enforces the stochastic transport of by the Lie derivative
along a vector field, which is given by
with the sum over multiple Wiener processes. The angled brackets denote the spatial integral over the domain of the pairing of and ,
The CH equation (1.5) was derived at one order beyond the celebrated Korteweg-de Vries (KdV) equation in the asymptotic expansion of the Euler fluid equations for non-linear shallow water waves on a free surface propagating under the restoring force of gravity.
Unlike the Burgers equation in (1.1), evolution by the CH equation (1.5) does not create shocks. Instead, CH creates a type of singular solution known as a peakon which possesses a sharp peak at the apex of its velocity profile.
The formation of the sharp peak in the singular peakon solution does not occur by wave breaking and the creation of a vertical slope in the velocity profile. Instead, numerical simulations reveal that peakons arise an intriguing process in which a pair of inflection points of opposite slope are created during the evolution of a spatially confined smooth initial profile. Then the two leading inflection points rise together as the height of the velocity between them increases and thereby the flow leaves the remainder of the profile behind,. The two leading inflection points continue to rise until they collide to create the peakon’s sharp peak.
The shape of the peakon profile turns out to be the Green’s function for the Helmholtz operator which relates the velocity and the momentum density in equation (1.5). This shape may be explained by recalling that the peakon solutions for the momentum density in (1.5) may be expressed by the singular momentum map ,
is the Green’s function for the Helmholtz operator on the real line with homogeneous boundary conditions. Remarkably, the purely discrete spectrum of the isospectral eigenvalue equation which demonstrates that the CH equation is a completely integrable Hamiltonian soliton system also implies thatonly the singular solutions in (1.6) will persist.
The aim of this paper is to study robustness of peakon formation by the stochastic Camassa–Holm (SCH) equation under the following stochastic perturbation,
See [2, 7] for more discussion of the derivation and analysis of the SCH equation in (1.7) and see  for discussions of its solution behaviour and the development of computational algorithms for numerical simulations of its solution behaviour.
While the introduction of stochastic transport into SCH cannot be expected to preserve the complete integrability of the unperturbed CH equation, one may ask whether the initial value problem for SCH still produces peakons on the real line. The present paper answers that question affirmatively via numerical simulation.
In this paper, we extend the work of , who originally derived the stochastic Camassa-Holm equation (1.17
) in the EPDiff regime. The authors then presented a discretisation of the stochastic ordinary differential equations describing the evolution of peakons, and used this discretisation to explore the interaction of peakons. The other main work which we develop is that of, who showed the probability of peakon formation in this regime is positive, though not necessarily unity. This can compared with the deterministic case, in which peakons always form from an inflection point upon a negative slope within finite time. We attempt to extend this by investigating peakon formation numerically.
Here is the paper’s structure. In Section 2, we verify that peakons do indeed satisfy the stochastic Camassa-Holm equation via writing it in hydrodynamic form. In Section 3 we present a finite element discretisation for the stochastic Camassa-Holm equation in the EPDiff regime, showing its numerical convergence properties in Section 4. In Section 5, we numerically investigate the steepening lemma of  using our discretisation.
1.2 Forms of the Deterministic Camassa-Holm Equation
In this paper, we will build on the work of  and  in exploring the properties of the Camassa-Holm equation within this stochastic framework.
The Camassa-Holm equation was introduced in , and
describes one-dimensional dispersive shallow-water waves.
It has a bi-Hamiltonian structure and possesses special soliton solutions, which propagate at constant velocity without changing shape or amplitude, as non-linear and dispersive effects cancel out.
Here we briefly present the deterministic equation in three different forms.
The first form of the equation, for velocity and parameter is
with subscripts and denoting partial derivatives with respect to spatial coordinate and time respectively.
The Hamiltonian structure of (1.8) is commonly exploited to write it as the coupled equations
where can be treated as a momentum that is related to via a Helmholtz operator.
A third form of (1.8), which we call the hydrodynamic form, can be derived by first writing (1.9b) as an operation upon , which combined with the definition of gives the following equation for :
The left hand side describes the action of two operators, and we find the result of alternating their order by introducing the commutator, so that
Inspection of the commutator shows that it can be written as the gradient of a potential. Applying the inverse Helmholtz operator to both sides, which commutes through the gradient, gives
Noticing that the non-linear term on the left can also be written as a gradient, and using , gives the final hydrodynamic form of the Camassa-Holm equation,
The significance of this form of the equation is that no second derivative of directly appears in the equation.
Further, this equation admits weak peakon solutions, as will be discussed in Section 2.
Equations (1.8), (1.9) and (1.13) are equivalent ways of expressing the Camassa-Holm equation.
1.3 The Stochastic Camassa-Holm Equation
The deterministic equation (1.9) can be found by exploiting the Hamiltonian structure:
with the Hamiltonian
in which case . A similar approach can be used to find the stochastic Camassa-Holm equation, as used in . In this case, we introduce the stochastic Hamiltonian
The respective variational derivative is then , with from (1.3). The resulting stochastic equation is then
which can be thought of as the stochastic case of (1.9). This equation also possesses a hydrodynamic form, which can also be obtained by considering the commutator
Separating this into deterministic and stochastic parts gives
which results in
2 Peakon Solutions to the Stochastic Camassa-Holm equation
In the hydrodynamic form (1.20), the equation admits weak peakon solutions, of the form
in which represents the position of the peakon and represents its momentum, and together they form a pair of canonical coordinates satisfying a Hamiltonian system. As stated in , the Hamiltonian of this stochastic system of peakons can be written as
The evolution of these canonical coordinates is described by the pair of coupled stochastic differential equations
Considering a single peakon and writing , gives
In  it was argued that substitution of (2.3) into (2.1) yields (1.17). However, the peakon solution (2.1) only makes sense as a weak solution to (1.20), the verification of which is the focus of this section. In other words, they are solutions to
The operation is performed upon some as
Taking the domain as and combining these elements to compute the integrals, both right and left hand sides of (2.6) return
3 A Finite Element Discretisation for the Stochastic Camassa-Holm Equation
Here we present a mixed finite element discretisation of the stochastic Camassa-Holm equation, inspired by the form (1.20), to have confidence in the discrete representation of peakons.
Considering the space of continuous linear functions, we seek to find the
that satisfy, for all , the equations
where is the time step, is the value of at the -th time level, and we have used the implicit midpoint rule to discretise in time, so that . The stochastic velocity is given by
4 Convergence of the Discretisation to Peakon Solutions
In this section, we show that when describing the evolution of a peakon, our discretisation presented in Section 3 converges to the equations
To do this, we set up a periodic one-dimensional domain of length , and taking , we specify an initial condition of
First we establish a numerical solution of (4.1) that is well-resolved temporally, by solving (4.1) using a simple forward Euler timestepping scheme, with initial condition of and and with . For the stochastic basis functions, we used only a single function:
Then we performed two tests, one showing the strong convergence of the discretisation (3.2) to this high resolution solution of (4.1) as approached .
This was performed at fixed spatial resolution with .
The second test showed convergence of (3.2) to (4.1) as , using .
When solving our discretisation (3.2) with a larger time step, say , to ensure that the realisation of the noise corresponded to those with , the random number used for that larger time step would be , where are the random numbers corresponding to the smaller . At a given time step, the and calculated from (4.1) were translated into the field corresponding to a peakon in the discretisation of the PDE using (2.1), which could be compared with the found from solving (3.2).
Results showing the strong temporal convergence of the discretisation are shown in Figure 1 (left), whilst Figure 1 (right) displays the results of the spatial convergence of the discretisation to the underlying stochastic differential equations for the evolution of the peakon.
5 Numerical Investigations of the Steepening Lemma
One of the key results of  was the investigation of the steepening lemma for the stochastic Camassa-Holm system.
In the deterministic case, a peakon will always form in finite time from an inflection point on a negative slope, which was one of the key results of .
In , it was shown that there was a non-zero probability of a peakon forming in these conditions in the stochastic Camassa-Holm system.
In this section we attempt to investigate, numerically, whether this probability is simply non-zero or almost unity.
To add to this discussion, we perform numerical simulations of the stochastic Camassa-Holm equation, using the discretisation of Section (3), under many different realisations of the noise, and record the formation of peakon solutions. The first step in this investigation is to determine a numerical diagnostic for whether a solution is in fact a peakon. Our diagnostic used is the distance between inflection points in in the discretisation, which we denote as . For a smooth peak, as , will converge to some non-zero value, whilst for a peaked solution, will converge to zero as .
As a peakon forms, values of calculated at different resolutions diverge, as illustrated in Figure 2, which shows the evolution of a peakon within the discretisation of Section 3 at differing resolutions. This figure was produced in a domain of length , with a time step of , and with the initial condition set as
with . We have applied a low-pass filter to the values so as to reduce error in the measurement of the positions of the inflection points.
This data can be used to approximate a time for the formation of the peakon, which we take to be the time at which the gradient of the best fit line of as a function of at a given time breaches a certain threshold, which we take to be 0.4.
Extending this approach to stochastic simulations, again using the initial condition (5.1), the histograms showing the evolution of over many different realisations (as a function of time and resolution ) are shown in Figure 3. This experiment was performed for 1000 realisations of the noise, where the stochastic basis functions were of
Figure 4 clearly shows that the inflection points move closer together as both time evolves, and . The histogram showing the peakon formation times from these realisations of the noise is shown in Figure 3.
In all these realisations of the noise, a peakon was adjudged to have formed, providing supporting evidence for the hypothesis that peakons form with a probability of unity in the stochastic Camassa-Holm equation from a smooth initial condition with an inflection point on a negative slope, which is clearly not something that can be explicitly proved experimentally. In fact, the evidence from Figures 3 and 4 is that the stochastic noise prompts the peakon solution to emerge more quickly in the deterministic case.
We have extended the discussion of peakons within the stochastic Camassa-Holm equation begun by  and . This first result was a demonstration that peakons satisfying the equations (2.3) derived in  do indeed satisfy the stochastic Camassa-Holm equation (1.20) when it is cast in hydrodynamic form. Next, we presented a new finite element discretisation for (1.20). We then showed numerically that this discretisation converges both strongly in time and also converges to (4.1) as . This discretisation then provided a tool with which to investigate the steepening lemma of  for the stochastic Camassa-Holm system. Starting with smooth initial conditions, we investigated the formation of peakons under different realisations of the stochastic noise. No examples without peakon formation were found, and the noise was observed to tend to lead to faster peakon formation.
TMB was supported by the EPSRC Mathematics of Planet Earth Centre for Doctoral Training at Imperial College London and the University of Reading, with grant number EP/L016613/1. TMB and CJC were also supported by EPSRC grant EP/N023781/1.
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