2. Evolution and conserved quantities

Taking the mean value of the local model (1.7) for the C ¹-smooth 2π-periodic solutions $\eta \in C^1(\mathbb{R} \times \mathbb{T})$ and integrating by parts yields the constraint

(2.1)\begin{equation} \oint \left[ \eta + (\partial_x \eta)^2 \right] dx = 0, \end{equation}

where $\oint$ denotes the integral of a periodic function on $\mathbb{T}$ over its period. The integral does not depend on the choice in the limits of integration.

Let $\Pi_0 : L^2(\mathbb{T}) \rightarrow L^2(\mathbb{T}) |_{\{1\}^{\perp}}$ be a projection operator to the periodic functions with zero mean. The local model (1.7) can be written in the evolution form

(2.2)\begin{equation} 2 c \partial_t \eta = (c^2 - 2 \eta) \partial_x \eta + \Pi_0 \partial_x^{-1} \Pi_0 \left[ (\partial_x \eta)^2 + \eta \right], \end{equation}

where $\Pi_0 \partial_x^{-1} \Pi_0 : L^2(\mathbb{T}) \rightarrow L^2(\mathbb{T}) |_{\{1\}^{\perp}}$ is uniquely defined on the periodic functions under the zero-mean constraint. The local well-posedness result suitable for solutions with peaked profiles is given by the following theorem.

Proof. The evolution equation (2.2) is a nonlocal version of the inviscid Burgers equation $2c \partial_t \eta = (c^2 - 2 \eta) \partial_x \eta$. Since

\begin{align*} \| \Pi_0 \partial_x^{-1} \Pi_0 \left[ \eta + (\partial_x \eta)^2 \right] \|_{L^2} &\leq \| \eta + (\partial_x \eta)^2 \|_{L^2} \leq \| \eta \|_{L^2} + \| \partial_x \eta \|_{L^{\infty}} \| \partial_x \eta \|_{L^2}, \\ \| \Pi_0 \partial_x^{-1} \Pi_0 \left[ \eta + (\partial_x \eta)^2 \right] \|_{L^{\infty}} &\leq \| \eta + (\partial_x \eta)^2 \|_{L^1} \leq \sqrt{2\pi} \| \eta \|_{L^2} + \| \partial_x \eta \|^2_{L^2},\\ \| \partial_x \Pi_0 \partial_x^{-1} \Pi_0 \left[ \eta + (\partial_x \eta)^2 \right] \|_{L^2 \cap L^{\infty}} &\leq \| \eta + (\partial_x \eta)^2 \|_{L^2 \cap L^{\infty}} \leq \| \eta \|_{L^2 \cap L^{\infty}} + \| \partial_x \eta \|_{L^{\infty}} \| \partial_x \eta \|_{L^2 \cap L^{\infty}}, \end{align*}

the nonlocal term $\Pi_0 \partial_x^{-1} \Pi_0 \left[ \eta + (\partial_x \eta)^2 \right]$ is a bounded operator from a ball in $H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$ to $H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$. Local well-posedness in $H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$ follows by using the method of characteristics.

The mass, momentum and energy conservation of the evolution equation (2.2) are obtained for the smooth solution $\eta \in C^0((-\tau_0,\tau_0),H^s_{\rm per}(\mathbb{T}))$ with $s \gt \frac{3}{2}$. Multiplying (1.1) by $\partial_x \eta$, and integrating over the period, yields conservation of the momentum

(2.3)\begin{equation} Q(\eta) := \frac{1}{2} \oint (\partial_x \eta)^2 dx, \end{equation}

and in view of the constraint (2.1), also conservation of the mass

(2.4)\begin{equation} M(\eta) := \oint \eta dx. \end{equation}

Furthermore, writing (2.2) in the Hamiltonian form

(2.5)\begin{equation} 2c \partial_t \eta = - \Pi_0 \partial_x^{-1} \Pi_0 \left[ c^2 Q'(\eta) - H'(\eta) \right], \end{equation}

where

(2.6)\begin{equation} H(\eta) := \frac{1}{2} \oint \left[ \eta^2 + 2 \eta (\partial_x \eta)^2 \right] dx, \end{equation}

yields conservation of the energy $H(\eta)$.

3. Travelling wave with the smooth and peaked profiles

A travelling wave with the speed c and the smooth profile $\eta \in C_{\rm per}^{\infty}(\mathbb{T})$ corresponds to the time-independent solution of the local model (1.1) found from the second-order differential equation

(3.1)\begin{equation} (c^2-2 \eta) \eta'' - (\eta')^2 + \eta = 0, \quad x \in \mathbb{T}. \end{equation}

This equation is integrable with the first-order invariant

(3.2)\begin{equation} E(\eta,\eta') = \frac{1}{2} (c^2-2 \eta) (\eta')^2 + \frac{1}{2} \eta^2 = \mathcal{E}, \end{equation}

the value of which is independent of x.

It was shown in [Reference Locke and Pelinovsky36] that the family of smooth 2π-periodic solutions $\eta \in C_{\rm per}^{\infty}(\mathbb{T})$ exists for $c \in (1,c_*)$ with $c_* := \frac{\pi}{2 \sqrt{2}}$. The peaked profile $\eta_* \in C^0_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$ corresponds to $c = c_*$ and is given explicitly as

(3.3)\begin{equation} \eta_*(x) = \frac{1}{16} ( \pi^2 - 4 \pi |x| + 2 x^2), \qquad x \in [-\pi,\pi], \end{equation}

extended as a 2π-periodic function on $\mathbb{T}$. It is easy to verify the validity of $\eta_*$ in (3.3) as a solution of (3.1) for $x \in [-\pi,0) \cup (0,\pi]$ with

\begin{equation*} \max_{x \in \mathbb{T}} \eta_*(x) = \eta_*(0) = \frac{c_*^2}{2}. \end{equation*}

The peaked profile $\eta_* \in C^0_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$ corresponds to the marginal value of $\mathcal{E}_c := \frac{c^4}{8}$ in (3.2), which separates the smooth profiles for $\mathcal{E}\in (0,\mathcal{E}_c)$ and the cusped profiles for $\mathcal{E} \in (\mathcal{E}_c,\infty)$. The value of $c = c_*$ is selected by setting the period of the peaked profile to 2π. The slope of the peaked profile $\eta_*$ has a finite jump discontinuity at x = 0 since

(3.4)\begin{equation} \eta_*'(x) = -\frac{1}{4} (\pi - |x|) {\rm sign}(x) \qquad x \in [-\pi,\pi], \end{equation}

which implies that $\eta_*'(0^+) - \eta_*'(0^-) = -\frac{\pi}{2}$. By using the Dirac delta distribution δ ₀ at x = 0, we can express the finite jump discontinuity of $\eta'_*(x)$ as the Dirac delta singularity of the second derivative at x = 0:

(3.5)\begin{equation} \eta_*''(x) = \frac{1}{4} - \frac{\pi}{2} \delta_0, \qquad x \in [-\pi,\pi]. \end{equation}

It can be checked through explicit computations that the periodic solution with the peaked profile (3.3) satisfies the constraint (2.1).

Figure 1 presents the periodic profiles η of the travelling waves for two values of c in $(1,c_*)$ and for $c = c_*$ (left) as well as the dependence of the wave amplitude $\| \eta \|_{L^{\infty}}$ versus c (right). The wave profiles were approximated numerically, see section 8. The peaked profile $\eta_*$ is shown by a dashed line on the left panel, and the corresponding value $c_*$ is shown by a dashed vertical line on the right panel. The family of periodic waves is continued past $c = c_*$ with the cusped profiles for $c \in (c_*,c_{\infty})$ which satisfy

(3.6)\begin{equation} \eta(x) = \frac{c^2}{2} - A(c) |x|^{2/3} + \mathcal{O}(|x|), \quad \mbox{as} \;\; |x| \to 0, \end{equation}

where $A(c) \gt 0$ is a numerical coefficient and the value of $c_{\infty}$ is numerically computed (dashed-dotted line on figure 1) [Reference Locke and Pelinovsky36].

4. Linearisation at the smooth and peaked travelling waves

Let $\eta \in C_{\rm per}^{\infty}(\mathbb{T})$ be the spatial profile of the smooth travelling waves for $c \in (1,c_*)$. The second-order equation (3.1) is equivalent to the Euler–Lagrange equation

\begin{equation*} H'(\eta) - c^2 Q'(\eta) = 0. \end{equation*}

Adding a perturbation $\zeta(t,x)$ to $\eta(x)$ and linearising the evolution equation (2.5), we obtain the linearised equation in the form

(4.1)\begin{equation} 2c\partial_t \zeta = - \Pi_0 \partial_x^{-1} \Pi_0 \mathcal L \zeta, \quad \quad \mathcal L := -\partial_x (c^2-2\eta)\partial_x + (2\eta'' -1), \end{equation}

where $\mathcal L : H_\mathrm{per}^2(\mathbb T) \subset L^2 (\mathbb T)\to L^2(\mathbb T)$ is the Hessian operator for $c^2 Q(\eta) - H(\eta)$ at the profile $\eta \in C_{\rm per}^{\infty}(\mathbb{T})$. As $c \to c_*$ and $\eta \to \eta_* \in C^0_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$, the Hessian operator becomes singular since

\begin{equation*} 2 \eta_*''(x) - 1 = -\frac{1}{2} - \pi \delta_0, \quad x \in [-\pi,\pi]. \end{equation*}

This suggests that the linearised equation (4.1) defined along the family of the smooth travelling waves breaks at the peaked travelling wave. We need to be careful to linearise the evolution equation (2.5) about the travelling wave with the peaked profile $\eta_*$ by working in the function space $H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$, where the local well-posedness is established by theorem 1.

To get the proper linearisation near the peaked profile $\eta_*$, we note the following result, which is obtained verbatim from the analysis of [Reference Madiyeva and Pelinovsky39, Reference Natali and Pelinovsky41].

Lemma 1. [Reference Madiyeva and Pelinovsky39, Reference Natali and Pelinovsky41]

Consider a local solution $\eta \in C^0((-\tau_0,\tau_0),H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T}))$ of theorem 1, and assume that there exists $\xi(t)$ such that

\begin{equation*} \lim_{x \to \xi(t)^-} \partial_x \eta(t,x) \neq \lim_{x \to \xi(t)^+} \partial_x \eta(t,x), \quad t \in (-\tau_0,\tau_0). \end{equation*}

Then, $\xi \in C^1((-\tau_0,\tau_0))$ satisfies

(4.2)\begin{equation} \xi'(t) = -\frac{1}{2c} (c^2 - 2 \eta(t,\xi(t))), \quad t \in (-\tau_0,\tau_0). \end{equation}

In order to consider a local solution $\eta \in C^0((-\tau_0,\tau_0),H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T}))$ of the evolution equation (2.2) in a local neighbourhood of the travelling wave with the peaked profile (3.3), we define the decomposition

(4.3)\begin{equation} \eta(t,x) = \eta_*(x-\xi(t)) + \zeta(t,x-\xi(t)), \end{equation}

where we assume that the only peak of $\eta(t,\cdot)$ on $\mathbb{T}$ is located at $x = \xi(t)$ for some $t \in (-\tau_0,\tau_0)$. The peak location $\xi(t)$ moves with the local characteristic speed of the inviscid Burgers equation, as in Lemma 1. Substituting (4.3) into (2.2) with $c = c_*$ and using (4.2) yields the evolution problem for the perturbation term $\zeta(t,x)$:

(4.4)\begin{equation} 2 c_* \partial_t \zeta = (c_*^2 - 2 \eta_*) \partial_x \zeta -2 (\zeta - \zeta |_{x = 0}) (\eta_*' + \partial_x \zeta) + \Pi_0 \partial_x^{-1} \Pi_0 \left[ \zeta + 2 \eta_*' \partial_x \zeta + (\partial_x \zeta)^2 \right], \end{equation}

where we have translated $x - \xi(t)$ into x on $\mathbb{T}$. Truncation of the evolution equation (4.4) by the linear terms in ζ yields the linearised equation

(4.5)\begin{equation} 2 c_* \partial_t \zeta = (c_*^2 - 2 \eta_*) \partial_x \zeta -2 \eta_*' (\zeta - \zeta |_{x = 0}) + \Pi_0 \partial_x^{-1} \Pi_0 \left[ \zeta + 2 \eta_*' \partial_x \zeta \right], \end{equation}

subject to the linearised constraint

(4.6)\begin{equation} \oint [\zeta + 2 \eta_*' \partial_x \zeta ] dx = 0. \end{equation}

The next result gives the equivalent form of the linearised evolution.

Lemma 2. The linearised equation (4.5) is equivalently written in the form

(4.7)\begin{equation} 2 c_* \partial_t \zeta = (c_*^2 - 2 \eta_*) \partial_x \zeta - \frac{1}{\pi} \oint \eta_*' \zeta dx + \frac{1}{2} \Pi_0 \partial_x^{-1} \Pi_0 \zeta, \end{equation}

where both $\oint \zeta dx$ and $\zeta |_{x = 0}$ are constant in t and satisfies the constraint

(4.8)\begin{equation} \zeta |_{x = 0} = -\frac{1}{2\pi} \oint \zeta dx, \end{equation}

Proof. The constraint (4.8) is obtained by substituting (3.4) into (4.6), and integrating the second term in (4.6) by parts. To simplify the linearised equation (4.5), we use (3.5) and write

(4.9)\begin{equation} \zeta + 2 \eta_*' \partial_x \zeta = 2 \partial_x (\eta_*' \zeta) + \frac{1}{2} \zeta + \pi \delta_0 \zeta. \end{equation}

This transforms the linearised equation (4.5) to the form

(4.10)\begin{equation} 2 c_* \partial_t \zeta = (c_*^2 - 2 \eta_*) \partial_x \zeta - \frac{1}{\pi} \oint \eta_*' \zeta dx + 2 \eta_*' \zeta |_{x = 0} + \frac{1}{2} \Pi_0 \partial_x^{-1} \Pi_0 \zeta + \pi \zeta |_{x = 0} \Pi_0 \partial_x^{-1} \Pi_0 \delta_0. \end{equation}

By using Fourier series, we get

\begin{equation*} \delta_0 = \frac{1}{2\pi} \sum_{n \in \mathbb{Z}} e^{i n x}, \end{equation*}

so that

\begin{equation*} \Pi_0 \partial_x^{-1} \Pi_0 \delta_0 = \Pi_0 \partial_x^{-1} \left( \delta_0 -\frac{1}{2\pi} \right) = \sum_{n \in \mathbb{Z}\backslash \{0\}} \frac{1}{2 \pi i n} e^{i n x}. \end{equation*}

On the other hand, it follows from (3.5) that

\begin{equation*} \eta_*''(x) = \frac{1}{4} - \frac{\pi}{2} \delta_0 = -\frac{1}{4} \sum_{n \in \mathbb{Z}\backslash \{0\}} e^{i n x}, \end{equation*}

which yields

\begin{equation*} \eta_*'(x) = -\sum_{n \in \mathbb{Z}\backslash \{0\}} \frac{1}{4in} e^{i n x}. \end{equation*}

Hence, the two terms with $\zeta |_{x=0}$ in (4.10) cancels out as

(4.11)\begin{equation} 2 \eta_*' + \pi \Pi_0 \partial_x^{-1} \Pi_0 \delta_0 = 0, \end{equation}

and the linearised equation (4.10) can be written in the form (4.7).

Finally, we show that both $\oint \zeta dx$ and $\zeta |_{x = 0}$ are constant in t. The conservation of $\oint \zeta dx$ follows by taking the mean value of (4.7), with the account of the projection term $-\frac{1}{\pi} \oint \eta_*' \zeta dx$. The conservation of $\zeta |_{x = 0}$ is shown by taking the limit x → 0 since if $\zeta = \sum_{n \in \mathbb{Z}} \zeta_n e^{inx}$, then

(4.12)\begin{align} \frac{1}{\pi} \oint \eta_*' \zeta dx &= \frac{1}{\pi} \sum_{n \in \mathbb{Z}\backslash \{0\}} \zeta_n \oint \eta_*'(x)e^{inx} dx \nonumber \\ &= -\frac{1}{\pi}\sum_{n \in \mathbb{Z}\backslash \{0\}} \zeta_n \left( \sum_{m \in \mathbb{Z}\backslash \{0\}} \frac{1}{4im} \oint e^{i(n+m)x} dx \right)\nonumber \\ &= \sum_{n \in \mathbb{Z} \backslash \{0\}} \frac{\zeta_n}{2in} \end{align}

and

(4.13)\begin{align} \lim_{x \to 0} \frac{1}{2} \Pi_0 \partial_x^{-1} \Pi_0 \zeta = \lim_{x \to 0} \frac{1}{2} \sum_{n \in \mathbb{Z}\backslash \{0\}} \zeta_n \Pi_0 \partial_x^{-1} e^{inx} = \sum_{n \in \mathbb{Z}\backslash \{0\}} \frac{\zeta_n}{2in}, \end{align}

from which it follow that $2c \lim\limits_{x \to 0} \partial_t \zeta(t,x) = 0$ and the value of $\zeta |_{x=0}$ is preserved in time.

The linearised equation (4.7) of lemma 2 is defined by the linearised operator $A : {\rm Dom}(A) \subset L^2(\mathbb{T}) \to L^2(\mathbb{T})$ given by

(4.14)\begin{equation} A f := (c_*^2 - 2 \eta_*) \partial_x f - \frac{1}{\pi} \oint \eta_*' f dx + \frac{1}{2} \Pi_0 \partial_x^{-1} \Pi_0 f, \end{equation}

where

(4.15)\begin{equation} {\rm Dom}(A) := \left\{f \in L^2(\mathbb{T}) : \;\; (c_*^2 - 2 \eta_*) f' \in L^2(\mathbb{T}) \right\} \equiv \mathcal{D}. \end{equation}

We denote the spectrum of $A : \mathcal{D} \subset L^2(\mathbb{T}) \to L^2(\mathbb{T})$ by $\sigma(A)$. According to the standard definition (definition 6.1.9 in [Reference Búhler and Salamon4]), the spectrum $\sigma(A)$ is further divided into three disjoint sets of the point spectrum $\sigma_p(A)$, the residual spectrum $\sigma_r(A)$, and the continuous spectrum $\sigma_c(A)$ with the resolvent set denoted by $\rho(A) = \mathbb{C} \backslash \sigma(A)$.

Remark 5. By using (2.1), (3.4) and (3.5), as well as $c_* = \frac{\pi}{2 \sqrt{2}}$, we obtain

\begin{align*} A \eta_*' &= (c_*^2 - 2 \eta_*) \eta_*'' - \frac{1}{\pi} \oint (\eta_*')^2 dx + \frac{1}{2} \Pi_0 \eta_* \\ &= -\frac{\pi}{2} (c_*^2 - 2 \eta_*) \delta_0 + \frac{1}{4} (c_*^2 - 2 \eta_*) - \frac{1}{\pi} \oint (\eta_*')^2 dx + \frac{1}{2} \eta_* - \frac{1}{4\pi} \oint \eta_* dx \\ &= -\frac{\pi}{2} (c_*^2 - 2 \eta_*) \delta_0 + \frac{\pi^2}{32} - \frac{3}{4 \pi} \oint (\eta_*')^2 dx \\ &= -\frac{\pi}{2} (c_*^2 - 2 \eta_*) \delta_0, \end{align*}

so that $A \eta_*' = 0$ in $L^2(\mathbb{T})$. Similarly, we have $\eta_*' \in \mathcal{D}$ so that $0 \in \sigma_p(A)$. However, $\eta_*' \notin C^0_{\rm per}(\mathbb{T})$, hence $\eta_*' \notin H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$. Thus, $H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$ is embedded into $\mathcal{D}$ but is not equivalent to $\mathcal{D}$. The spectral theory of the linear operator $A : \mathcal{D} \subset L^2(\mathbb{T}) \to L^2(\mathbb{T})$ is developed in a wider space of functions than the space needed for the local well-posedness of the evolution equation (2.2).

5. Truncated linearised equation

The following lemma allows us to truncate the linearised operator (4.14) and (4.15).

Proof. By the Fourier theory, we have

\begin{equation*} \sigma(K) = \sigma_p(K) = \left\{\frac{1}{2n}, \;\; n \in \mathbb{Z} \backslash \{0\} \right\}. \end{equation*}

Since eigenvalues of $\sigma_p(K)$ are square summable, $K : L^2(\mathbb{T}) \to L^2(\mathbb{T})$ is a compact (Hilbert–Schmidt) operator.

Using lemma 3, we see that $A = A_0 + K$, where K is a compact perturbation of the unbounded truncated operator $A_0 : {\rm Dom}(A_0) \subset L^2(\mathbb{T}) \to L^2(\mathbb{T})$ given by

(5.1)\begin{equation} A_0 f := (c_*^2 - 2 \eta_*) \partial_x f - \frac{1}{\pi} \oint \eta_*'(x) f(x) dx, \end{equation}

with the same ${\rm Dom}(A_0) = {\rm Dom}(A) = \mathcal{D}$. The constraint in the definition of A ₀ ensures that

(5.2)\begin{equation} \oint (A_0f)(x) dx = 0 \quad \mbox{if} \;\; f \in \mathcal{D}. \end{equation}

The spectrum of A ₀ can be analysed similarly to the work [Reference Geyer and Pelinovsky22]. In fact, since

\begin{equation*} c_*^2 - 2 \eta_*(x) = \frac{1}{4} [\pi^2 - (\pi - |x|)^2], \quad x \in [-\pi,\pi], \end{equation*}

extended as a 2π-periodic function on $\mathbb{T}$, we define the change of coordinates $x \mapsto z$ by

(5.3)\begin{equation} \frac{dx}{dz} = \frac{1}{4} x (2\pi - x), \quad x \in [0,2\pi], \end{equation}

where the interval $[0,2\pi]$ is located between the two consequent peaks on the periodic domain $\mathbb{T}$. Solving the differential equation (5.3) with $x(0) = \pi$ yields

(5.4)\begin{equation} x(z) = \pi + \pi \tanh\left( \frac{\pi z}{4} \right), \end{equation}

which is an invertible mapping $\mathbb{R} \ni z \mapsto x \in [0,2\pi]$. The following lemma shows that the spectrum of the operator $A_0 : \mathcal{D}\subset L^2(\mathbb{T}) \to L^2(\mathbb{T})$ can be found from the spectrum of a simpler linear operator defined on the infinite line $\mathbb{R}$.

Lemma 4. The spectrum of the truncated operator $A_0 : \mathcal{D}\subset L^2(\mathbb{T}) \to L^2(\mathbb{T})$ coincides with the spectrum of the linear operator $D_0 : H^1(\mathbb{R}) \subset L^2(\mathbb{R}) \to L^2(\mathbb{R})$ given by

(5.5)\begin{equation} D_0 h := \partial_z h + \frac{\pi}{4} \tanh\left(\frac{\pi z}{4} \right) h + \frac{\pi}{4} w(z) \int_{\mathbb{R}} w'(z) h(z) dz, \end{equation}

where $w(z) := {\rm sech}\left(\frac{\pi z}{4} \right)$. The constraint (5.2) coincides with the constraint $\langle w, D_0 h \rangle = 0$, which holds for every $h \in H^1(\mathbb{R})$, where $\langle \cdot, \cdot \rangle$ is the standard inner product in $L^2(\mathbb{R})$ with the induced norm $\| \cdot \|$.

Proof. Using the transformation (5.4), we obtain $A_0 f = B_0 g$, where $g(z) = f(x)$ and $B_0 : {\rm Dom}(B_0) \subset L_w^2(\mathbb{R}) \to L_w^2(\mathbb{R})$ is given by

(5.6)\begin{equation} B_0 g := \partial_z g + \frac{\pi}{4} \int_{\mathbb{R}} w(z) w'(z) g(z) dz \end{equation}

and

\begin{equation*}\mathrm D\mathrm o\mathrm m(B_0):=\left\{g\in L_w^2(\mathbb{R}):\;g'\in L_w^2(\mathbb{R})\right\}\equiv H_w^1(\mathbb{R}),\end{equation*}

with the weight $w(z) := {\rm sech}\left(\frac{\pi z}{4} \right)$. Here the exponentially weighted spaces $L^2_w(\mathbb{R})$ and $H^1_w(\mathbb{R})$ are defined with the squared norms:

\begin{equation*} \| g \|_{L^2_w}^2 = \int_{\mathbb{R}} w^2(z) |g(z)|^2 dz, \qquad \| g \|_{H^1_w}^2 = \int_{\mathbb{R}} w^2(z) \left( |g'(z)|^2 + |g(z)|^2 \right) dz \end{equation*}

and the inner product in $L^2_w(\mathbb{R})$ is defined by

\begin{equation*} \langle g_1,g_2 \rangle_{L^2_w} = \int_{\mathbb{R}} w^2(z) g_1(z) g_2(z) dz. \end{equation*}

The constraint (5.2) coincides with the constraint $\langle 1, B_0 g \rangle_{L^2_w} = 0$, which holds for every $g \in H^1_w(\mathbb{R})$. By using the change of variables $h(z) = w(z) g(z)$, we get $B_0 g = w^{-1} D_0 h$ and $\langle 1, B_0 g \rangle_{L^2_w} = \langle w, D_0 h \rangle = 0$, where $D_0 : H^1(\mathbb{R}) \subset L^2(\mathbb{R}) \to L^2(\mathbb{R})$ is given by (5.5). Due to the transformations above, the spectrum of $A_0 : \mathcal{D}\subset L^2(\mathbb{T}) \to L^2(\mathbb{T})$ coincides with the spectrum of the linear operator $D_0 : H^1(\mathbb{R}) \subset L^2(\mathbb{R}) \to L^2(\mathbb{R})$.

The following theorem prescribes the spectrum of the truncated operator $A_0 : \mathcal{D}\subset L^2(\mathbb{T}) \to L^2(\mathbb{T})$ given by (5.1).

Theorem 2. The spectrum of $A_0 : \mathcal{D} \subset L^2(\mathbb{T}) \to L^2(\mathbb{T})$ completely covers the closed vertical strip given by

(5.7)\begin{equation} \sigma(A_0) = \left\{\lambda \in \mathbb{C} : \quad -\frac{\pi}{4} \leq {\rm Re}(\lambda) \leq \frac{\pi}{4} \right\}. \end{equation}

Proof. We obtain $\sigma_p(A_0)$ and $\rho(A_0)$ as

(5.8)\begin{align} \sigma_p(A_0) &= \left\{\lambda \in \mathbb{C} : \quad -\frac{\pi}{4} \lt {\rm Re}(\lambda) \lt \frac{\pi}{4} \right\}, \end{align}

(5.9)\begin{align} \rho(A_0) &= \left\{\lambda \in \mathbb{C} : \quad |{\rm Re}(\lambda)| \gt \frac{\pi}{4} \right\}. \end{align}

Since $\sigma(A_0)$ is a closed set and $\rho(A_0)$ is an open set, the closure of the open region (5.8) yields (5.7).

$\underline{\sigma_p(A_0)}$: By lemma 4, it is equivalent to consider $\sigma_p(D_0)$, where $D_0 : H^1(\mathbb{R}) \subset L^2(\mathbb{R}) \to L^2(\mathbb{R})$ is given by (5.5). Let $h \in H^1(\mathbb{R})$ be a solution of $D_0 h = \lambda h$ for some $\lambda \in \mathbb{C}$. Then, $h = h(z)$ satisfies

(5.10)\begin{equation} h'(z) + \frac{\pi}{4} \tanh\left(\frac{\pi z}{4} \right) h(z) + \frac{\pi}{4} w(z) \langle w', h \rangle = \lambda h(z), \quad z \in \mathbb{R}, \end{equation}

subject to the orthogonality condition $\lambda \langle w, h \rangle = 0$. Substitution $h(z) = \tilde{h}(z) w(z)$ reduces (5.10) to the form

(5.11)\begin{equation} \tilde{h}'(z) + \frac{\pi}{4} \langle w w', \tilde{h} \rangle = \lambda \tilde{h}(z), \quad z \in \mathbb{R}, \end{equation}

subject to the orthogonality condition $\lambda \langle w^2, \tilde{h} \rangle = 0$.

For λ = 0, the general solution of equation (5.11) is $\tilde{h}(z) = c_1 + c_2 z$, where $(c_1,c_2)$ are arbitrary constants. This yields the general solution $h(z) = (c_1 + c_2 z) w(z)$ of equation (5.10) for λ = 0. Since $h \in H^1(\mathbb{R})$, then $0 \in \sigma_p(D_0)$.

For λ ≠ 0, the general solution of equation (5.11) is a scalar multiplier of the particular solution

(5.12)\begin{equation} \tilde{h}(z) = e^{\lambda z} + \frac{\pi}{4 \lambda} \langle w w', e^{\lambda z} \rangle, \end{equation}

where the inner product is defined for $|{\rm Re}(\lambda)| \lt \frac{\pi}{2}$. Since

\begin{equation*} \frac{\pi}{8} \| w \|^2 = \frac{1}{2} \int_{\mathbb{R}} {\rm sech}^2(z) dz = 1, \end{equation*}

the orthogonality condition $\lambda \langle w^2, \tilde{h} \rangle = 0$ is satisfied for the solution (5.12). Integration by parts and transformation back to h yields the solution

\begin{equation*} h(z) = e^{\lambda z} w(z) - \frac{\pi}{8} w(z) \langle w^2, e^{\lambda z} \rangle, \end{equation*}

which show that $h \in H^1(\mathbb{R})$ if and only if $|{\rm Re}(\lambda)| \lt \frac{\pi}{4}$, so that $\sigma_p(A_0) = \sigma_p(D_0)$ is given by (5.8).

$\underline{\rho(A_0)}$: By lemma 4, it is equivalent to consider $\rho(D_0)$. Let $h \in H^1(\mathbb{R})$ be a solution of $(D_0 - \lambda) h = f$ for some $\lambda \in \mathbb{C}$ and $f \in L^2(\mathbb{R})$. Then, $h = h(z)$ satisfies

(5.13)\begin{equation} h'(z) + \frac{\pi}{4} \tanh\left(\frac{\pi z}{4} \right) h(z) + \frac{\pi}{4} w(z) \langle w', h \rangle = \lambda h(z) + f(z), \quad z \in \mathbb{R}. \end{equation}

Since $\langle w, D_0 h \rangle = 0$, we have $\lambda \langle w, h \rangle + \langle w, f \rangle = 0$. Multiplying equation (5.13) by $\bar{h}$, integrating over $\mathbb{R}$, adding complex conjugation and dividing by 2 yields

\begin{equation*} \frac{\pi}{4} \langle \tanh\left(\frac{\pi z}{4} \right) h,h \rangle + \frac{\pi}{4} {\rm Re} \langle h,w \rangle \langle w',h \rangle = {\rm Re}(\lambda) \| h \|^2 + {\rm Re} \langle h,f \rangle. \end{equation*}

By Cauchy–Schwarz inequality and the constraint $\bar{\lambda} \langle h,w \rangle + \langle f,w \rangle = 0$, we obtain

\begin{align*} \left( {\rm Re}(\lambda) - \frac{\pi}{4} \right) \| h \|^2 &\leq {\rm Re}(\lambda) \| h \|^2 - \frac{\pi}{4} \langle \tanh\left(\frac{\pi z}{4} \right) h,h \rangle \\ &= - {\rm Re} \langle h,f \rangle - {\rm Re} \frac{\pi}{4 \bar{\lambda}} \langle f,w \rangle \langle w',h \rangle \\ &\leq \left(1 + \frac{\pi \| w \| \|w'\|}{4 |\lambda|} \right) \| h \| \| f \| \end{align*}

and

\begin{align*} \left( -{\rm Re}(\lambda) - \frac{\pi}{4} \right) \| h \|^2 &\leq -{\rm Re}(\lambda) \| h \|^2 + \frac{\pi}{4} \langle \tanh\left(\frac{\pi z}{4} \right) h,h \rangle \\ &= {\rm Re} \langle h,f \rangle + {\rm Re} \frac{\pi}{4 \bar{\lambda}} \langle f,w \rangle \langle w',h \rangle \\ &\leq \left(1 + \frac{\pi \| w \| \|w'\|}{4 |\lambda|} \right) \| h \| \| f \|. \end{align*}

This yields the bound

\begin{equation*} \| h \| \leq \left( 1 + \| w \| \|w'\| \right) \frac{\| f \|}{|{\rm Re}(\lambda)| - \frac{\pi}{4}}, \quad \text{for} \;\; |{\rm Re}(\lambda)| \gt \frac{\pi}{4}. \end{equation*}

Hence, $\{\lambda \in \mathbb{C}: |{\rm Re}(\lambda)| \gt \frac{\pi}{4} \}$ belongs to $\rho(D_0)$, but since $\sigma(D_0)$ is a closed set and $\rho(D_0)$ is an open set, then $\{\lambda \in \mathbb{C}: |{\rm Re}(\lambda)| \gt \frac{\pi}{4} \}$ is equivalent to $\rho(D_0)$ in view of the location of $\sigma_p(D_0)$. This completes the proof of $\rho(A_0) = \rho(D_0)$ given by (5.9).

6. Full linearised evolution

The full linearised evolution is defined by the linear operator $A : \mathcal{D} \subset L^2(\mathbb{T}) \to L^2(\mathbb{T})$ given by (4.14). The following theorem prescribes the spectrum of A.

Theorem 3. The spectrum of $A : \mathcal{D} \subset L^2(\mathbb{T}) \to L^2(\mathbb{T})$ completely covers the closed vertical strip given by

(6.1)\begin{equation} \sigma(A) = \left\{\lambda \in \mathbb{C} : \quad -\frac{\pi}{4} \leq {\rm Re}(\lambda) \leq \frac{\pi}{4} \right\}. \end{equation}

Proof. Again, we obtain $\sigma_p(A)$ and $\rho(A)$ as

(6.2)\begin{align} \sigma_p(A) &= \left\{\lambda \in \mathbb{C} : \quad -\frac{\pi}{4} \lt {\rm Re}(\lambda) \lt \frac{\pi}{4} \right\} = \sigma_p(A_0), \end{align}

(6.3)\begin{align} \rho(A) &= \left\{\lambda \in \mathbb{C} : \quad |{\rm Re}(\lambda)| \gt \frac{\pi}{4} \right\} = \rho(A_0). \end{align}

Since $\sigma(A)$ is a closed set and $\rho(A)$ is an open set, the closure of the open region (6.2) yields (6.1).

$\underline{\sigma_p(A)}$:Let $f \in \mathcal{D}$ be a solution of $Af = \lambda f$ for some $\lambda \in \mathbb{C}$. Then, $f = f(x)$ satisfies

(6.4)\begin{equation} \frac{1}{4} x(2\pi - x) f'(x) + \frac{1}{4 \pi} \int_{0}^{2\pi} (\pi-x) f(x) dx + \frac{1}{2} \Pi_0 \partial_x^{-1} \Pi_0 f = \lambda f(x), \quad 0 \lt x \lt 2\pi. \end{equation}

subject to the orthogonality condition $\lambda \int_0^{2\pi} f(x) dx = 0$.

If $f \in \mathcal{D}$, then $f \in L^2(\mathbb{T})$ and $(c_*^2-2\eta_*) f' \in L^2(\mathbb{T})$, whereas $\lim\limits_{x \to 0^+} f(x)$ and $\lim\limits_{x \to 2\pi^-} f(x)$ may not be defined. Nevertheless, since $c_*^2 - 2 \eta_*(x) \neq 0$ for $x \in (0,2\pi)$, we have $f \in C^0(0,2\pi)$. Bootstrapping arguments for equation (6.4) imply that $f \in C^{\infty}(0,2\pi)$. Differentiating (6.4) in x yields the second-order differential equation

(6.5)\begin{equation} \frac{1}{4} x(2\pi - x) f''(x) + \frac{1}{2} (\pi - x) f'(x) + \frac{1}{2} f(x) - \frac{1}{4\pi} \int_0^{2\pi} f(x) dx = \lambda f'(x), \quad 0 \lt x \lt 2\pi. \end{equation}

If λ ≠ 0, then $\int_0^{2\pi} f(x) dx = 0$ so that equation (6.5) can be rewritten in the form

(6.6)\begin{equation} \frac{1}{4} x(2\pi - x) f''(x) + \frac{1}{2} (\pi - x) f'(x) + \frac{1}{2} f(x) = \lambda f'(x), \quad 0 \lt x \lt 2\pi. \end{equation}

One solution of (6.6) is obtained explicitly as $f_1(x) = 2 \lambda - \pi + x$. The second linearly independent solution $f_2(x)$ of (6.6) is obtained from the Wronskian

(6.7)\begin{equation} f_1(x) f_2'(x) - f_1'(x) f_2(x) = W(x), \end{equation}

where W(x) satisfies the first-order differential equation by Abel’s theorem:

(6.8)\begin{equation} W'(x) = \frac{2 (2 \lambda - \pi + x)}{x(2\pi-x)} W(x). \end{equation}

Integrating (6.8) yields

(6.9)\begin{equation} W(x) = \frac{\pi^2}{x (2\pi-x)} \left( \frac{x}{2 \pi - x} \right)^{\frac{2\lambda}{\pi}}, \end{equation}

where the constant of integration has been normalised by the condition $W(\pi) = 1$. It follows from (6.7) and (6.9) that if λ ≠ 0, then $f_2(x)$ satisfies the following asymptotic limits

\begin{align*} f_2(x) \sim \left\{\begin{array}{ll} x^{\frac{2\lambda}{\pi}}, \quad & \lambda \neq \frac{\pi}{2}, \\ 1, \quad & \lambda = \frac{\pi}{2} \end{array} \right. \quad \text{as } \;\; x \to 0^+ \end{align*}

and

\begin{align*} f_2(x) \sim \left\{\begin{array}{ll} (2\pi - x)^{-\frac{2\lambda}{\pi}}, \quad & \lambda \neq -\frac{\pi}{2}, \\ 1, \quad & \lambda = -\frac{\pi}{2} \end{array} \right. \quad \text{as } \;\; x \to (2\pi)^-. \end{align*}

On the other hand, $f_1,f_2 \in C^{\infty}(0,2\pi)$ for every $\lambda \in \mathbb{C}$.

If $|{\rm Re}(\lambda)| \lt \frac{\pi}{4}$, then $f_2 \in L^2(\mathbb{T})$ and $(c_*^2-\eta_*) f_2' \in L^2(\mathbb{T})$, so that $f(x) = c_1 f_1(x) + c_2 f_2(x)$ belongs to $\mathcal{D}$ for every $(c_1,c_2) \in \mathbb{R}^2$. Satisfying the constraint $\int_0^{2\pi} f(x) dx = 0$ for λ ≠ 0 yields a one-parameter family of solutions $f \in \mathcal{D}$ of (6.4) for every $\lambda \in \sigma_p(A) \backslash \{0\}$, where $\sigma_p(A)$ is given by (6.2).

If λ = 0, then equation (6.5) contains a constant term. Without the constant term $-\frac{1}{4\pi} \int_0^{2\pi} f(x) dx$, the two homogeneous solutions $f_1,f_2 \in \mathcal{D}$ are given by

\begin{equation*} f_1(x) = x - \pi, \quad f_2(x) = \frac{1}{2\pi} (x-\pi) \ln \frac{x}{2\pi - x} - 1, \end{equation*}

However, only f ₁ satisfies (6.5) since $\int_0^{2\pi} f_1(x) dx = 0$. On the other hand, it is easy to see that $f(x) = 1$ is also a solution of (6.5). Thus, there exists a two-parameter family of solutions $f = c_1 (x-\pi) + c_2 \in \mathcal{D}$ of equation (6.4) for λ = 0, so that $0 \in \sigma_p(A)$.

Finally, if $|{\rm Re}(\lambda)| \geq \frac{\pi}{4}$, then $f_2 \notin L^2(\mathbb{T})$ due to the asymptotic limits as $x \to 0^+$ and $x \to (2\pi)^-$. On the other hand, $\int_0^{2\pi} f_1(x) dx = 4\pi \lambda \neq 0$ for λ ≠ 0, so that there exist no nonzero solutions $f \in \mathcal{D}$ of equation (6.4) for every $\lambda \notin \sigma_p(A)$. This completes the proof of $\sigma_p(A)$ given by (6.2).

$\underline{\rho(A)}$: Let $f \in \mathcal{D}$ be a solution of the resolvent equation $(A - \lambda) f = g$ for $g \in L^2(\mathbb{T})$ rewritten in the form:

(6.10)\begin{equation} (c_*^2 - 2 \eta_*) \partial_x f -\frac{1}{\pi} \oint \eta_*' f dx + \frac{1}{2} \Pi_0 \partial_x^{-1} \Pi_0 f - \lambda f = g, \quad x \in \mathbb{T}. \end{equation}

Since $\oint Af dx = 0$, we get $-\lambda \oint f dx = \oint g dx$. Taking into account that $\Pi_0 \partial_x^{-1} \Pi_0$ is a skew-adjoint operator in $L^2(\mathbb{T})$, we multiply (6.10) by $\bar{f}$, integrate over $\mathbb{T}$, add a complex conjugate equation and divide by 2 to obtain

\begin{equation*} {\rm Re} \langle f, (c_*^2 - 2 \eta_*) \partial_x f \rangle - {\rm Re}(\lambda) \| f \|^2 -\frac{1}{\pi} {\rm Re} \langle f, 1 \rangle \langle \eta_*', f \rangle = {\rm Re} \langle f,g \rangle. \end{equation*}

Integrating by parts in the first term and using $-\bar{\lambda} \langle f, 1 \rangle = \langle g, 1 \rangle$ for λ ≠ 0, we obtain

\begin{equation*} \langle \eta_*' f,f \rangle - {\rm Re}(\lambda) \| f \|^2 + {\rm Re} \frac{\langle \eta_*', f \rangle \langle g, 1 \rangle}{\pi \bar{\lambda}} = {\rm Re} \langle f,g \rangle. \end{equation*}

Since $-\frac{\pi}{4} \leq \eta_*'(x) \leq \frac{\pi}{4}$ for $x \in [0,2\pi]$, we get by Cauchy–Schwarz inequality that

\begin{align*} \left( {\rm Re}(\lambda) - \frac{\pi}{4} \right) \| f \|^2 &\leq {\rm Re}(\lambda) \| f \|^2 - \langle \eta_*' f,f \rangle \\ &= -{\rm Re} \langle f,g \rangle + {\rm Re} \frac{\langle \eta_*', f \rangle \langle g, 1 \rangle}{\pi \bar{\lambda}} \\ &\leq \left(1 + \frac{\sqrt{2\pi} \| \eta_*' \|}{\pi |\lambda|} \right) \| g \| \| f \| \end{align*}

and

\begin{align*} \left( -{\rm Re}(\lambda) - \frac{\pi}{4} \right) \| f \|^2 &\leq -{\rm Re}(\lambda) \| f \|^2 + \langle \eta_*' f,f \rangle \\ &={\rm Re} \langle f,g \rangle - {\rm Re} \frac{\langle \eta_*', f \rangle \langle g, 1 \rangle}{\pi \bar{\lambda}} \\ &\leq \left(1 + \frac{\sqrt{2\pi} \| \eta_*' \|}{\pi |\lambda|} \right) \| g \| \| f \|. \end{align*}

This yields the bound

\begin{equation*} \| f \| \leq \left(1 + \frac{4 \sqrt{2\pi} \| \eta_*' \|}{\pi^2} \right) \frac{\| g \|}{|{\rm Re}(\lambda)| - \frac{\pi}{4}}, \quad \text{for} \;\; |{\rm Re}(\lambda)| \gt \frac{\pi}{4}. \end{equation*}

Hence, $\{\lambda \in \mathbb{C}: |{\rm Re}(\lambda)| \gt \frac{\pi}{4} \}$ belongs to $\rho(A)$, but since $\sigma(A)$ is a closed set and $\rho(A)$ is an open set, then $\{\lambda \in \mathbb{C}: |{\rm Re}(\lambda)| \gt \frac{\pi}{4} \}$ is equivalent to $\rho(A)$ in view of the location of $\sigma_p(A)$ in (6.2). This completes the proof of $\rho(A)$ given by (6.3).

7. Nonlinear evolution

We shall now derive the nonlinear instability result for the peaked travelling wave by using the nonlinear evolution equation (4.4). With the help of equations (4.9) and (4.11) in lemma 2, we rewrite the nonlinear evolution equation (4.4) in the equivalent form:

(7.1)\begin{equation} 2 c_* \partial_t \zeta = (c_*^2 - 2 \eta_*) \partial_x \zeta -2 (\zeta - \zeta |_{x = 0}) \partial_x \zeta -\frac{1}{\pi} \oint \eta_*' \zeta dx + \frac{1}{2} \Pi_0 \partial_x^{-1} \Pi_0 \left[ \zeta + 2 (\partial_x \zeta)^2 \right]. \end{equation}

After applying the transformation (4.3) and integrating by parts using (3.5), the original constraint (2.1) becomes

(7.2)\begin{align} 0 = \oint \left[ \zeta + 2 \eta_*' \partial_x \zeta + (\partial_x \zeta)^2 \right] dx = \frac{1}{2} \oint \left[ \zeta + 2 (\partial_x \zeta)^2 \right] dx + \pi \zeta|_{x=0}. \end{align}

The following lemma identifies two conserved quantities of the evolution equation (7.1), whose sum yields the constraint (7.2).

Lemma 5. Consider a local solution $\zeta \in C^0((-\tau_0,\tau_0),H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T}))$ of the evolution equation (7.1) for some $\tau_0 \gt 0$. Then

(7.3)\begin{equation} \oint \zeta dx \quad \mbox{and} \quad \zeta|_{x=0} + \frac{1}{\pi} \oint (\partial_x \zeta)^2 dx \end{equation}

are conserved for $t \in (-\tau_0,\tau_0)$.

Proof. The conservation of $\oint \zeta dx$ is shown by taking the mean value of the evolution equation (7.1), while the conservation of $\zeta|_{x=0} + \frac{1}{\pi} \oint (\partial_x \zeta)^2 dx$ follows from the constraint (7.2). We can also show the latter conservation directly as follows. For smooth solutions, we differentiate equation (7.1) and obtain

\begin{align*} \notag 2 c_* \partial_t \partial_x \zeta &= (c_*^2 - 2 \eta_*) \partial^2_x \zeta -2 \eta_*' \partial_x \zeta - 2 (\zeta - \zeta |_{x = 0}) \partial^2_x \zeta - 2(\partial_x \zeta)^2 \end{align*}

(7.4)\begin{align} & \quad + \frac{1}{2} (\zeta + 2 (\partial_x \zeta)^2) - \frac{1}{4\pi} \oint (\zeta + 2 (\partial_x \zeta)^2) dx \end{align}

Multiplying (7.4) by $\partial_x \zeta$ and integrating in x over $\mathbb{T}$ yields

(7.5)\begin{equation} c_* \frac{d}{dt} \oint (\partial_x \zeta)^2 dx = - \oint \eta_*' (\partial_x \zeta)^2 dx. \end{equation}

Furthermore, taking the limit x → 0 in equation (7.1) as in (4.12) and (4.13), we get

(7.6)\begin{equation} 2 c_* \lim_{x\to 0} \partial_t \zeta = \lim_{x \to 0} \Pi_0 \partial_x^{-1} \Pi_0 (\partial_x \zeta)^2 = \frac{2}{\pi} \oint \eta_*' (\partial_x \zeta)^2 dx. \end{equation}

By adding (7.5) divided by π and (7.6) divided by 2, we verify conservation of $\zeta|_{x=0} + \frac{1}{\pi} \oint (\partial_x \zeta)^2 dx$. By Sobolev’s embedding of $H^1_{\rm per}(\mathbb{T})$ into $C^0_{\rm per}(\mathbb{T})$, we have well-defined $\zeta|_{x=0} \in C^0(-\tau_0,\tau_0)$. Hence, the conserved quantities (7.3) are well-defined for the local solution $\zeta \in C^0((-\tau_0,\tau_0),H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T}))$.

Based on the conserved quantity $\zeta|_{x=0} + \frac{1}{\pi} \oint (\partial_x \zeta)^2 dx$, we obtain the nonlinear instability of the peaked travelling wave given by the following theorem.

Theorem 4. For every δ > 0 there exists $\zeta_0 \in H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$ satisfying

(7.7)\begin{equation} \| \zeta_0 \|_{H^1_{\rm per}} \leq \delta^2, \quad \| \zeta_0 \|_{W^{1,\infty}} \leq \delta, \end{equation}

such that the unique local solution $\zeta \in C^0((-\tau_0,\tau_0),H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T}))$ of the evolution equation (7.1) with $\zeta|_{t = 0} = \zeta_0$ satisfies

(7.8)\begin{equation} \| \zeta(t_0,\cdot) \|_{W^{1,\infty}} = 1, \end{equation}

for some $t_0 \in (0,\tau_0)$.

Proof. The proof follows by the method of characteristics which works for every local solution $\zeta \in C^0((-\tau_0,\tau_0),H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T}))$ of the evolution equation (7.1). Let $x = X(t,s)$ be the family of characteristic curves for $(t,s) \in (-\tau_0,\tau_0) \times (0,2\pi)$ obtained from

(7.9)\begin{equation} \left\{\begin{array}{l} 2 c_* \partial_t X(t,s) = -(c_*^2 - 2 \eta_*(X)) + 2 (\zeta(t,X) - \zeta(t,0) ), \\ X(0,s) = s. \end{array} \right. \end{equation}

Since the vector field of the initial-value problem (7.9) is Lipschitz for the local solution $\zeta \in C^0((-\tau_0,\tau_0),H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T}))$, there is a unique solution for $X \in C^1((-\tau_0,\tau_0) \times (0,2\pi))$ such that $X(t,0) = 0$ and $X(t,2\pi) = 2 \pi$. Since $[0,2\pi]$ is the fundamental period of $\mathbb{T}$, we also get $\zeta(t,0) = \zeta(t,2\pi)$. By solving the linear equation for $\partial_s X(t,s)$, we get

\begin{equation*} \partial_s X(t,s) = \exp\left( \frac{1}{c_*} \int_0^t \left[ \eta_*'(X(t',s)) + \partial_x \zeta(t',X(t',s)) \right] dt' \right), \end{equation*}

from which it follows that $\partial_s X(t,s) \gt 0$ for every $t \in (-\tau_0,\tau_0)$ and $s \in (0,2\pi)$. Hence the mapping $[0,2\pi] \ni s \mapsto X(t,s) \in [0,2\pi]$ is a diffeomorphism for $t \in (-\tau_0,\tau_0)$.

To proceed further, we consider the restriction of $\zeta_0 \in H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T}) \cap C^1(0,2\pi)$. Along the family of characteristic curves, we can now define $Z(t,s) := \zeta(t,X(t,s))$ and $V(t,s) := \partial_x \zeta(t,X(t,s))$. By using the evolution equations (7.1) and (7.4) along the family of characteristic curves satisfying (7.9), we obtain the following initial-value problems:

(7.10)\begin{equation} \left\{\begin{array}{l} 2 c_* \partial_t Z(t,s) = -\frac{1}{\pi} \langle \eta_*', \zeta \rangle + \frac{1}{2} \Pi_0 \partial_x^{-1} \Pi_0 (\zeta + 2 (\partial_x \zeta)^2), \\ Z(0,s) = \zeta_0(s), \end{array} \right. \end{equation}

and

(7.11)\begin{equation} \left\{\begin{array}{l} 2 c_* \partial_t V(t,s) = -2\eta_*'(X) V - V^2 + \frac{1}{2} (Z(t,s) + Z(t,0)), \\ V(0,s) = \zeta_0'(s), \end{array} \right. \end{equation}

where we have used the constraint (7.2). If $\zeta_0 \in H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T}) \cap C^1(0,2\pi)$, then the solutions of the initial-value problems (7.9), (7.10) and (7.11) are defined in the class of functions

\begin{equation*} \begin{cases} X \in C^1((-\tau_0,\tau_0) \times (0,2\pi)), \\ Z \in C^1((-\tau_0,\tau_0) \times (0,2\pi)), \\ V \in C^1((-\tau_0,\tau_0),C^0(0,2\pi)), \end{cases} \end{equation*}

respectively, with the bounded one-sided limits as $s \to 0^+$ and $s \to (2\pi)^-$. Due to the conservation law in (7.3), we have

\begin{equation*} \zeta|_{x=0} + \frac{1}{\pi} \oint (\partial_x \zeta)^2 dx = C_0, \end{equation*}

from which $\zeta |_{x=0} \leq C_0$ with a time-independent positive constant C ₀. Since $X(t,0) = 0$, we get $Z(t,0)\leq C_0$ so that the evolution problem (7.11) in the limit $s \to 0^+$ yields for $V_0(t) := \lim\limits_{s \to 0^+} V(t,s)$:

\begin{align*} 2c_* V_0'(t) = \frac{\pi}{2} V_0(t) - V_0^2(t) + Z(t,0) \leq \frac{\pi}{2} V_0(t) + C_0. \end{align*}

Iterating the inequality as

\begin{align*} 2c_* \frac{d}{dt} e^{-\frac{\pi t}{4c_*}} V_0(t) \leq C_0 e^{-\frac{\pi t}{4c_*}} \end{align*}

and integrating yields

\begin{align*} 2c_* \left[ e^{-\frac{\pi t}{4c_*}} V_0(t) - V_0(0) \right] \leq \frac{4 c_* C_0}{\pi} \left[ 1 - e^{-\frac{\pi t}{4 c_*}} \right] \leq \frac{4 c_* C_0}{\pi}. \end{align*}

This implies

\begin{equation*} V_0(t) \leq \left( V_0(0) + \frac{2}{\pi} C_0 \right) e^{\frac{\pi t}{4 c_*}}. \end{equation*}

If the initial bound (7.7) is true, we get by Sobolev embedding of $H^1_{\rm per}(\mathbb{T})$ to $L^{\infty}(\mathbb{T})$ that

\begin{equation*} |C_0| \leq \delta^2 + \frac{1}{\pi} \delta^2 \leq 2 \delta^2, \end{equation*}

so that the interval $(-\delta,-\frac{2}{\pi} |C_0|)$ is nonempty for all sufficiently small δ > 0. Selecting $-\delta \lt V_0(0) \lt -\frac{2}{\pi} |C_0|$ to satisfy the initial bound (7.7) and to ensure that $V_0(0) + \frac{2}{\pi} C_0 \lt 0$ yields the exponential divergence $V_0(t) \to -\infty$ as $t \to +\infty$. Since $\tau_0 \gt 0$ is the maximal existence time in $H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$ norm, there exists $t_0 \in (0,\tau_0)$ such that the instability bound (7.8) holds.

Remark 7. By incorporating the quadratic term in the bound

\begin{equation*} 2c_* V_0'(t) = \frac{\pi}{2} V_0(t) - V_0^2(t) + Z(t,0) \leq \frac{\pi}{2} V_0(t) - V_0^2(t) + C_0, \end{equation*}

one can find initial data $\zeta_0 \in H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$ for which $\| \zeta(t,\cdot) \|_{W^{1,\infty}}$ diverges in a finite time, see [Reference Madiyeva and Pelinovsky39] for a similar analysis of the Camassa–Holm equation. However, we do not have the bound on $\| \zeta(t,\cdot) \|_{H^1_{\rm per}}$ compared to the case of the Camassa–Holm equation, where the $H^1_{\rm per}$ norm of the perturbation does not grow in time, see [Reference Lenells32, Reference Lenells33, Reference Madiyeva and Pelinovsky39].

8. Numerical approximations

We approximate numerically the smooth profile $\eta \in C_{\rm per}^{\infty}(\mathbb{T})$ of the travelling waves from the second-order equation (3.1) and the eigenvalues of the Hessian operator $\mathcal L : H_\mathrm{per}^2(\mathbb T) \subset L^2 (\mathbb T)\to L^2(\mathbb T)$ which defines the linearised time evolution (4.1) for the smooth travelling waves. In both cases, we are interested to understand the convergence of numerical results to the peaked travelling wave with the profile $\eta_* \in C^0_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$ as $c \to c_*$. We show that the lowest eigenvalue of $\mathcal{L}$ diverges as $c \to c_*$, which suggests that the linearised equation (4.1) cannot be used for the peaked travelling wave. This explains why we had to derive a different linearised equation (4.5) for the peaked travelling wave.

To obtain the solutions $\eta \in C_{\rm per}^{\infty}(\mathbb{T})$ of the second-order equation (3.1) for $c \in (1,c_*)$, we use the first-order invariant (3.2) and define η from the boundary-value problem:

(8.1)\begin{equation} \left\{ \begin{array}{ll} \displaystyle \left (\frac{d\eta}{dx} \right )^2=\frac{2\mathcal E-\eta^2}{c^2-2\eta}, \\[7pt] \eta(\pm\pi)= -\sqrt{2\mathcal E}. \end{array} \right. \end{equation}

Since $\eta(-x) = \eta(x)$, we take the negative sign in the square root of (8.1) for $x \in [0,\pi]$ and obtain the solution profile $\eta(x)$ for $x \in [0,\pi]$ by finding the root of the integral equation

(8.2)\begin{equation} f(\eta(x))-x = 0 , \qquad f(\eta) := \int_{\eta/\sqrt{2\mathcal E}}^{1}\frac{\sqrt{c^2-2\sqrt{2\mathcal E}x}}{\sqrt{1-x^2}}\,dx, \end{equation}

for $\mathcal E \in (0,\mathcal E_c)$, where $\mathcal E_c := \frac{c^4}{8}$ is the value separating smooth and cusped profiles.

To determine the value of $\mathcal E$ for each $c\in (1,c_*)$, we consider the period function $T(\mathcal E,c)$ studied in [Reference Locke and Pelinovsky36]. The period function is represented by using the complete elliptic integral $E(\kappa)$ of the second kind with the elliptic modulus $\kappa \in (0,1)$, which is defined by $\mathcal{E}\in (0,\mathcal{E}_c)$ and $c \in (1,c_*)$ as follows:

(8.3)\begin{equation} T(\mathcal E,c)= 4E(\kappa)\sqrt{c^2+2\sqrt{2\mathcal E}}, \quad \kappa = \sqrt{\frac{4\sqrt{2\mathcal E}}{c^2+2\sqrt{2\mathcal E}}}. \end{equation}

The periodic profile $\eta \in C_{\rm per}^{\infty}(\mathbb{T})$ corresponds to the value of $\mathcal{E} = \mathcal{E}(c)$ found from the root of $T(\mathcal{E}(c),c) = 2\pi$.

To approximate the solution profile numerically, we let $x_j = jh,j\in \{0,\dots ,N\}$ be a fixed grid with the step size $h=\pi/N$ for a large integer N. By the fundamental theorem of calculus, the solution profile $\{(x_j,\eta_j)\}_{j=0}^{N}$ can be found by solving $f(\eta_j)-x_j=0$ for every j. We implement Newton–Raphson’s method as the root-finding algorithm under a specific tolerance ϵ > 0, such that

(8.4)\begin{equation} \eta_j^{(k+1)} = \eta_j^{(k)} - \frac{1}{f\prime\left (\eta_j^{(k)}\right )} \left [ f\left (\eta_j^{(k)}\right )-x_{j}\right ], \quad \quad \left | f\left (\eta_j^{(k)}\right )-x_{j} \right | \lt \epsilon, \end{equation}

where $k\in \mathbb N$ denotes iteration number, and we take $\eta_0^{(k)}(0)=\sqrt{2\mathcal E}$ and $\eta_N^{(k)}(\pi)= -\sqrt{2\mathcal E}$ as two boundary grid points for any k to avoid the singularities. Given a c-grid $\{c_i\}_{i=1}^M$ with M grid points, we compute $\mathcal{E}_i:=\mathcal E(c_i)$ from the period function (8.3) by solving numerically $T(\mathcal{E}_i,c_i) = 2\pi$. This outputs the sets of parameter pairs $\{(c_i, \mathcal E_i)\}_{i=1}^M$, which can be used in the root-finding algorithm (8.4). After the solution $\{(x_j,\eta_j)\}_{j=0}^{N}$ is obtained on $[0,\pi]$ for N + 1 grid points, the even reflection fills all $2N+1$ grid points on $[-\pi,\pi]$ domain. The solution points $\{(x_j,\eta_j)\}_{j=-N}^{N}$ are plotted in figure 1 (left) and the parameter pairs $\{(c_i,\sqrt{2{\mathcal E}_i})\}_{i=1}^M$ are plotted in figure 1 (right).

The solution $\{(x_j,\eta_j)\}_{j=-N}^{N}$ can be represented in Fourier space by using the discrete Fourier transform (DFT) with $2N+1$ modes:

(8.5)\begin{equation} \hat \eta_n = \frac{h}{2\pi} \sum_{j=-N}^{N-1}\eta_j e^{-in x_j},\quad \quad n \in \{-N,\dots, N\} \end{equation}

where one of the endpoints $x_N=\pi$ in the physical space is removed due to the 2π-periodicity. For the peaked wave profile $\eta_*$ at $c=c_*$, the solution (3.3) can be represented as the Fourier cosine series

(8.6)\begin{equation} \eta_* (x)= -\frac{\pi^2}{48}+\sum_{m=1}^{\infty}\frac{\cos(mx)}{2m^2}. \end{equation}

By applying DFT on the selected grid, the solution points $\{(n,\hat{\eta}_n)\}_{n=0}^{N}$ are plotted for the smooth profiles in figure 2. The black dashed line represents the Fourier series (8.6) for the peaked profile. We note that the fast convergence of the Fourier transform for the smooth waves is replaced by the slow convergence $\mathcal{O}(m^{-2})$ of the Fourier transform for the peaked waves.

Figure 2. The solution profiles $\hat{\eta}$ in Fourier space (8.5) in log-log coordinates for $c = 1.03,1.07$ with N = 300 grid points and $\epsilon = 10^{-14}$ tolerance. The black dashed line represents the peaked profile $\eta_*$ for $c = c_*$.

Next, we study numerically the spectrum of the Hessian operator $\mathcal{L}: H^2_{\rm per}(\mathbb{T}) \subset L^2(\mathbb{T}) \to L^2(\mathbb{T})$ which appears in the linearised equation (4.1). The spectrum of $\mathcal L$ can be computed by solving the spectral problem with the 2π-periodic conditions,

(8.7)\begin{equation} \mathcal L \gamma = \lambda \gamma, \quad \gamma \in H^2_{\rm per}(\mathbb{T}). \end{equation}

We will apply two numerical methods (the finite-difference method and the Fourier collocation method) to solve the spectral problem (8.7). We write $\mathcal L =\mathcal M + \mathcal W$ with $\mathcal M = -\partial_x (c^2-2\eta)\partial_x$ and $\mathcal W= (2\eta'' -1)$.

For the finite difference method, using the numerical approximation of the solution profile $\{(x_j,\eta_j)\}_{j=-N}^N$ and the central difference approximation of the second derivative $\mathcal M$, we construct the differentiation matrix for $\mathcal L$ acting on the eigenvector $\gamma = (\gamma_{-N},\dots , \gamma_{N-1})\in\mathbb R^{2N}$,

\begin{equation*} \mathcal L = \begin{bmatrix} \mathcal (W^0 +\mathcal M^0) (\eta_{-N}) & \mathcal M^{+1}(\eta_{-N}) & 0 & \cdots & \mathcal M^{-1}(\eta_N) \\ \mathcal M^{-1}(\eta_{-N+1}) & (\mathcal W^0 +\mathcal M^0)(\eta_{-N+1}) & \mathcal M^{+1}(\eta_{-N+1}) & \cdots & 0 \\ 0 & \mathcal M^{-1}(\eta_{-N+2}) & \mathcal (W^0 +\mathcal M^0) (\eta_{-N+2}) & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \mathcal M^{+1}(\eta_{N-1}) & 0 & 0 & \cdots & (\mathcal W^0 +\mathcal M^0)(\eta_{N-1}) \end{bmatrix}, \end{equation*}

where the boundary point $x_N = \pi$ is removed due to the 2π-periodicity. The diagonal elements $\mathcal{M}^0(\eta_j)$, $\mathcal W^0(\eta_j)$ and the off-diagonal elements $\mathcal M^{\pm 1}(\eta_j)$ for $j\in \{-N,\dots ,N-1\}$ can be written as

\begin{equation*} \mathcal M^{0}(\eta_j)= \tfrac{2c^2-2\eta_j-\eta_{j+1}-\eta_{j-1} }{h^2}, \quad \mathcal M^{\pm 1}(\eta_j)= -\tfrac{c^2-\eta_j-\eta_{j\pm1} }{h^2}, \end{equation*}

and

\begin{equation*} \mathcal W^0(\eta_j) = \frac{4\mathcal E + 2\eta_j^2 - 2c^2\eta_j}{\left (c^2-2\eta_j \right )^2}-1 \end{equation*}

for $c\in (1,c_*)$, where the differential equations (3.1) and (3.2) have been used to eliminate $\eta''(x)$. By numerically solving the eigenvalue problem, we obtain the first four eigenfunctions γ plotted in figure 3. The eigenfunctions display spikes near x = 0 in the limit of $c \to c_*$.

Figure 3. Eigenfunctions corresponding to the first four eigenvalues for five values of c in $(1,c_*)$. The grid in physical space is chosen to be N = 300. The solution profiles obtained from equation (8.4) are used, and all eigenfunctions are plotted on $[-\pi,\pi]$ with positive slope near $-\pi$.

For the Fourier collocation method, we use the discrete Fourier transform (8.5) and represent γ in the spectral problem (8.7) by $\hat \gamma = (\hat \gamma_{-N}, \dots, \hat \gamma_{N})\in \mathbb R^{2N+1}$. Thanks to the L ²-isomorphism between functions in physical and Fourier spaces, the eigenvalue problem in Fourier space $\widehat {\mathcal L} \hat \gamma = \lambda \hat \gamma$ shares the same eigenvalues as that in the physical space. We write again $\widehat {\mathcal L}= \widehat{\mathcal M} + \widehat{\mathcal W}$ with

\begin{equation*} \widehat{\mathcal M}=2(\mathcal D_1\hat \eta) \mathcal D_1-c^2\mathcal D_2+ 2\hat \eta \mathcal D_2, \qquad \widehat{\mathcal W} = 2\mathcal (D_2\hat \eta) - \pi I \end{equation*}

for $c \in (1,c_*)$, where the first and second derivative are represented by

\begin{equation*} \mathcal D_1 = i\,\mathrm{diag}(-N,\dots, N),\quad \quad \mathcal D_2 = \mathcal D_1^2 \end{equation*}

and $\hat{\eta}$ is the Toeplitz matrix for convolution with the Fourier modes for $m \in \{-N,\dots, N\}$,

\begin{equation*} \hat \eta = \begin{bmatrix} \hat \eta_0 & \cdots & \hat \eta_{-N} & \cdots & 0 \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ \hat \eta_N & \cdots & \hat \eta_0 & \cdots & \hat \eta_{-N} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ 0 & \cdots & \hat \eta_N & \cdots & \hat \eta_0 \end{bmatrix}. \end{equation*}

By numerically solving the eigenvalue problem in the Fourier space, we obtain the first four eigenfunctions plotted in figure 4. Convergence of eigenfunctions in Fourier space for large m becomes worse as $c \to c_*$.

Figure 4. The absolute value of eigenfunctions corresponding to the first four eigenvalues in Fourier space is plotted versus $m\in \{1,\dots, N\}$ for five values of c in $(1,c_*)$. The grid in physical space is chosen to be N = 300, and the solution profiles $\hat \eta$ are obtained from equations (8.4) and (8.5).

Figure 5 shows the first four eigenvalues obtained by the finite-difference method (left) and the Fourier collocation method (right). The lowest eigenvalue diverges to $-\infty$ as $c \to c_*$. After λ ₁, the even-numbered eigenvalues $\lambda_2, \lambda_4, \dots$ correspond to eigenfunctions of even parity in x, whereas the odd-numbered eigenvalues $\lambda_3,\lambda_5,\dots$ correspond to eigenfunctions of odd parity. Convergence of eigenvalues as $c \to c_*$ is low in both physical and Fourier space for the fixed truncation number N. The grey shaded region highlights the loss of precision in the numerical approximations of eigenvalues with poor convergence to the eigenvalues of the limiting peaked wave for N = 300.

Figure 5. The dependence of the first four eigenvalues of the spectral problem (8.7) is plotted versus c for $c \in (1,c_*)$ obtained with the finite-difference method (left) and with the Fourier collocation method (right). Eigenvalues computed for the peaked profile with $c = c_*$ in the finite-difference method (left) are marked as circles. The grid in physical space is chosen to be N = 300.

To compute eigenvalues for the limiting wave with the peaked profile $\eta_*$ at $c = c_*$ in the finite-difference method, we use

\begin{equation*} c = c_* : \quad \mathcal{W}^0(\eta_j) = -\frac{1}{2} - \pi (\delta_0)_j. \end{equation*}

The Dirac delta distribution is approximated by the Gaussian pulse with a small parameter $\alpha = \pi/N$ as

\begin{equation*} \delta_0(x) \approx \frac{1}{\sqrt{\pi \alpha^2}}e^{-x^2/\alpha^2}, \quad x \in \mathbb{T}. \end{equation*}

As is shown in the left panel of figure 5, eigenvalues do not converge well as $c \to c_*$ to the limiting eigenvalues at $c = c_*$ for the fixed value of N = 300.

To illustrate further the low convergence of eigenvalues as $c \to c_*$, we plot the dependence of the third eigenvalue (which is theoretically zero, see [Reference Locke and Pelinovsky36]) versus c in figure 6 (left). For computations with different methods and for different N, we observe the growth $|\lambda_3|$, which is a numerical way to detect the loss of accuracy of numerical computations. Similarly, we show in figure 6 (right) the L ² norms of the residual terms

\begin{equation*} \left\{ \begin{array}{ll} \displaystyle \Vert \mathcal L\gamma_3 - \lambda_3 \gamma_3 \Vert_{L^2} = \left( \sum_{j=-N}^{N-1} | \mathcal L \gamma_3(x_j) - \lambda_3 \gamma_3(x_j) |^2 \Delta x \right)^{1/2}, \\[7pt] \displaystyle \Vert \widehat {\mathcal L} \hat \gamma_3 - \lambda_3 \hat \gamma_3 \Vert_{L^2} = \left( \sum_{k=-N}^{N} | \widehat{\mathcal L} \hat \gamma_3 (m_k) - \lambda_3 \hat{\gamma}_3(m_k)|^2 \Delta m \right)^{1/2}. \end{array} \right. \end{equation*}

Figure 6. (a) The third eigenvalue $|\lambda_3|$ plotted versus c to show its departure from 0 as $c\to c_*^-$ for different grids $N=100,200,300$ by the finite difference (CD) and Fourier collocations (Fourier) methods. (b) The same plots but for the L ²-norm of the residual terms $\Vert \mathcal L\gamma_3 - \lambda_3 \gamma_3 \Vert_{L^2}$ and $\Vert \widehat {\mathcal L} \hat \gamma_3 - \lambda_3 \hat \gamma_3 \Vert_{L^2}$.

The residual terms grow as $c\to c_*$ in agreement with other figures, and the growth becomes visible for larger values of c if the truncation number N is increased.

We conclude that the spectral stability problem for the travelling wave with the peaked profile $\eta_* \in C^0_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$ cannot be analysed by working with the spectral stability problem for the family of travelling waves with the smooth profiles $\eta \in C_{\rm per}^{\infty}(\mathbb{T})$ in the limit $c \to c_*$. The lack of convergence is fundamental, both at the levels of functional analysis and numerical approximations. On the analysis side, the linearised system (4.1) for the smooth waves is invalid for the peaked waves and needs to be replaced by the linearised system (4.7). On the numerics side, the eigenvalues of the Hessian operator $\mathcal{L}$ do not converge well as $c \to c_*$ to the limiting eigenvalues at $c = c_*$ for the fixed value of N. Due to the lack of convergence, we have a striking discrepancy between the spectral stability of the smooth waves [Reference Locke and Pelinovsky36] and the spectral instability (theorem 3) of the peaked waves.