Waves of maximal height for a class of nonlocal equations with inhomogeneous symbols

Our object of study is the following nonlocal equation

where $L$ denotes the Fourier multiplier operator with symbol $m(\xi)=(1+\xi^{2})^{-\alpha/2}$, $\alpha>1$. Equation (1.1) is known as the fractional Korteweg–de Vries equation. We are looking for a $2\pi$-periodic traveling-wave solution $u(t,x)=\phi(x-\mu t)$, where $\mu>0$ denotes the speed of the right-propagating wave. In this moving frame, equation (1.1) takes the form

where the equation has been integrated once, and the constant of integration set to zero. In fact, there is no loss of generality in doing so due to the Galilean change of variables:

for any $\gamma\in\mathbb{R}$, which maps solutions of $-\mu\phi+L\phi+\frac{1}{2}\phi^{2}=B$ to solutions of a new equation of the same form. In this paper, the above multiplier $m(\xi)$ has order of $-\alpha$, which is less than $-1$. This type of dispersive terms tends to behave smoothly at far field and worse at the origin. Solutions to the type of equation (1.1) with negative orders feature wave breaking and peaked periodic waves; see discussions in [13, 15]. It should be mentioned that for a linear operator of order strictly smaller than $-1$, the regularity of the highest, periodic, traveling-wave solution does not depend on the order operator; see Theorem 4.5. Instead, the dispersive term may affect the angle at the wave crest.

Investigation on highest waves for various types of water-waves equations has a long history. One of the earliest works to study peaked waves was Stokes, who [19] conjectured that for a fixed wavelength and the gravitational acceleration, the Euler equation has a highest, periodic traveling-wave with a sharp crest of angle $2\pi/3$. However, attempts to prove the Stokes conjecture had failed until the work by Amick, Fraekel, and Toland [1]. Whitham [21] also made an assumption that the solution $\phi$ to the Whitham equation reached its critical height when $\phi=\mu$ (the notation has been changed to match equation (1.2)). Recently, Ehrnström and Wahlén [12] confirmed the existence and regularity of a highest and periodic traveling-wave solution for the Whitham equation, and hence successfully proved Whitham conjecture. This solution features a sharp crest and is exactly $C^{1/2}$-Hölder continuous at the peak and smooth between its crest and trough on the half-period. After this breakthrough work, there have been an increasing number of studies on nonlocal equations using their approach. One of the earliest papers in this direction is to justify the convexity of the wave profile between consecutive crests using a computer-assisted proof [14]. A similar result as in [12] but for solitary waves was also obtained by Truong, Wahlén, and Wheeler [20] using a center manifold theorem approach. Moreover, Ehrnström, Johnson, and Claasen [9] proved the existence and regularity of a highest wave for the bidirectional Whitham equation with cubic nonlinearity and a Fourier multiplier symbol $m(\xi)=\tanh(\xi)/\xi$. They showed that the highest cusped wave has a singularity at its crest of the form $|x\log(|x|)|$. Utilizing the same approach as in [12], Arnesen [3] considered the Degasperis–Procesi equation. This equation has a nonlocal form with a quadratic nonlinearity and a Fourier multiplier symbol $m(\xi)=(1+\xi^{2})^{-1}$. Using the method in [12], Arnesen constructed a family of highest, peaked, periodic traveling waves and showed that any highest, even, periodic wave of the Degasperis–Procesi equation is exactly Lipschitz at its crest. The author, therefore, excluded the existence of even, periodic, cusped traveling-wave solutions.

Since the unidirectional Whitham equation is nonlocal, the equation (1.1) is obtained, up to a scaling factor, by replacing the Whitham dispersion relation by a more regular dispersive term. Building on the integral kernel $K$, we prove the existence and regularity properties of a highest, $2\pi$-periodic traveling-wave solution of equation (1.2). Our most direct influence are the works by Ehrnström and Wahlén [12] and by Bruell and Dhara [5]. In the former work, the investigation started with the Whitham equation

where $L$ is the Fourier multiplier operator given by the symbol

This multiplier has order greater than $-1$, is inhomogeneous and completely monotone. The main result is the existence of a highest, cusped, $P$-periodic traveling-wave solution, which is even, strictly decreasing, smooth on each half-period, and belongs to the Hölder space $C^{\frac{1}{2}}(\mathbb{R})$. The proof is based on the regularity and monotonicity properties of the convolution kernel induced by $m$. On the other hand, Bruell and Dhara [5] considered the fractional KdV equation as in equation (1.1), where $L$ is the Fourier multiplier operator given by the homogeneous symbol

This multiplier has order less than $-1$, is homogeneous, and completely monotone. The authors showed the existence of a highest, $2\pi$-periodic traveling waves solution, which is Lipschitz continuous. Their method is based on the nonlocal approach in the paper [12] but adapt to treat homogeneous symbols. The convolution kernel associated with this homogeneous symbol cannot be identified with a positive, decaying function on the real line. To compensate for this lack of positivity, solutions are assumed to have the zero mean, and hence the constant of integration $B$ in equation (1.2) now becomes $\widehat{\phi}(0)$.

In the present paper, we will combine the approach in [12] to treat inhomogeneous symbols, and the derivation for an estimate of a priori traveling-wave solution to nonlocal equations as in [5]. We recall that the symbol $m$ for the fractional KdV equation (1.1) is Bessel potential function, which is inhomogeneous and of order strictly smaller than $-1$. We also do not make the assumption of zero mean solution as in [5]. Since the kernel operator is positive, we choose the constant of integration in equation (1.2) to be $B=0$. Moreover, even though our kernel $K$ is even and smooth for all $x\in\mathbb{R}$ with rapid decay derivatives, we do not use a closed formula for $K$ in the physical space, as well as for its periodization as in [5, 12]. It is also worth to mention that the kernel $K$ does not exhibit the completely monotone feature, which plays a main role for the analysis in [5, 12]. Instead, our investigation only relies on its qualitative behaviors such as positivity of $K$ and the signs of its derivatives; see Section 2 and 3. The properties of the convolution kernel are proved in the ongoing work by Bruell, Ehrnström, Johnson, and Wahlén [6].

We now briefly discuss the strategies we used to derive the results. We will be using a global bifurcation theory as in the paper [7] to build a global, locally analytic curve of periodic smooth waves where the solution maximum approaches $\mu$. We further show that the only possibility along the main bifurcation curve is that $\max\phi$ must approach $\mu$. Moreover, we investigate the solution $\phi$ satisfying $\max\phi=\mu$. Using the properties of the convolution operator $K$, we obtain Lipschitz continuous at its crest.

Let us formulate our main existence theorem which constructs solutions bifurcating from nontrivial, even, smooth, and $2\pi$-periodic traveling-wave solutions of equation (1.1). At the end of the bifurcation curve, the limiting solution reaches its highest and is precisely Lipschitz continuous.

The outline of our study is as follows. In Section 2 we introduce notations and functional-analytic settings that we encounter in the analysis. We also inspect the convolution kernel $K$ corresponding to the symbol $m(\xi)$ and show that the action of the operator $K$ on periodic functions on the real line is in fact similar to the periodized kernel $K_{P}$ acts on these functions in one period. In Section 3, we discuss some interesting analytic features of the periodized kernel $K_{P}$. In fact, both $K$ and $K_{P}$ share many properties.

Section 4 is the heart of this paper. We provide some general estimates of an even solution to equation (1.2). In particular, using the regularity and monotonicity properties of the convolution kernel $K_{P}$, we prove that if a traveling-wave solution, which is even, periodic, and monotone on half the period, attains its maximum equal to the wave speed $\mu$, then it is Lipschitz continuous. Finally, Section 5 proved our main theorem, Theorem 1.1. It consists of the bifurcation analysis from [5, 11, 10], in which we rule out certain possibilities for the bifurcation curve. We, therefore, construct a global curve of sinusoidal, periodic smooth waves along which $\max\phi\to\mu$, and an analysis of solutions at their maximum $\max\phi=\mu$.

Setting $\mathbb{T}:=[-\pi,\pi]$, we denote by $\mathcal{D}(\mathbb{T})$ the space of test functions on $\mathbb{T}$, whose dual space, the space of distributions on $\mathbb{T}$, is $\mathcal{D}^{\prime}(\mathbb{T})$. If $\mathcal{S}(\mathbb{Z})$ is the space of rapidly decaying functions from $\mathbb{Z}$ to $\mathbb{C}$ and $\mathcal{S}^{\prime}(\mathbb{Z})$ denotes its dual space, let $\mathcal{F}:\mathcal{D}^{\prime}(\mathbb{T})\to\mathcal{S}^{\prime}(\mathbb{Z})$ be the Fourier transformation on the torus defined by the duality on $\mathcal{D}(\mathbb{T})$ via

To simplify our notations, throughout this paper, if $f$ and $g$ are elements in an ordered Banach space, we write $f\lesssim g$ $(f\gtrsim g)$ if there exists a constant $c>0$ such that $f\leq cg$ $(f\geq cg)$. Moreover, the notation $f\eqsim g$ is used whenever $f\lesssim g$ and $f\gtrsim g$. We also denote $\mathbb{R}_{+}:=[0,\infty]$ and by $\mathbb{N}_{0}$ the set of natural numbers including zero. The space $\mathcal{L}(X;Y)$ denotes the set of all bounded linear operators from $X$ to $Y$.

Next, let us investigate how the convolution $K*$ acts on periodic functions. Suppose that $f\in L^{\infty}(\mathbb{R})$ is periodic and even. Since $K$ is in $L^{1}(\mathbb{R})$, we can write:

It is clear from the definition of $K_{P}(x)$ that it is $2\pi$-periodic, even, and continuous on $\mathbb{T}$. Moreover, Minkowski’s inequality shows that $K_{P}(x)$ belongs to $L^{p}(-\pi,\pi)$, for $1\leq p<2$. Therefore, Carleson–Hunt Theorem [17] implies that $K_{P}(x)$ can be approximated pointwise by its Fourier series:

and the Fourier coefficients of $K_{P}$ are given by

Thus, the periodic problem is given by the same multiplier as the problem on the line, so we have the representation:

The periodized operator $K_{P}$ is introduced to aid the analysis of periodic solutions satisfying certain sign conditions in a half-period. From the above analysis, $K_{P}$ can be expressed as the Fourier series

and the relationship between $K$ and $K_{P}$ is

Note that this sum is absolutely convergent since $K$ has rapid decay.

This section is devoted to summarize some properties of the kernel $K_{P}$. We recall the Fourier representation of $K_{P}$ (2.2) and the relationship (2.3) between $K$ and $K_{P}$. Recalling that $K=\mathcal{F}^{-1}(m)$ and the Fourier multiplier is given by the Bessel potential $m(\xi)=(1+\xi^{2})^{-\alpha/2}$ for $\alpha>1$, we record some properties of the periodized operator $K_{P}$.

We know from the above theorem that $K_{P}$ is even and decreasing on $(0,\pi)$. The next lemma provides monotonicity property for the operator $L=K*$ based on $K_{P}$.

In this section, we provide some basic properties of solutions to the main equation (1.2) such as its a priori estimate, monotonicity, and regularity. When we say $\phi$ is a solution, $\phi$ must be real-valued, bounded, and satisfy equation (1.2) pointwise. Fixing $\alpha>1$, to aid our analysis, we re-write equation (1.2) as

Our first result is a rough bound for a solution according to the wave speed $\mu$.

We come to the first set of results in this section, which we need to rule out the closed loop possibility of the solution in the global bifurcation analysis. It shows that any nontrivial, even, periodic $C^{1}$ solution, which is nonincreasing on $(0,\pi)$, must be strictly decreasing and less than $\mu$ on $(0,\pi)$. Thus, we conclude that $\max\phi=\phi(0)=\mu$.

In both papers [5, 12], a solution to the nonlocal equation is smooth when it is below its maximum value. In the next theorem, we show that the same conclusion is reached when the solution approaches the maximum value $\phi(0)=\mu$ from below. Here we only rely on the boundedness of the solution.

As a motivation, we shall see that when the order of the dispersive term is strictly smaller than $-1$, any decrease of the order does not affect the regularity of the wave solution.

In the next theorem, we investigate the regularity of a solution when it touches $\mu$ from below. It was shown that the solution has $\frac{1}{2}$-Hölder regularity at $x=0$ for an inhomogeneous multiplier of order greater than $-1$ [12], and Lipschitz continuous at $x=0$ for a homogeneous multiplier of order less than $-1$ [5]. For our case with inhomogeneous symbol of order less than $-1$, we, indeed, obtain a Lipschitz regularity at $x=0$.