A Direct Construction of Solitary Waves for a Fractional Korteweg–de Vries Equation With an Inhomogeneous Symbol

Abstract

We construct solitary waves for the fractional Korteweg–de Vries (fKdV) type equation: $\begin{aligned} u_{t} + (Λ^{- s} u + u^{2})_{x} = 0, \end{aligned}$ where $Λ^{- s}$ denotes the Bessel potential operator $(1 + | D |^{2})^{- s / 2}$ for $s \in (0, \infty)$ . The approach is to parameterize the known periodic solution curves through the relative wave height. Using a priori estimates, we show that the periodic waves locally uniformly converge to waves with negative tails, which are transformed to the desired branch of solutions. The obtained branch reaches a highest wave, the behavior of which varies with $s$ . The work is a generalization of recent work by Ehrnström–Nik–Walker, and is, as far as we know, the first simultaneous construction of small, intermediate, and highest solitary waves for the complete family of (inhomogeneous) fKdV equations with negative-order dispersive operators. The obtained waves display exponential decay rate as $| x | \to \infty$ .

Keywords

inhomogeneous dispersion negative-order dispersive equations solitary waves

1. Introduction

This work focuses on the direct construction of a full family of solitary waves for a fractional Korteweg–de Vries (fKdV) equation of the form:

\begin{aligned} u_{t} + (Λ^{- s} u + u^{2})_{x} = 0. \end{aligned}

(1.1)

Here,

u (t, x)

is a real-valued function representing the deflection of a water fluid surface from its rest position at time

t \geq 0

and position

x \in R

. The symbol

Λ^{- s}

is a Fourier multiplier operator of order

- s

defined through

Λ^{- s} : f \mapsto F^{- 1} [(1 + ξ^{2})^{- s / 2} \hat{f} (ξ)],

for all positive values of

s

. Sometimes referred to as a Bessel potential operator, this operator can also be expressed via convolution as

Λ^{- s} f = K_{s} * f

, where the convolution kernel

K_{s}

is given by

\begin{aligned} K_{s} (x) = F^{- 1} ((1 + ξ^{2})^{- s / 2}) (x) = \frac{1}{2 π} \int_{R} (1 + ξ^{2})^{- s / 2} e^{i x ξ} d ξ . \end{aligned}

(1.2)

The last equality defines the Fourier transform. We shall consider traveling-wave solutions

u (x, t) = φ (x - μ t)

, where

μ

is the (rightward) wave speed. In this setting, equation (1.1) reads as

- μ φ^{'} + (φ^{2})^{'} + (Λ^{- s} φ)^{'} = 0.

By integrating the above equation, we obtain the steady equation

\begin{aligned} - μ φ + φ^{2} + Λ^{- s} φ = 0, \end{aligned}

(1.3)

where the right-hand side can be set to zero by the Galilean transformation,

φ \mapsto φ + ρ, μ \mapsto μ + ρ,

with

ρ

chosen such that

ρ (1 - μ - (ρ / 2))

cancels the integration constant.

Equation (1.1) is an inhomogeneous fKdV equation, and therefore a Whitham-type equation (Whitham, 1967). It originates from the KdV equation by replacing the polynomial dispersion relation $m_{KdV} (ξ) = 1 - (1 / 6) ξ^{2}$ expressed by the Fourier multiplier $1 + (1 / 6) \partial_{x}^{2}$ with more exact approximations of the Euler dispersion relation $m_{Euler} (ξ) = \sqrt{\tanh ξ / ξ}$ , or, as in the case of (1.1), with full scales of equations describing varying strength and behavior of the dispersion with respect to nonlinear effects. In the case of the Whitham equation, the symbol is exactly $m_{Euler}$ , describing the right-going branch of the dispersion relation for gravity water waves on finite depth. The modeling of bidirectional motion can be captured in Boussinesq equations (Broer, 1964), including full-dispersion Whitham–Boussinesq systems with symbol $m_{Euler}^{2} (ξ) = \tanh ξ / ξ$ , where the linear wave speed is the positive or negative root of this expression (Lannes, 2013). Many such models quantitatively well describe the propagation of small-amplitude waves in shallow water over moderate time spans (Emerald, 2021).

A direct construction of a full set of solitary waves for the Whitham equation with symbol $m_{Euler}$ was recently given in Ehrnström, Nik, et al. (2023). In contrast to the KdV equation, the Whitham equation features dispersion that allows for both solitary, breaking, and highest waves. Ehrnström and Wahlén (2019) were the first to prove that there exist periodic traveling-wave solutions to the Whitham equation having exactly $1 / 2$ -Hölder regularity. In the more recent paper (Truong et al., 2022), the authors construct global curves of solitary waves up to the highest wave for the same problem using a new nonlocal variant of the center manifold theorem and global bifurcation. Whereas small solitary solutions for (1.1) are known from Ehrnström et al. (2012), Hildrum (2020), and Stefanov and Wright (2020), global curves of solitary solutions up to the highest wave have not been proven. We construct them using the direct method from Ehrnström, Nik, et al. (2023).

In the periodic setting of (1.1), more is known. Ørke (2024) established the existence and regularity of the highest periodic waves for equation (1.1) when $s \in (0, 1)$ , and proved that the non-constant solutions (say $φ$ ) to the steady equation are everywhere smooth except where the wave height equals half the wave speed, $φ (0) = μ / 2$ . As in the Whitham case, the highest periodic waves show $s$ -Hölder regularity at the crest attained in the form of a cusp. Le (2022) studied a similar model for even weaker dispersion $s > 1$ and proved that the highest waves in that case are Lipschitz continuous. Arnesen (2019) considered the Degasperis–Procesi equation with symbol $(1 + ξ^{2})^{- 1}$ and used the nonlocal method to prove the existence of highest traveling-wave solutions. As we shall prove, when $s = 1$ , the highest waves are not Lipschitz but exhibit a cusp with logarithmic Lipschitz regularity near the origin. A similar behavior was observed by Ehrnström et al. (2019) in the bidirectional Whitham equation, which has symbol $m_{Euler}^{2}$ , which corresponds to $s = 1$ . Very recently, Ehrnström, Mæhlen, et al. (2023) completed this investigation by providing exact leading-order asymptotics for the highest waves to this and more general nonlocal equations. In summary, the regularity of the highest periodic wave solutions varies on $s$ for values $s < 1$ ; for $s = 1$ it is log-Lipschitz, while for $s > 1$ it is always exactly Lipschitz. However, $s$ affects the angle, or “cusple,” at the wave crest.

Other authors have investigated various generalizations of the KdV equation with homogeneous symbols. The homogeneous equation corresponding to (1.1) is

\begin{aligned} u_{t} + | D |^{- s} u_{x} + u u_{x} = 0, \end{aligned}

(1.4)

with

| D |^{s}

given by

F (| D |^{s} f) (ξ) = | ξ |^{s} \hat{f} (ξ)

. Bruell and Dhara (2021) showed the existence and regularity of periodic traveling-wave solutions of (1.4) when

s > 1.

Dahne and Gómez-Serrano (2023) studied the same for

s = 1

with an impressive combination of analysis and interval arithmetics. Later, Dahne (2024) was able to show the regularity and existence of the highest traveling-wave solutions for the whole family

s \in (0, 1)

, with exact leading asymptotics at the origin [matching that of Ehrnström, Mæhlen, et al. (2023) in comparable cases]. Hildrum and Xue (2023) earlier studied (1.4) for the same interval of

s

, but for more general nonlinearities, in some cases, finding waves with simultaneous singularities at both the crests and troughs.

The purpose of this study is to establish the existence of a full family of solitary waves for the fKdV equation (1.1) with an inhomogeneous symbol $Λ^{- s}$ , for the full range of values $s \in R^{+}$ . This contrasts with the homogeneous case (1.4), which does not allow for solitary waves in $H^{1}$ in the regime we are considering here (Linares et al., 2014). For $Λ^{s}$ with positive $s$ , solitary waves, when they exist, can be constructed variationally with arbitrary height (see Klein et al., 2018; Mæhlen, 2020; Marstrander, 2024). In this paper, we follow the same method as in Ehrnström, Nik, et al. (2023), but have to take care of the separate cases $s \in (0, 1)$ , $s = 1$ and $s > 1$ , where the latter case is entirely new. The approach is built upon using a priori bounds obtained from the periodic theory, and in some cases, we perform this theory for the first time.

The organization of the rest of the paper is as follows: In Section 2, we study the convolution kernel $K_{s}$ , and introduce periodized kernels $K_{s, P}$ . In Section 3, we discuss the properties of the solution of (1.3), including its regularity at the highest point. We construct a sequence of periodic waves in Section 4 showing the convergence to the traveling waves, which will eventually be transformed into solitary waves. In the last subsection, we discuss the decay rate of the obtained solitary waves. Here, we mention the work by Arnesen (2022), which more generally considers the decay and symmetry of nonlocal dispersive wave equations.

2. Bessel Potential and Convolution Kernel

In this section, we discuss some properties, estimates, and findings related to the Bessel potential operator $Λ^{- s}$ and the associated kernel $K_{s}$ . The kernel $K_{s}$ , defined by (1.2), is a widely recognized inverse Fourier transform of $(1 + ξ^{2})^{- s / 2}$ and is given by

K_{s} (x) = \frac{1}{2^{(s - 1) / 2} \sqrt{π} Γ (s / 2)} G_{(1 - s) / 2} (| x |) | x |^{(s - 1) / 2}, for s > 0,

where

G_{(1 - s) / 2}

denotes the modified Bessel function of the second kind (Aronszajn & Smith, 1961, Chapter 2, Section 4), which provides the rationale for referring to

Λ^{- s}

as a Bessel potential operator. This operator serves as a generalization of the Riesz potential function, which has the limitation that “

s

” must be less than the dimension

n

of the Euclidean space. In contrast,

Λ^{- s}

operates effectively for all positive values of

s

. The majority of the properties concerning the kernel

K_{s}

and the operator

Λ^{- s}

discussed here can be found in the works of Aronszajn and Smith (1961, Chapter 2) and Grafakos (2009, Section 1.2.2). The first part of this section focuses on the convolution kernel

K_{s}

, which will be useful in proving the results for the operator

Λ^{- s}

. However, we shall refrain from presenting most of the proofs, as similar results have been discussed in Le (2022) and Ørke (2024).

We say $X ≲ Y$ whenever $X \leq c Y$ for some positive constant $c$ and $X ≃ Y$ when we have $Y ≲ X ≲ Y$ . We use the symbols $≲_{s}$ and $≃_{s}$ when the constants depend on a variable $s$ . $O (Y)$ represents any quantity $X$ for which $| X | ≲ Y$ , while $o (Y)$ indicates $X / Y \to 0$ as $Y \to 0$ . The general definitions of the sets and spaces mentioned in this paper are taken from the book by Triebel (1992, Chapter 1).

Lemma 2.1 Grafakos, 2009

The kernel $K_{s}$ can be written in the following integral expression:

K_{s} (x) = \frac{1}{\sqrt{4 π} Γ \frac{s}{2}} \int_{0}^{\infty} e^{- t - \frac{x^{2}}{4 t}} t^{\frac{s - 3}{2}} d t,

provided

s > 0

K_{s}

is even, positive, continuous, and smooth on

R ∖ {0}

K_{s}

is decreasing for

x > 0

and integrable such that

‖ K_{s} ‖_{L^{1} (R)} = 1

. Moreover, it satisfies the following asymptotic estimates:

(i)
for $| x | \geq 1$ ,
$K_{s} (x) ≲_{s} e^{- | x |},$

(ii)
for $| x | \leq 1$ ,
$\begin{aligned} K_{s} (x) ≃_{s} {\begin{array}{lr} | x |^{s - 1} - C_{s} + O (| x |^{s + 1}), & 0 < s < 1, \\ \log \frac{1}{| x |} + c_{s} - O (| x |^{2 - ϵ}), & s = 1, \\ 1 - o (| x |), & s > 1, \end{array} \end{aligned}$
(2.1)
where $ϵ$ is a small positive real number. $C_{s}$ and $c_{s}$ are some positive constants depending on $s$ .
(iii)
When $| x | \geq 1$ , we can define $K_{s}^{'}$ as the first derivative of $K_{s}$ , and
$K_{s}^{'} (x) ≲_{s} | x | e^{- | x |} .$

The following lemma establishes additional properties of $K_{s}$ . Since $K_{s}$ is completely monotone for $s \in (0, 1)$ , it is strictly decreasing and strictly convex on $(0, \infty)$ . In contrast, the behavior of $K_{s}$ for $1 < s < 2, s = 2$ , and $s > 2$ reflects the characteristics of the solution $φ$ in the subsequent analysis.
Lemma 2.2
(i)
The function $K_{s} (x)$ is completely monotone on $(0, \infty)$ for $0 < s \leq 2$ , that is,
${(- \frac{d}{d x})}^{j} K_{s} (x) > 0, for j = 0, 1, 2, \dots .$

(ii)
$K_{s}$ belongs to the Sobolev space $W^{α, 1}$ for all $α < s$ . In particular, if $s > 1$ , $K_{s}$ is in the space $W^{1, 1}$ , that is, the first derivative (in the distributional sense) of absolutely integrable functions is also absolutely integrable.
(iii)
If $1 < s < 2$ , $K_{s}$ is $β$ -Hölder continuous with $β \in (0, s - 1)$ . When $s = 2$ , $K_{s}$ is Lipschitz continuous and for $s > 2$ , $K_{s}$ is continuously differentiable.

The statement (i) is established for $s \in (0, 1)$ in Ørke (2024), but, based on Lemma 2.12 in Ehrnström et al. (2019), the result extends to all $s$ up to 2. Statement (ii) can be derived using the definition (1.2) of $K_{s}$ , while (iii) follows directly from Sobolev embeddings.

Now, we define periodized kernel $K_{s, P}$ as
$\begin{aligned} K_{s, P} (x) = \sum_{n \in Z} K_{s} (x + n P), P \in (0, \infty), \end{aligned}$
(2.2)
assuming that $K_{s, P}$ approaches $K_{s}$ as $P \to \infty$ . Since $K_{s}$ is integrable with $‖ K_{s} ‖_{L^{1} (R)} = 1$ and is monotonically decreasing in $(0, \infty)$ , (2.2) is well defined and absolutely convergent. We can also define $K_{s, P}$ in terms of Fourier series
$K_{s, P} (x) = \sum_{k \in Z} C_{k} \exp (\frac{2 π i k x}{P}),$
where
$\begin{aligned} C_{k} = \frac{1}{P} \int_{0}^{P} K_{s, P} (x) \exp (\frac{- 2 π i k x}{P}) d x . \end{aligned}$
$C_{k}$ can be calculated by substituting $K_{s, P}$ from (2.2), and computing the integral using a change of variable. Then, we can explicitly rewrite $K_{s, P}$ as
$\begin{aligned} K_{s, P} (x) = \frac{1}{P} \sum_{k \in Z} {(1 + {(\frac{2 π k}{P})}^{2})}^{- \frac{s}{2}} \exp (\frac{2 π i k x}{P}), \end{aligned}$
(2.3)
which converges absolutely for $s > 1$ .

The definition of $K_{s, P}$ allows it to inherit many properties from the convolution kernel $K_{s}$ , but restricted to a specific domain. The following lemma lists these properties:
Lemma 2.3
$K_{s, P}$ is even, positive, periodic with period $P$ , and strictly increasing on $(- P / 2, 0)$ . For $0 < s \leq 2$ , $K_{s, P}$ is completely monotone on $(0, P / 2)$ and convex on $(0, P)$ . For $s > 1$ , $K_{s, P}$ is in $W^{1, 1}$ , in particular, $K_{s, P}^{'}$ is integrable.
Remark 2.4
The behavior of $K_{s, P}$ near the origin is similar to that of $K_{s}$ (Lemma 2.1(ii)). If we split $K_{s, P}$ in its regular and singular parts as $K_{s, P} = R_{s, P} + S_{s, P}$ , where $R_{s, P}$ is a real analytic function in $(- P / 2, P / 2)$ and $S_{s, P}$ has the singularity at the origin, then for $s > 0$ and $| x | \leq 1$ , we have
$\begin{aligned} S_{s, P} (x) ≃_{s} {\begin{array}{lr} | x |^{s - 1}, & s \neq 2 k - 1, \\ | x |^{s - 1} \log \frac{1}{| x |}, & s = 2 k - 1, \end{array} k \in Z^{+} . \end{aligned}$

The following results on $Λ^{- s}$ are direct consequences of the fact that the kernels $K_{s}$ and $K_{s, P}$ are even and positive on the real line, and therefore on the interval $(- P / 2, P / 2)$ for $P > 0$ .
Proposition 2.5
(i)
The operator $Λ^{- s}$ is parity preserving and strictly monotone on $C (R)$ .
(ii)
Let $f$ be a continuous, odd, $P$ -periodic function that is non-negative and nonzero on $(- P / 2, 0)$ . Then $Λ^{- s} f$ is positive on $(- P / 2, 0)$ .

The operator $Λ^{- s}$ is a classical symbol of order $- s$ , that is, it satisfies the following inequality:
$\begin{aligned} | D_{ξ}^{k} (1 + ξ^{2})^{- \frac{s}{2}} | ≲_{s, k} (1 + | ξ |)^{- s - k}, k \in Z^{+} . \end{aligned}$
(2.4)

We shall discuss the coefficients in (2.4) later in Subsection 4.1.

Let $B_{p, r}^{m} (R)$ represents a class of Besov spaces with $m \in R$ , $1 \leq p, r \leq \infty$ . According to Bahouri et al. (2011, Section 2.5), $L^{\infty} (R)$ is embedded in $B_{\infty, \infty}^{0} (R)$ . Moreover, $B_{\infty, \infty}^{s} (R)$ is equivalent to the Zygmund space $C^{s} (R)$ of bounded functions for any positive real number $s$ . $C^{s} (R)$ is equivalent to the Hölder spaces $C^{[s], s - [s]}$ for $s \in R^{+} ∖ N$ and are thus often referred to as the Hölder–Zygmund spaces. Additionally, $C^{s} (R)$ are continuously embedded in the Hölder spaces $C^{α}$ for all $α < s$ . Here, $[s]$ represents the integer part of $s$ such that $[s] < s$ . We have the following nice result for $Λ^{- s}$ on the space $B_{p, r}^{m} (R)$ .
Proposition 2.6 Bahouri et al., 2011, Proposition 2.78

The operator $Λ^{- s}$ is continuous from $B_{p, r}^{m} (R)$ to $B_{p, r}^{m + s} (R)$ for $s > 0$ , $m \in R$ , and $1 \leq p, r \leq \infty$ .

A particular case of the above proposition, which we use later, can be stated as: “The operator $Λ^{- s}$ is bounded from $L^{\infty} (R)$ to $C^{α} (R)$ for some $α \in (0, 1)$ less than $s > 0$ .”

3. Properties of Solutions

This section contains a set of a priori properties associated with the traveling wave solution (say $φ$ ) to (1.3). These properties will be used in the subsequent section, particularly for constructing solitary waves as the period $P$ tends toward infinity. Our approach closely follows that of Ehrnström, Nik, et al. (2023). Most of the proofs align with those in Ehrnström, Nik, et al. (2023), given the shared properties between $K$ in their work and $K_{s}$ in our study. For conciseness, several proofs are omitted and will only be provided when they differ from those in Ehrnström, Nik, et al. (2023).

Proposition 3.1 Ehrnström, Nik, et al., 2023, Proposition 3.1

Let $φ$ be a bounded solution of (1.3) and $μ \geq 0$ be the wave speed such that $lim_{x \to \infty} φ (x) = γ$ , then $γ$ will also be the solution of (1.3) with the wave speed $μ$ , and attains the value either $μ - 1$ or $0$ .

Proposition 3.2
The solution $φ \in L^{1} (- P / 2, P / 2)$ of the steady equation (1.3) also belongs to $L^{2} (- P / 2, P / 2)$ for any $μ \in R$ , $P > 0$ . In fact,
$‖ φ ‖_{L^{2} (- \frac{P}{2}, \frac{P}{2})}^{2} = (μ - 1) \int_{- \frac{P}{2}}^{\frac{P}{2}} φ (x) d x .$
Moreover, $φ$ has a negative mean when $μ < 1$ and a positive mean when $μ > 1$ .

The above result can be proved by integrating the steady equation (1.3) from $- P / 2$ to $P / 2$ .

The next proposition predicts the range of the wave speed $μ$ for the traveling-wave solution to be non-negative and non-decreasing. The proof is similar to the one given in Ehrnström, Nik, et al. (2023, Proposition 3.2); however, we present a simpler version here.
Proposition 3.3
Let $φ$ and $μ$ be the same as in Proposition 3.1. Additionally, if $φ$ is even, non-negative, non-constant, and non-decreasing on $(- \infty, 0)$ , then $μ \geq 1$ .
Proof.
We prove this proposition by contradiction. Assume that $μ < 1$ . Proposition 3.1 promptly gives us $γ = lim_{x \to - \infty} φ (x) = 0$ , as the assumption $μ - 1 < 0$ shows that the other possible value of $γ = μ - 1$ cannot hold due to the non-negativity of $φ$ . Now, from (1.3), we have
$(μ - φ) φ = Λ^{- s} φ .$
For $x < 0$ , the convolution formula gives
$\begin{aligned} (μ - φ (x)) φ (x) = & \int_{R} K_{s} (x - y) φ (y) d y \\ = & \int_{{| y | < | x |}} K_{s} (x - y) φ (y) d y + \int_{{| y | \geq | x |}} K_{s} (x - y) φ (y) d y . \end{aligned}$
(3.1)
For the first integral of (3.1), since $φ$ is non-decreasing on $(- \infty, 0)$ , non-negative, and even; and $K_{s}$ is positive on $R ∖ {0}$ , we get
$\begin{aligned} \int_{x}^{- x} K_{s} (x - y) φ (y) d y \geq φ (x) \int_{0}^{- 2 x} K_{s} (y) d y . \end{aligned}$
(3.2)
For the second integral, choose a variable $σ > 0$ such that
$\begin{aligned} \frac{1}{2} > \int_{0}^{σ} K_{s} (x) d x > μ - \frac{1}{2} + ε, ε > 0. \end{aligned}$
(3.3)
Again, since $φ$ is non-negative and $K_{s}$ is even and positive, we get
$\begin{aligned} \int_{{| y | \geq | x |}} K_{s} (x - y) φ (y) d y \geq \int_{x}^{x + σ} K_{s} (x - y) φ (y) d y . \end{aligned}$
(3.4)
Thus, substituting (3.2) and (3.4) into (3.1), we have
$(μ - φ (x)) φ (x) \geq φ (x) \int_{0}^{- 2 x} K_{s} (y) d y + \int_{x}^{x + σ} K_{s} (x - y) φ (y) d y,$
which we can rewrite using the monotonicity of $φ$ as
$\begin{aligned} (μ - φ (x)) φ (x) & \geq φ (x) \int_{0}^{- 2 x} K_{s} (y) d y + φ (x) \int_{0}^{σ} K_{s} (z) d z, \\ ⟹ μ - φ (x) & \geq \int_{0}^{- 2 x} K_{s} (y) d y + \int_{0}^{σ} K_{s} (z) d z . \end{aligned}$

Now, assume $h (x) := \int_{0}^{- 2 x} K_{s} (y) d y \leq 1 / 2$ as $‖ K_{s} ‖_{L^{1} (R)} = 1$ , then there exists a sequence of negative real numbers ${x_{n}}_{n \in N}$ with $lim_{n \to \infty} x_{n} = - \infty$ such that $h (x_{n}) \to 1 / 2$ and $φ (x_{n}) \to 0$ as $n \to \infty$ , it follows that
$μ - φ (x_{n}) \geq h (x_{n}) + (μ - \frac{1}{2} + ε) ⟹ μ \geq \frac{1}{2} + (μ - \frac{1}{2} + ε),$
which is a contradiction to itself. Hence, $μ \geq 1$ under the assumed conditions.

The following corollary addresses a challenge that arises when examining solutions that may not belong to $L^{1} (R)$ . It provides insight into the solutions of (1.3) under certain conditions and establishes a relationship between the solution $φ$ and the behavior of convolutions involving $φ$ and specific test functions.
Corollary 3.4 Ehrnström, Nik, et al., 2023

Let $φ \in L^{\infty} (R)$ be a solution of (1.3) with a wave speed $μ \geq 0$ , where $φ$ is non-negative and possesses a limit as $x$ tends to positive or negative infinity. Then, for every nonzero smooth and compactly supported test function $ψ$ with $ψ \geq 0$ , the integral

\int_{R} ψ * (φ ((μ - 1) - φ)) d x = 0,

holds. Specifically, this implies that only solutions that vanish everywhere have a unit speed.

Lemma 3.5
Any solution $φ$ to the steady fKdV equation satisfying $φ \leq μ / 2$ is smooth across every open set where $φ$ is less than $μ / 2$ .

A very brief proof of the this lemma is given in Ørke (2024, Lemma 3.5), which is eventually satisfied for all positive $s$ as we need the map $Λ^{- s} : L^{\infty} (R) \to C^{s} (R)$ to be linear and bounded (Proposition 2.6). Detailed proof can be found in Ehrnström and Wahlén (2019, Theorem 5.1).
4. The Construction of Solitary Waves

In this section, we aim to construct a full family of solitary waves with the decay property. For this, we parameterize our solution $φ$ , which also depends on $μ$ , and consider $(φ, μ)$ as a solution pair. Let $λ$ be the relative wave height of a solution given by

\begin{aligned} φ_{λ} (0) = \frac{λ μ}{2}, 0 < λ \leq 1. \end{aligned}

Corresponding to

λ = 1

, the wave attains maximum height. The family of crests is placed continuously between the zero solution and the highest wave. The following theorem shows the existence of the highest periodic waves for each period

P

and relative wave height

λ

, and is the combination of results proved in Ørke (2024), Ehrnström, Mæhlen, et al. (2023), and Le (2022) for the cases

0 < s < 1

s = 1

, and

s > 1

, respectively. The case

s = 1

is similar to the case of the bidirectional Whitham equation studied by Ehrnström et al. (2019), where the kernel

K_{P}

has logarithmic blowup at

x = 0

, and is a particular case of a model thoroughly studied by Ehrnström, Mæhlen, et al. (2023). Section 3 in Ehrnström, Mæhlen, et al. (2023) comes with few assumptions on the kernel and solution of the considered problem, which aligns well with our case. Whereas Section 4 in Ehrnström, Mæhlen, et al. (2023) studies the logarithmic kernel, its bound, and global regularity of highest waves.

Let $φ_{P, λ}$ be a periodic solution of (1.3) with the periodized kernel $K_{s, P}$ .

Theorem 4.1
For each finite $P > 0$ and each $λ \in (0, 1]$ , (i)
the steady equation (1.3) has an even, $P$ -periodic solution $φ_{P, λ}$ with wave speed $μ_{P, λ}$ and relative wave height $λ$ such that $φ_{P, λ}$ is strictly increasing on $(- P / 2, 0)$ .
(ii)
$φ_{P, λ}$ is smooth on $R ∖ P Z$ . For $λ < 1$ , $φ_{P, λ} \in C^{\infty} (R)$ .
(iii)
For the highest periodic wave solution $φ_{P, λ}$ , where $λ = 1$ , the regularity of $φ_{P, λ}$ is as follows: $φ_{P, λ} \in C^{s} (R)$ for $s \in (0, 1)$ ; $φ_{P, λ}$ is log-Lipschitz when $s = 1$ ; and $φ_{P, λ}$ is Lipschitz for $s > 1$ .
(iv)
The solutions have subcritical wave speed $0 < μ_{P, λ} \leq 1$ and are uniformly bounded by
$\begin{aligned} μ_{P, λ} - 1 \leq φ_{P, λ} \leq φ_{P, λ} (0) = \frac{λ μ_{P, λ}}{2} . \end{aligned}$
(4.1)

The existence of such solutions $φ_{P, λ}$ when $s \in (0, 1)$ is given in Section 3.3 of Ørke (2024) and is proved by using bifurcation theory. Section 5 of Le (2022) follows the same approach for $s > 1$ . The case $s = 1$ for the existence of the desired solutions is addressed in Ehrnström et al. (2019, Section 5). The smoothness of $φ_{P, λ}$ away from the origin is ensured by Lemma 3.5. Third part (iii) demonstrates the behavior of solutions near the origin. The highest traveling wave has $s$ -Hölder regularity near origin when $s \in (0, 1)$ as nicely demonstrated in Theorem 3.7 in Ørke (2024). For $s > 1$ , Theorem 4.6 in Le (2022) proves Lipschitz continuity of the solution near the origin for equation (1.3). Ehrnström, Mæhlen, et al. (2023) deal with the kernel having a logarithmic singularity around zero. We discuss this case here to show that our case with $s = 1$ falls in the same settings; however, here we restrict ourselves to the periodic solutions.
Proof.
(iii) (For the case $s = 1$ ) For the steady periodic solution $φ_{P}$ of (1.1), we have
$\begin{aligned} K_{s} * φ_{P} = μ φ_{P} - φ_{P}^{2} . \end{aligned}$
(4.2)
Let $f (t) = μ t - t^{2}$ such that $- (1 / 2) f^{″} (t) = 1$ . The function $f$ attains a maximum at $μ / 2$ , and
$f (\frac{μ}{2}) - f (t) = (- \frac{1}{2} f^{″} (t)) {(\frac{μ}{2} - t)}^{2} = {(\frac{μ}{2} - t)}^{2} .$
Let $ω = (μ / 2) - φ_{P}$ , then $ω > 0$ satisfies the above equation in the following sense:
$\begin{aligned} ω^{2} (x) & = K_{s} * ω (x) - K_{s} * ω (0) \\ = \int_{- \frac{P}{2}}^{\frac{P}{2}} K_{s, P} (x - y) ω (y) d y - \int_{- \frac{P}{2}}^{\frac{P}{2}} K_{s, P} (y) ω (y) d y \\ = \int_{0}^{\frac{P}{2}} [K_{s, P} (x + y) + K_{s, P} (x - y) - 2 K_{s, P} (y)] ω (y) d y \\ = \int_{0}^{\frac{P}{2}} Δ_{x}^{2} K_{s, P} (y) ω (y) d y, \end{aligned}$
(4.3)
where $Δ_{x}^{2} f (t)$ is the usual second-order difference. Since, $K_{s, P} (x)$ comes with logarithmic singularity $\log (1 / | x |)$ for $s = 1$ , we divide (4.3) by $1 / [x^{2} \log (1 / x)^{2}]$ and split the integral for $0 < x < (P / 2) < \infty$ , we have
$\begin{aligned} (g (x))^{2} := {(\frac{ω (x)}{x \log (1 / x)})}^{2} = \frac{1}{(x \log (1 / x))^{2}} (\int_{0}^{x} + \int_{x}^{\frac{P}{2}}) Δ_{x}^{2} K_{s, P} (y) ω (y) d y . \end{aligned}$
(4.4)
Following the same steps as in Ehrnström, Mæhlen, et al. (2023, Lemma 4.1), we get the bounds for $g (x)$ as follows:
$\begin{aligned} o (1) g (x) + \frac{1}{2} max_{t \in (0, r]} g (t) \geq g {(x)}^{2} \geq o (1) g (x) + \frac{1}{2} min_{t \in (0, r]} g (t), \end{aligned}$
(4.5)
provided $R_{s, P}^{″} \in L^{1} (- P / 2, P / 2)$ , where $R_{s, P}$ is the regular part of the kernel $K_{s, P}$ given by Remark 2.4. For $x > 0$ , $R_{s, P}^{″} (x) = K_{s, P}^{″} (x) - S_{s, P}^{″} (x)$ , and
$\begin{aligned} \int_{- \frac{P}{2}}^{\frac{P}{2}} | R_{s, P}^{″} (x) | d x & = \int_{- \frac{P}{2}}^{\frac{P}{2}} | K_{s, P}^{″} (x) - S_{s, P}^{″} (x) | d x \\ \leq \sum_{n \in Z} \int_{- \frac{P}{2 + n P}}^{\frac{P}{2 + n P}} | K_{s}^{″} (x) - S_{s}^{″} (x) | d x \\ = 2 \int_{0}^{\infty} | K_{s}^{″} (x) - S_{s}^{″} (x) | d x, \end{aligned}$
where $S_{s} (x)$ is the singular part in $K_{s}$ and is $(1 / π) \log (1 / | x |)$ near the origin when $s = 1$ . $K_{s} (x) = (1 / π) G_{0} (x)$ , where $G_{0} (x) = \int_{1}^{\infty} (e^{- x t} / \sqrt{t^{2} - 1}) d t$ is the modified Bessel function of second kind for $s = 1$ , and $G_{0}^{″} (x) = \int_{1}^{\infty} e^{- x t} (t^{2} / \sqrt{t^{2} - 1}) d t,$ this gives
$\begin{aligned} \int_{- \frac{P}{2}}^{\frac{P}{2}} | R_{s, P}^{″} (x) | d x & \leq \frac{2}{π} \int_{0}^{\infty} | G_{0}^{″} (x) - \frac{1}{x^{2}} | d x \\ = \frac{2}{π} \int_{0}^{\infty} | \int_{1}^{\infty} e^{- x t} \frac{t^{2}}{\sqrt{t^{2} - 1}} d t - \int_{0}^{\infty} t e^{- x t} d t | d x \\ \leq \frac{2}{π} \int_{0}^{\infty} (\int_{1}^{\infty} e^{- x t} (\frac{t^{2}}{\sqrt{t^{2} - 1}} - t) d t + \int_{0}^{1} t e^{- x t} d t) d x \\ = \frac{2}{π} (\int_{1}^{\infty} (\frac{t}{\sqrt{t^{2} - 1}} - 1) d t + \int_{0}^{1} 1 d t) = \frac{4}{π} . \end{aligned}$
Since $φ_{P}$ is a periodic, even solution of the steady equation (4.2) such that it is decreasing and smooth on an interval $(0, r), r > 0$ , and $φ_{P} (0) = μ / 2$ , then from (4.5)
$\begin{aligned} φ_{P} = \frac{μ}{2} - (\frac{1}{2} + o (1)) x \log (1 / x) as x ↘ 0. \end{aligned}$
(4.6)

As $φ_{P}$ is continuously differentiable and decreasing in $(0, P / 2)$ , following the same idea as above, we have
$\begin{aligned} φ_{P}^{'} = - (\frac{1}{2} + o (1)) \log (1 / x) as x ↘ 0. \end{aligned}$
(4.7)
A detailed proof of (4.7) can be found in Ehrnström, Mæhlen, et al. (2023, Proposition 4.4).

Lemma 4.6 of Ehrnström, Mæhlen, et al. (2023) proves that for an absolutely continuous function $φ$ on an open interval $I$ containing zero, and for a modulus of continuity $w$ such that
$\begin{aligned} ess lim_{x \to 0} sup \frac{| φ^{'} (x) |}{w^{'} (| x |)} < \infty, \end{aligned}$
(4.8)
$φ$ admits $M w$ as a modulus of continuity on the same interval $I$ , that is,
$| φ (x) - φ (y) | \leq M w (| x - y |), for all x, y \in I,$
for a positive constant $M$ , depending only on (4.8) and properties of $w$ . Since $φ_{P}$ is an absolutely continuous, even, and $P$ -periodic solution of (4.2) given by (4.6) and (4.7), which is smooth and decreasing on a small interval $(0, r)$ then by choosing $w (x) = x \log (1 + 1 / x)$ , we have
$\begin{aligned} | φ_{P} (x) - φ_{P} (y) | \leq M | x - y | \log (1 + \frac{1}{| x - y |}), \end{aligned}$
(4.9)
on an open interval $I$ containing zero. Since the solution is uniformly bounded and smooth away from the origin, with estimates only depending on the $L_{\infty}$ -norm of the solution (Lemma 3.5), we get (4.9) as a bound, uniformly in $P$ .

(iv) For the inequality (4.1), we assume that there exist such solutions described by (i). Let $φ_{1}$ be the super-solution of (1.3), that is, $- μ φ_{1} + Λ^{- s} φ_{1} + φ_{1}^{2} \leq 0$ . We have
$\begin{aligned} - μ φ_{1} + φ_{1}^{2} & \leq - Λ^{- s} φ_{1}, \\ ⟹ {(- \frac{μ}{2} + φ_{1})}^{2} \leq \frac{μ^{2}}{4} & - Λ^{- s} φ_{1} \leq \frac{μ^{2}}{4} - inf φ_{1}, \end{aligned}$
as $Λ^{- s}$ is a monotone operator (Proposition 2.5) and $Λ^{- s} c = c$ for positive constant $c$ . In particular,
${(- \frac{μ}{2} + inf φ_{1})}^{2} \leq \frac{μ^{2}}{4} - inf φ_{1} ⟹ inf φ_{1} (inf φ_{1} - (μ - 1)) \leq 0.$
If $inf φ_{1} \geq 0$ , this gives $inf φ_{1} \leq μ - 1$ , and this implies $μ \geq 1.$ Hence, for the case $μ \leq 1$ , $μ - 1 \leq inf φ_{1}$ . Any P-periodic solution $φ_{P}$ of equation (1.3) must satisfy the inequality obtained for the super-solution that is, $μ - 1 \leq inf φ_{P}$ .

From equation (1.3), $Λ^{- s} φ_{P}^{'} = (μ - 2 φ_{P}) φ_{P}^{'}$ , and since $φ_{P}^{'}$ satisfies all the properties of the function described by Proposition 2.5 (ii), hence, $Λ^{- s} φ_{P}^{'} > 0$ implies that $μ - 2 φ_{P} > 0$ and we have, $φ_{P} < μ / 2$ . Let us parameterize the family by assuming $λ$ as a relative wave height of the solution such that $φ_{P, λ} (0) = λ μ_{P, λ} / 2, λ \in (0, 1]$ , and combining the results, we have (4.1).

In the following theorem, we shall establish a limit for the solution as the period $P$ approaches infinity.
Theorem 4.2
Let $(φ_{P, λ}, μ_{P, λ})$ be as in the previous theorem then for all $P \geq 1$ , $λ \in (0, 1]$ , there exists a positive constant $δ$ such that for $| x | \leq δ$ , $φ_{P, λ}$ will satisfy the following inequalities depending upon the value of $s$ ,
$\begin{aligned} \frac{μ_{P, λ}}{2} - φ_{P, λ} (x) \geq δ | x |^{σ}, \end{aligned}$
(4.10)
where $σ = s$ when $s \in (0, 1)$ , and $σ = 1$ for $s > 1$ . For $s = 1$ , we have
$\begin{aligned} \frac{μ_{P, λ}}{2} - φ_{P, λ} (x) \geq δ | x \log | x | | . \end{aligned}$
(4.11)
Proof.
We shall only consider the case $x < 0$ as $φ_{P, λ}$ is even. Now, the periodicity and evenness of $K_{s, P}$ and $φ_{P, λ}$ give
$\begin{aligned} (Λ^{- s} φ_{P, λ}) (x + h) - (Λ^{- s} φ_{P, λ}) (x - h) \\ = \int_{- \frac{P}{2}}^{0} ((K_{s, P} (y - x) - K_{s, P} (y + x)) (φ_{P, λ} (y + h) - φ_{P, λ} (y - h))) d y . \end{aligned}$
(4.12)
By Lemmas 2.1 and 2.3, we have
$K_{s, P} (y - x) - K_{s, P} (y + x) > 0 for - \frac{P}{2} < x, y < 0.$
By Theorem 4.1 (i),
$φ_{P, λ} (y + h) - φ_{P, λ} (y - h) \geq 0 for - \frac{P}{2} < y < 0, 0 < h < \frac{P}{2} .$
For $- (P / 2) < x < 0$ , by (4.12) and Fatou’s lemma, we get
$\begin{aligned} (\frac{μ_{P, λ}}{2} - φ_{P, λ} (x)) φ_{P, λ}^{'} (x) & = lim_{h \to 0} \frac{(Λ^{- s} φ_{P, λ}) (x + h) - (Λ^{- s} φ_{P, λ}) (x - h)}{4 h} \\ \geq \frac{1}{2} \int_{- \frac{P}{2}}^{0} (K_{s, P} (y - x) - K_{s, P} (y + x)) φ_{P, λ}^{'} (y) d y . \end{aligned}$
(4.13)

Case 1 when $s \in (0, 1)$ : Let us fix two points say $p_{1}$ , $p_{2}$ from the interval $(- P / 2, 0)$ such that $p_{2} < p_{1}$ . Choose $x \in (p_{2}, p_{1})$ and $ς \in (- P / 2, p_{2}]$ . Then from (4.13), and the monotonicity of $φ_{P, λ}$ on $(- P / 2, 0)$ , we have
$\begin{aligned} (\frac{μ_{P, λ}}{2} - φ_{P, λ} (ς)) φ_{P, λ}^{'} (x) & \geq (\frac{μ_{P, λ}}{2} - φ_{P, λ} (x)) φ_{P, λ}^{'} (x) \\ \geq \frac{1}{2} \int_{- \frac{P}{2}}^{0} (K_{s, P} (y - x) - K_{s, P} (y + x)) φ_{P, λ}^{'} (y) d y \\ \geq \frac{1}{2} \int_{p_{2}}^{p_{1}} (K_{s, P} (y - x) - K_{s, P} (y + x)) φ_{P, λ}^{'} (y) d y \\ = \frac{1}{2} \int_{p_{2}}^{p_{1}} (- 2 x) K_{s, P}^{'} (y + τ) φ_{P, λ}^{'} (y) d y \\ \geq - p_{1} K_{s, P}^{'} (2 p_{2}) (φ_{P, λ} (p_{1}) - φ_{P, λ} (p_{2})), \end{aligned}$
where $| τ | < | x |$ follows from the convexity of $K_{s, P}$ on $(- P / 2, 0)$ and the mean value theorem. Integrating from $p_{2}$ to $p_{1}$ on $x$ and dividing by $(φ_{P, λ} (p_{1}) - φ_{P, λ} (p_{2}))$ gives
$\begin{aligned} \frac{μ_{P, λ}}{2} - φ_{P, λ} (ς) \geq - p_{1} K_{s, P}^{'} (2 p_{2}) (p_{1} - p_{2}) . \end{aligned}$
(4.14)
Now, from (2.2) and Lemma 2.1, we have
$\begin{aligned} K_{s, P}^{'} (x) & = K_{s}^{'} (x) + \sum_{n \in Z ∖ {0}} K_{s}^{'} (x + n P) \\ \leq c_{1} (s - 1) | x |^{s - 2} + O (| x |^{s}) + c_{2} \sum_{n \in Z ∖ {0}} | x + n P | e^{- | x + n P |}, \end{aligned}$
where $c_{1}$ and $c_{2}$ are positive constants depending on $s$ only, and $x$ is certainly negative. By rewriting the above relation as
$- K_{s, P}^{'} (x) \geq - c_{1} (s - 1) | x |^{s - 2} - O (| x |^{s}) - c_{2} \sum_{n \in Z ∖ {0}} | x + n P | e^{- | x + n P |} .$
For $- (P / 4) < p_{2} < 0$ ,
$\begin{aligned} - K_{s, P}^{'} (2 p_{2}) & \geq - c_{3} | p_{2} |^{s - 2} - O (| 2 p_{2} |^{s}) - c_{2} \sum_{n \in Z ∖ {0}} | 2 p_{2} + n P | e^{- | 2 p_{2} + n P |} \\ \geq - c_{3} | p_{2} |^{s - 2} - O (| 2 p_{2} |^{s}) - c_{2} \sum_{n \in Z ∖ {0}} (\frac{2 | n | + 1}{2}) P e^{- (\frac{2 | n | + 1}{2}) P} \\ \geq - c_{3} | p_{2} |^{s - 2}, \end{aligned}$
where $c_{3}$ is another positive constant depending on $s$ . By using the above inequality in (4.14),
$\frac{μ_{P, λ}}{2} - φ_{P, λ} (ς) \geq - p_{1} c_{3} | p_{2} |^{s - 2} (p_{1} - p_{2}),$
and taking $ς = p_{2} = x$ and $p_{1} = x / 2$ with $- (P / 4) < x < 0$ , we have
$\begin{aligned} \frac{μ_{P, λ}}{2} - φ_{P, λ} (x) \geq c^{'} | x |^{s} . \end{aligned}$
(4.15)
Now, by picking $0 < δ < c^{'}$ small enough, we get
$\frac{μ_{P, λ}}{2} - φ_{P, λ} (x) \geq δ | x |^{s},$
uniformly for $P \geq 1$ , $0 < λ \leq 1$ and $| x | < δ$ by continuity.

Case 2 when $s > 1$ : The idea of proof given here is taken from Bruell and Dhara (2021). From Theorem 4.1, we know that the $φ_{P, λ}$ is Lipschitz at crest, that is,
$\begin{aligned} μ - φ_{P, λ} ≲ | x | for | x | ≪ 1, \end{aligned}$
(4.16)
which sets the upper bound for $(μ - φ_{P, λ})$ . To establish the lower bound for $| x | ≪ 1$ , we divide (4.13) by (4.16),
$\begin{aligned} φ_{P, λ}^{'} (x) ≳ \frac{1}{2} \int_{- \frac{P}{2}}^{0} \frac{K_{s, P} (y - x) - K_{s, P} (y + x)}{| x |} φ_{P, λ}^{'} (y) d y . \end{aligned}$
(4.17)
Since, for some $y \in (- p / 2, 0)$ , we can write
$lim_{x \to 0^{-}} \frac{K_{s, P} (y - x) - K_{s, P} (y + x)}{| x |} = 2 K_{s, P}^{'} (y) .$
By substituting this value in (4.17) and applying Fatou’s lemma, we get
$\begin{aligned} \underset{x \to 0}{lim inf} φ_{P, λ}^{'} (x) ≳ \int_{- \frac{P}{2}}^{0} K_{s, P}^{'} (y) φ_{P, λ}^{'} (y) d y . \end{aligned}$
(4.18)
As both $K_{s, P}^{'}$ and $φ_{P, λ}^{'}$ are continuous and bounded in $(- p / 2, 0)$ , the integral in the left-hand side is a constant. Now, by applying the mean value theorem on $φ_{P, λ}$ for interval $(x, 0)$ for $x < 0$ ,
$\frac{φ_{P, λ} (0) - φ_{P, λ} (x)}{| x |} = φ_{P, λ}^{'} (ξ) for some | ξ | ≪ 1.$
Since, $φ_{P, λ} (0) = λ μ_{P, λ} / 2$ , or $φ_{P, λ} (0) \leq μ_{P, λ} / 2$ , by placing the lower bound from (4.18), we have the desired result.

Case 3 when $s = 1$ : We discussed this case in the previous theorem, which provides the stronger version; however, to get the uniform upper bound, let’s restrict ourselves to the one inequality from (4.5) only,
$\begin{aligned} g {(x)}^{2} \geq o (1) g (x) + \frac{1}{2} min_{t \in (0, r]} g (t), \end{aligned}$
(4.19)
where
$g (x) = \frac{\frac{μ_{P, λ}}{2} - φ_{P, λ} (x)}{x \log (1 / x)}$
is positive and monotonic in $(0, r]$ , for any small $δ > 0$ ,
$\frac{\frac{μ_{P, λ}}{2} - φ_{P, λ} (x)}{| x \log (1 / x) |} \geq δ$
$(μ_{P, λ} / 2) - φ_{P, λ} (x) \in C ([0, P])$ is bounded and non-negative, providing $φ_{P, λ}$ is periodic solution of (1.3), and $φ_{P, λ} (0) = λ μ_{P, λ} / 2$ . Also, $| x \log (1 / x) |$ in increasing function in $(0, r]$ , (4.11) holds true for $s = 1$ .

We now construct a sequence ${(φ_{P_{n}, λ}, μ_{P_{n}, λ})}$ of periodic solution pairs obtained from the previous two theorems. We shall show that there exists a subsequence converging to a non-periodic solution which might not decay to $0$ as $P_{n} \to \infty$ . The limiting solution will eventually be the solution we desire, and after the Galilean transform, it gives the solitary wave solution.
Theorem 4.3
The sequence ${(φ_{P_{n}, λ}, μ_{P_{n}, λ})}$ of solution pairs from Theorem 4.1 has a subsequence converging locally uniformly to a solution pair $(ϕ_{λ}, ϑ_{λ}) \in C (R) \times (0, 1]$ as $P_{n} \to \infty$ with relative height $λ \in (0, 1]$ such that
$ϑ_{λ} - 1 \leq ϕ_{λ} \leq \frac{λ ϑ_{λ}}{2} .$
This non-periodic limiting solution $ϕ_{λ}$ is bounded, non-constant, even, and strictly increasing on $(- \infty, 0)$ . Moreover, $ϕ_{λ}$ is smooth on $R$ for $λ \in (0, 1)$ and $ϑ_{λ} > 0$ .

Further, the Galilean transformation
$\begin{aligned} Φ_{λ} = ϕ_{λ} + 1 - ϑ_{λ}, μ_{λ} = 2 - ϑ_{λ}, \end{aligned}$
(4.20)
provide a solution $Φ_{λ}$ of (1.3), which satisfies all the properties of $ϕ_{λ}$ and gives the desired solitary wave solution providing $lim_{x \to \pm \infty} Φ_{λ} (x) = 0$ with supercritical wave speed $μ_{λ} \in (1, 2)$ such that
$0 < Φ_{λ} \leq μ_{λ} - 1 + λ (1 - \frac{μ_{λ}}{2}) .$
$Φ_{λ}$ is smooth everywhere for $λ \in (0, 1)$ and for $λ = 1$ , $Φ_{λ}$ is smooth on $R ∖ {0}$ such that $Φ_{λ} \in C^{s}$ for $s \in (0, 1)$ , $Φ_{λ}$ is log-Lipschitz for $s = 1$ and Lipschitz for $s > 1$ .
Proof.
The approach of the proof is similar to the one presented in Ehrnström, Nik, et al. (2023); however, here we deal with the operator $Λ^{- s}$ and different cases of $s > 0$ . We assume that there exists a subsequence ${P_{n}}_{n}$ such that $lim_{P_{n} \to \infty} μ_{P_{n}, λ} = ϑ_{λ} .$ It can be easily shown that
$\begin{aligned} [φ_{P, λ} (x) - φ_{P, λ} (y)]^{2} \leq | Λ^{- s} φ_{P, λ} (x) - Λ^{- s} φ_{P, λ} (y) |, \end{aligned}$
for all $x, y \in R$ . From Proposition 2.6 and the following remark, $Λ^{- s}$ is bounded from $L^{\infty} (R)$ to $C^{α} (R)$ for any $α \in (0, 1)$ ,
$\frac{| Λ^{- s} φ_{P, λ} (x) - Λ^{- s} φ_{P, λ} (y) |}{| x - y |^{α}} \leq ‖ Λ^{- s} φ_{P, λ} ‖_{C^{α} (R)} \leq ‖ Λ^{- s} ‖_{L (L^{\infty} (R), C^{α} (R))} ‖ φ_{P, λ} ‖_{\infty},$
for all $x, y \in R$ . This shows that ${φ_{P_{n}, λ}}_{n}$ is uniformly bounded in $C^{α / 2} (R)$ , for some $α \in (0, 1)$ . Since ${φ_{P_{n}, λ}}_{n}$ is equicontinuous, by Arzelà–Ascoli theorem, there exists a subsequence converging locally uniformly to a function $ϕ_{λ} \in C (R)$ .

Also, $| φ_{P_{n}, λ} | \leq 1$ and $‖ K_{s} ‖_{L^{1} (R)} = 1$ , Lebesgue’s dominated convergence theorem gives
$Λ^{- s} φ_{P_{n}, λ} (x) = K_{s} * φ_{P_{n}, λ} (x) \to K_{s} * ϕ_{λ} (x) = Λ^{- s} ϕ_{λ} (x),$
as $P_{n} \to \infty$ for all $x \in R$ . Thus, $ϕ_{λ}$ solves (1.3) with wave speed $ϑ_{λ}$ .

From Theorem 4.2, we have a uniform bound for the desired solution $ϕ_{λ} (x)$ near the origin as
$\begin{aligned} ϕ_{λ} (x) \leq {\begin{array}{lr} \frac{ϑ_{λ}}{2} - δ | x |^{s}, & 0 < s < 1, \\ \frac{ϑ_{λ}}{2} - δ | x \log | x | |, & s = 1, \\ \frac{ϑ_{λ}}{2} - δ | x |, & s > 1. \end{array} \end{aligned}$
(4.21)

Recall that the only constant solutions to (1.3) are $ϕ_{λ} = 0$ and $ϕ_{λ} = ϑ_{λ} - 1$ . When $ϕ_{λ} = 0$ , $ϕ_{λ} (0) = λ ϑ_{λ} / 2$ gives $ϑ_{λ} = 0$ , and this contradicts (4.21). For $ϕ_{λ} = ϑ_{λ} - 1$ , we have $λ (ϑ_{λ} / 2) = ϑ_{λ} - 1$ , that is, $ϑ_{λ} = 2 / (2 - λ)$ . Since $0 < λ \leq 1$ , this contradicts $0 < ϑ_{λ} \leq 1$ . Hence, $ϕ_{λ}$ is a non-constant function.

The evenness of $ϕ_{λ}$ is inherited from ${φ_{P_{n}, λ}}_{n}$ , and it is easy to show that $ϕ_{λ}$ is strictly increasing on $(- \infty, 0)$ , one can follow the same steps as in Ehrnström, Nik, et al. (2023, Theorem 4.3). The smoothness of $ϕ_{λ}$ on $R$ for $0 < λ < 1$ follows from Lemma 3.5. The property of $ϕ_{λ}$ of being non-constant with monotonicity shows that it is non-periodic. Again, since $ϕ_{λ}$ is bounded, Proposition 3.1 tells us that $lim_{x \to \infty} ϕ_{λ} (x)$ could be $0$ or $ϑ_{λ} - 1$ . Assuming $lim_{x \to \infty} ϕ_{λ} (x) = 0$ immediately gives $ϕ_{λ}$ is non-negative. Now, from Proposition 3.3 and our assumption here, it yields $ϑ_{λ} = 1$ . However, from Corollary 3.4, only trivial solutions can have unit wave speed. Hence, $lim_{x \to \infty} ϕ_{λ} (x) = ϑ_{λ} - 1 < 0.$

The limiting non-periodic solution inherits its regularity from the corresponding periodic wave. As a particular case, when $s = 1$ , we may use the triangle inequality and the earlier estimate (see (4.9)) to write
$\begin{aligned} | ϕ_{λ} (x) - ϕ_{λ} (y) | & \leq | ϕ_{λ} (x) - ϕ_{λ, P} (x) | + | ϕ_{λ} (y) - ϕ_{λ, P} (y) | + | ϕ_{λ, P} (x) - ϕ_{λ, P} (y) | \\ \leq | ϕ_{λ} (x) - ϕ_{λ, P} (x) | + | ϕ_{λ} (y) - ϕ_{λ, P} (y) | + M | x - y | \log (1 + \frac{1}{| x - y |}) . \end{aligned}$
Now fix a compact interval $K$ and consider $x, y \in K$ . By letting $P \to \infty$ , because of the uniform convergence, the first two terms on the right-hand side converge to $0$ . Therefore,
$| ϕ_{λ} (x) - ϕ_{λ} (y) | \leq M | x - y | \log (1 + \frac{1}{| x - y |}),$
for all $x, y \in K$ . But since $K$ is arbitrary, this estimate holds for all $x, y \in R$ .

Now, we only have to exclude the negative tails to obtain positive solutions, say $Φ_{λ}$ as $x$ approaches $\infty$ . For this, we apply the Galilean transformation (4.20), which gives the solitary wave solution with wave speed $μ_{λ} \in (1, 2)$ . We eliminate the possibility that $μ_{λ} = 2$ as then again from Corollary 3.4, it would lead us to a constant solution $Φ_{λ} \equiv 0$ or $Φ_{λ} \equiv 1$ . All other properties of $Φ_{λ}$ are eventually satisfied, including smoothness.

In the following subsection, we discuss the decay rate of solitary waves for equation (1.1) in accordance with the work of Arnesen (2022). Several other recent works in this area for different equations can be found in Bruell et al. (2017), Eychenne and Valet (2023), and Pei (2020).
4.1. Exponential Decay

It has been shown that the decay rate of solitary wave solutions is related to the decay rate of the associated convolution kernel (see Bona & Li, 1997; Bruell et al., 2017). Rewriting the steady equation (1.3) as

φ (μ - φ) = H_{μ} * φ^{2}, where H_{μ} = F^{- 1} (\frac{m}{μ - m}),

shifts the focus to

H_{μ}

. The exponential decay of

H_{μ}

implies the exponential decay of the solution, with the exact rate depending on the wave speed

μ

and

s

. For this section, we use

m (ξ) = (1 + ξ^{2})^{- s / 2}

for conciseness.

Arnesen (2022) studies the exact decay rate and symmetry of solitary waves for a class of nonlocal dispersive wave equations $u_{t} + (L u + G (u))_{x} = 0$ , reduced to the steady form

\begin{aligned} - μ φ + L φ + G (φ) = 0. \end{aligned}

(4.22)

where

L

is a Fourier multiplier operator with a symbol

m

defined by

\hat{L f} (ξ) = m (ξ) \hat{f} (ξ)

, and

G

is a nonlinear function. Under certain conditions on

m

and

G

(as detailed in the theorem below), solitary waves for this class of equations exhibit exponential decay, have only one crest, and are, in general, symmetric. We provide only the main theorem here; for deeper insights, one can refer to Arnesen (2022).

Theorem 4.4 Arnesen, 2022

Let $m (ξ)$ be an even, real analytic function, and bounded above by $μ$ for all $ξ \in R$ . Let $G : R \to R$ be bounded on every compact set, and satisfy $| G (u) | ≲ | u |^{t}$ for all small values of $u$ and for some $t > 1$ . If the function $m (ξ)$ follows the inequality

\begin{aligned} | m^{(k)} (ξ) | \leq C_{k} (1 + | ξ |)^{- s - k}, k = 0, 1, 2, \dots, \end{aligned}

(4.23)

for some

s > 0

, where

C_{k}

are positive constants depending on

k

such that

\begin{aligned} lim_{k \to \infty} \frac{C_{k + 1} / (k + 1)!}{C_{k} / k!} = l, for some l \geq 0, \end{aligned}

(4.24)

then for a non-trivial solution

u \in L^{\infty} (R)

with

lim_{| x | \to \infty} u (x) = 0

, there exists a number

δ_{μ} > 0

depending upon

m

and

μ

such that

e^{δ | . |} u (.) \in L^{1} \cap L^{\infty} (R),

for all

δ \in (0, δ_{μ}) .

Our case, involving the fKdV equation with the inhomogeneous symbol $(1 + ξ^{2})^{- s / 2}$ , aligns with the assumptions made in Theorem 4.4. We have $G (u) = u^{2}$ , which immediately fulfills all the requirements on $G$ , and the symbol $m (ξ) = (1 + ξ^{2})^{- s / 2}$ is even, real analytic, and bounded above by $μ$ (as $μ$ is the supercritical wave speed obtained in Theorem 4.3). We only need to prove the limit (4.24). For this, we use Faà di Bruno’s formula to calculate the following derivatives:

D_{ξ}^{k} (1 + ξ^{2})^{- \frac{s}{2}} = \sum_{j = 0}^{[\frac{k}{2}]} \frac{(- 1)^{k - j} 2^{k - 2 j} k!}{j! (k - 2 j)!} [\frac{s}{2} (\frac{s}{2} + 1) \dots (\frac{s}{2} + k - j - 1)] ξ^{k - 2 j} (1 + ξ^{2})^{- \frac{s}{2} - (k - j)} .

Suppose

M_{s, k}

is a uniform upper bound (depending on

s

and

k

) for the convergent series

\sum_{j = 0}^{[\frac{k}{2}]} \frac{(- 1)^{j}}{j! (k - 2 j)!} \frac{1}{2^{k + 2 j}} [\frac{s}{2} (\frac{s}{2} + 1) \dots (\frac{s}{2} + k - j - 1)] \frac{ξ^{k - 2 j}}{(1 + ξ^{2})^{\frac{k - 2 j}{2}}},

such that

| M_{s, k} | = \frac{1}{(k - 1)! 2^{k + 1}} [\frac{s}{2} (\frac{s}{2} + 1) \dots (\frac{s}{2} + k - 1)] .

This leads to

\begin{aligned} | D_{ξ}^{k} (1 + ξ^{2})^{- s / 2} | & \leq 2^{2 k} k! | M_{s, k} | (1 + ξ^{2})^{- \frac{s + k}{2}} \\ < 2^{2 k} k! | M_{s, k} | (2^{\frac{s + k}{2}} (1 + | ξ |)^{- s - k}) . \end{aligned}

Therefore, for a fixed $s > 0$ , the coefficients $C_{k}$ in (4.23) are given by

C_{k} = 2^{(\frac{s - k}{2})} (\frac{k}{2}) [\frac{s}{2} (\frac{s}{2} + 1) \dots (\frac{s}{2} + k - 1)],

which establishes the expression in (4.24).

Now, $m (z)$ is analytic for all $z \in C ∖ {- i, i}$ and along the imaginary axis, we have $m (i b) = (1 - b^{2})^{- s / 2}$ which is even and a bijection from $[0, 1)$ to $[1, \infty)$ . This implies that there exists $δ_{μ, s} \in (0, 1)$ satisfying $(1 - δ_{μ, s}^{2})^{- s / 2} = μ$ for $μ > 1$ such that

\begin{aligned} H_{μ} (x) = e^{δ_{μ, s} | x |} (v (x) + \frac{2 \sqrt{π} μ (1 - δ_{μ, s}^{2})^{\frac{s + 2}{2}}}{s δ_{μ, s}}), \end{aligned}

(4.25)

for some even function

v \in L^{p} ({x \in R : | x | \leq 1})

for all

P \in (0, \infty)

with

\begin{aligned} v (x) ≃_{s} {\begin{array}{lr} | x |^{s - 1}, & 0 < s < 1, \\ \log \frac{1}{| x |}, & s = 1, \\ 1, & s > 1, \end{array} for | x | \leq 1. \end{aligned}

We need the expression (4.25) for

H_{μ}

to prove Theorem 4.4 and to find

δ_{μ, s}

(see Arnesen 2022, Lemma 3.4). And here we conclude that the obtained solitary solutions

Φ_{λ}

of (1.3) with wave speeds

μ_{λ} \in (1, 2)

and

λ \in (0, 1)

such that

lim_{x \to \pm \infty} Φ_{λ} (x) = 0

and

0 \leq Φ_{λ} \leq μ_{λ} - 1 + λ (1 - (μ_{λ} / 2))

, decay exponentially with

e^{δ | . |} Φ_{λ} (.) \in L^{1} \cap L^{\infty} (R),

for all

δ \in (0, δ_{μ, s})

Footnotes

Acknowledgments

The authors express their gratitude to Mats Ehrnström and Douglas Seth for their continuous support and insightful discussions. The authors also thank Gabriele Brüll for discussions on the decay rate and Kristoffer Varholm for his willingness to explain the mechanics behind the work (Ehrnström, Mæhlen, et al., 2023).

ORCID iD

Swati Yadav

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: SY is supported by the ERCIM “Alain Bensoussan” Fellowship Programme at NTNU, Trondheim. The work is supported in part by the project IMod–Partial differential equations, statistics and data: An interdisciplinary approach to data-based modeling, project number 325114, from the Research Council of Norway.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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