# Riesz Potentials for Korteweg-de Vries Solitons and Sturm-Liouville Problems

Authors: Vladimir Varlamov
Journal: International Journal of Differential Equations
 (2010)

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## Abstract

Riesz potentials (also called Riesz fractional derivatives) and their Hilbert
transforms are computed for the Korteweg-de Vries soliton. They are expressed
in terms of the full-range Hurwitz Zeta functionsζ+(s,a) and ζ−(s,a).
It is proved that these Riesz potentials and their Hilbert transforms are linearly
independent solutions of a Sturm-Liouville problem. Various new
properties are established for this family of functions. The fact that the
Wronskian of the system is positive leads to a new inequality for the Hurwitz
Zeta functions.


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## Body

## 1. Introduction

In recent years the theory of fractional derivatives and integrals called Fractional Calculus has been steadily gaining importance for applications. Ordinary and partial differential equations of fractional order have been widely used for modeling various processes in physics, chemistry, and engineering (see, e.g., [1–3] and the references therein). Recent theoretical developments shed new light on the interpretation and properties of fractional derivatives. Having written the latter in the form of Stieltjes integrals, Podlubny [4] found new physical and geometric interpretation of these structures relating them to inhomogeneity of time. Extension of the classical maximum principle to the case of a time-fractional diffusion equation appeared in the recent work of Luchko [5]. In the present paper we are concerned with Riesz fractional derivatives (also called Riesz potentials; see [6, page 88], and [7, page 117]) that are defined as fractional powers of the Laplacian Dα=(-Δ)α/2 with α∈ℝ. They are well known for their role in investigating the solvability of nonlinear partial differential equations, and the Korteweg-de Vries equation (KdV henceforth) in particular (see, e.g., [7–11] and the references therein). In the current work, Riesz potentials of KdV solitons are computed and their relation to ordinary differential equations is established.We continue the study of Riesz fractional derivatives of solutions to Korteweg-de-Vries-type equations started in [12]. After appropriate rescaling, KdV can be written in the form(1.1)ut+uxxx+3(u2)x=0,x∈ℝ,t>0.
It is well known that the fundamental solution of the Cauchy problem for the linearized KdV is expressed in terms of the Airy function of the first kind Ai(x) and its Hilbert transform (conjugate) in terms of the Scorer function Gi(x). The papers [12–14] were devoted to the study of fractional properties of the Airy functions and their conjugates and to the establishing of related properties for KdV-type equations.Although there exists extensive literature on solitons, as far as we know, a study of their fractional properties is still missing. A preliminary investigation of Riesz potentials for a KdV soliton was carried out in [15]. In this paper the emphasis was put on the issue of whether solitons inherit fractional properties of fundamental solutions. Riesz potentials of a soliton, uα(X)=Dαu0(X), where u0(X)=2sech2X, X=x-4t, and Dα=(-∂x2)α/2, and their Hilbert transforms, vα(X)=-Huα(X), were obtained in terms of the Hurwitz Zeta function of a complex argument, ζ(s,z) with s=2+α and z=1/2+iX/π. It was proved in [15] that the zero mean properties hold for both uα(X) and vα(X) with α>0. This confirmed the predictions based on the properties of fundamental solutions of the linearized Cauchy problems.The goal of the current paper is to go further and to study Riesz potentials of solitons as solutions of differential equations. We intend to show that these functions and their Hilbert transforms form linearly independent systems of solutions for a second-order ordinary differential equation in a self-adjoint form. This fact may be helpful in understanding the issue of using these structures as intrinsic mode functions in signal processing (see [16, 17] and the references therein), that is, in using Riesz potentials for expansions. In this context it is interesting to point out that the graphs of the functions uα(X) reveal a striking similarity to those of the Airy wavelets generated by the function Ai′(x)Ai′(-x) (see [18, page 34], and Figure 3 below).For the analysis to follow; we employ the full-range Hurwitz Zeta functions:ζ+(s,a)=ζ(s,a)+ζ(s,1-a) and ζ-(s,a)=ζ(s,a)-ζ(s,1-a) (symmetric and antisymmetric combinations of ζ(s,a) and ζ(s,1-a)), recently introduced in [19] for a∈ℝ. We prove that the functions wα(X)=uα(X)+ivα(X), α>-1 are solutions of the Sturm-Liouville problem(1.2)-ddX(Pα(X)dwdX)+Qα(X)w=λρα(X)w,X∈ℝ,(1.3)limX→±∞w(X)=0,
for λ=1. Here Pα(X)>0, ρα(X)>0, and Qα(X) is a real function. The essential point consists in proving that the Wronskian of uα(X) and vα(X) is positive for all α>-1 and x∈ℝ. It allows one to prove that Pα(X) and ρα(X) are positive and to estimate the number of zeros of uα(X) and vα(X) on any bounded interval.The fact that this Wronskian is positive also leads to a new inequality for the Hurwitz Zeta functions(1.4)ζ(s,z̅)ζ(s+1,z)+ζ(s,z)ζ(s+1,z̅)>0,
where s>1, z=1/2+iX/π, and the bar over the letter denotes complex conjugation. As far as we know, there are no results in the literature on the arguments of the Hurwitz Zeta functions. However, (1.4) provides some information on this issue. Indeed, setting ζ(s,z)=rseiφs and ζ(s+1,z)=rs+1eiφs+1 in (1.4) allows one to deduce the relation cos(φs+1(z)-φs(z))>0 for s>1, z=1/2+iX/π, and X∈ℝ.The paper is organized as follows. In Section2, we provide the necessary information on the special functions involved. Section 3 is devoted to the study of Riesz potentials for KdV solitons and their Hilbert transforms. In Section 3.1, the main properties of these functions are summarized. Sturm-Liouville problem (1.2) is derived in Section 3.2. Section 3.3 deals with the properties of the Wronskian W[uα,vα]. Zeros of the functions uα and vα are studied in Section 3.4. In Section 4, the inequality (1.4) is discussed.


## 2. Preliminaries

Introduce the Fourier transform of the functionf:ℝ→ℝ by(2.1)f̂(ξ)=ℱ{f}(ξ)=∫-∞∞e-iξxf(x)dx,
and the inverse Fourier transform by
(2.2)f(x)=ℱ-1{f̂}(x)=12π∫-∞∞eiξxf̂(ξ)dξ.For realα and x∈ℝ, define Riesz potentials of a function f(x) by the formula (see [7, page 117])(2.3)Dαf(x)=12π∫-∞∞|ξ|αf̂(ξ)eiξxdξ,
provided that the integral in the right-hand side exists. Define derivatives of Dxαf(x) with respect to α by(2.4)∂αnDxαf(x)=12π∫-∞∞|ξ|αlnn|ξ|f̂(ξ)eiξxdξ,n∈ℕ,
provided that these integrals exist.Introduce the Hilbert transform of the functionf by (see [20, page 120])(2.5)Hf(x)=1πP.V.∫-∞∞f(y)y-xdy,
where x∈ℝ and P.V. denotes the Cauchy principal value of the integral. According to our choice of the Fourier transform (Hf̂)(ξ)=isgn(ξ)f̂(ξ), one can see that H2=-I on Lp(ℝ),p≥1, where I is the identity operator. Also, ∂x=H∘D and ∂x2=-D2, where the operator D is defined by (2.3).Next, introduce the Trigamma function by (see [21, page 260, 6.4.1])(2.6)ψ′(z)=d2dz2lnΓ(z)=∫0∞te-zt1-e-tdt,ℜ(z)>0.
Notice that (see [21], page 260, 6.4.7)(2.7)ψ′(z)+ψ′(1-z)=-πddz{cot(πz)}.
Also, the following asymptotic expansion holds:(2.8)ψ′(z)~1z+12z2+16z3+O(1z5)forz→∞,|argz|<π.The Hurwitz (generalized) Zeta function is defined by (see [22, page 88])(2.9)ζ(s,a)=∑n=0∞1(n+a)sforℜ(s)>1,a∈ℂ,a≠0,-1,-2,….
This implies that(2.10)∂aζ(s,a)=-sζ(s+1,a).
This function has the integral representation(2.11)ζ(s,a)=1Γ(s)∫0∞xs-1e-ax1-e-xdxforℜ(s)>1,ℜ(a)>0,
where Γ(x) is the Gamma function. In two particular cases, we have (see [22]) that(2.12)ζ(s,12)=(2s-1)ζ(s),(2.13)ζ(2,a)=ψ′(a),
where ζ(s) is the Riemann Zeta function.The singularity ofζ(s,a) as s→1 is given by the relation(2.14)lims→1{ζ(s,a)-1s-1}=-Γ′(a)Γ(a)=-ψ(a).
The asymptotic expansion of ζ(s,a) for large a is (see [23])(2.15)ζ(s,a)~12a-s+a1-ss-1+1Γ(s)∑k=1∞B2k(2k)!Γ(2k+s-1)a2k+s-1,|a|<∞,|arga|<π,
where Bn are the Bernoulli numbers.Introduce the full-range forms of Hurwitz Zeta functions (see [19])(2.16)ζ+(s,a)=∑n=-∞∞1[(n+a)2]s/2,(2.17)ζ-(s,a)=∑n=0∞1[(n+a)2]s/2-∑n=-∞-11[(n+a)2]s/2.
Representations (2.16) and (2.17) imply that(2.18)ζ+(s,a)=ζ(s,a)+ζ(s,1-a),ζ-(s,a)=ζ(s,a)-ζ(s,1-a).
Hence, follow the symmetric and antisymmetric properties of the functions,(2.19)ζ+(s,a)=ζ+(s,1-a),ζ-(s,a)=-ζ-(s,1-a).
It follows from (2.16) that ζ+(s,a) is a periodic function of a, with the unit period. It is even with respect to a=0 and a=1/2. The function ζ-(s,a) is odd about a=1/2 and ζ-(s,1/2)=0. These functions satisfy the functional differential equations(2.20)∂aζ+(s,a)=-sζ-(s+1,a),∂aζ-(s,a)=-sζ+(s+1,a).Consider now thats∈ℝ and denote by a¯ the complex conjugate of a. Then ζ(s,a̅)=ζ(s,a)¯. It implies that, for a=a1+ia2,(2.21)ζ+(s,a)=2ℜ{ζ(s,a)}=2Γ(s)∫0∞xs-1e-a1xcosa2x1-e-xdx.
In a similar way,(2.22)ζ-(s,a)=2iℑ{ζ(s,a)}=-2iΓ(s)∫0∞xs-1e-a1xsin(a2x)1-e-xdx.Denote byW[u,v] the Wronskian of the functions u(x) and v(x), that is, W[u,v]=|uvu′v′|. For reader's convenience, we present here [24, Theorem 5.3].Theorem 2.1.
Letp(t)>0, q(t) be real valued and continuous for 0≤t≤T. Let u and v be real valued solutions of the equation
(2.23)(pu′)′+qu=0,
satisfying the condition
(2.24)p(t)W[u(t),v(t)]=C0>0.
Let N be the number of zeros of u(t) on [0,T]. Then
(2.25)|πN-C0∫0Tdtp(t)[v2(t)+u2(t)]|≤π.In conclusion of this section, we would like to quote an interesting result concerning integrals over the real axis (see [25]).Theorem 2.2.
For any integrable functionf(x) and g(x)=x-c2/x with c=const∈ℝ,
(2.26)P.V.∫-∞∞f(g(x))dx=P.V.∫-∞∞f(x)dx.
Moreover, the above formula holds true if
(2.27)g(x)=x-∑j=1∞bjx-cj,
where {bj} is any sequence of positive constants, cj are any real constants, and the series is convergent.


## 3. Fractional Derivatives of A KdV Soliton and Their Conjugates

In this section, we consider Riesz fractional derivatives of a KdV soliton and their Hilbert transforms and establish their properties. We notice that all the graphs were obtained with theMathematica 6 software.We take the soliton solution of (1.1) in the form u0(X)=2sech2X with X=x-4t (see Figure 1) and introduce the function(3.1)wα(X)=uα(X)+ivα(X),
where(3.2)uα(X)=Dαu0(X),vα=-DαHu0(X).
Notice that the functions uα(X) and vα(X) form a conjugate pair (see [20, page 120]) since(3.3)uα(X)=Hvα(X),vα(X)=-Huα(X).Figure 1
Graph of the solitonu0(X).The next statement was proved in [15]. Using the functions (2.16) and (2.17), we rewrite it in a more convenient form.Theorem 3.1.
The functionsuα(X) and vα(X) have the following representations for α>-1 and X∈ℝ:
(3.4)uα(X)=2Γ(2+α)π2+αζ+(2+α,12+iXπ),(3.5)vα(X)=i2Γ(2+α)π2+αζ-(2+α,12+iXπ),
where ζ+(s,a) and ζ-(s,a) are the full-range Hurwitz Zeta functions (see (2.16) and (2.17)) and Γ(s) is the Gamma function.


### 3.1. Properties of the Functionsuα and vα

In this subsection, we collect the properties of the functionsuα and vα. Some of them were established in [15] and some are given for the first time as follows.Properties of the Functionsuα and vα
(1)
The functionsuα(X) and vα(X) satisfy the functional differential equations(3.6)uα′(X)=-vα+1(X),vα′(X)=uα+1(X),
where α>-1, X∈ℝ, and the prime denotes differentiation with respect to X. This follows from (3.3) and the relation d/dX=H∘D.(2)
The functionsuα(X) are even and the functions vα(X) are odd on ℝ (see [15] and Figures 1–4).(3)
The functionuα(X) is periodic with the period 𝒳 = iπ. It follows from the periodicity of ζ+(s,a) with the unit period.(4)
For allα>0 (see [15]),(3.7)∫-∞∞uα(X)dX=0,∫-∞∞vα(X)dX=0.
These properties are reflected on the graphs (see Figures 3 and 4).(5)
For allα>0 and n∈ℕ,(3.8)∂αn∫-∞∞uα(X)dX=0,∂αn∫-∞∞vα(X)dX=0.
These relations follow from the differentiation of the identities in (3.7).(6)
For allα>0 and c∈ℝ,(3.9)P.V.∫-∞∞uα(X-c2X)dX=0,P.V.∫-∞∞vα(X-c2X)dX=0.
Moreover,(3.10)P.V.∫-∞∞uα(X-∑j=1∞bjX-cj)dX=0,P.V.∫-∞∞vα(X-∑j=1∞bjX-cj)dX=0,
where bj is any sequence of positive constants, cj are any real constants, and the series converges. These relations follow from (3.7) and Theorem 2.2.(7)
The functional sequence of Riesz potentials{uα(X)} converges pointwise to the soliton u0(X), and the functional sequence {vα(X)} converges pointwise to the conjugate soliton v0(X) for α→0+ (see [15]). Notice that(3.11)∫-∞∞u0(X)dX=4,P.V.∫-∞∞v0(X)dX=0
(see Figures 1 and 2). Here the conjugate soliton v0(X) is given by
(3.12)v0(X)=2iπ2[ψ′(12+iXπ)-ψ′(12-iXπ)].
Equation (3.12) can be recovered from (3.5) with α=0 thanks to (2.13).Figure 2
Graph of the conjugate solitonv0(X).Figure 3
Graph of the fractional derivativeu3.8(X).Figure 4
Graph of the conjugate fractional derivativev3.8(X).Remark 3.2.
The conjugate soliton (3.12) is an algebraic solitary wave for extended KdV:
(3.13)vt+vxxx+3[2v·Hv-H(v2)]x=0,
obtained by applying the Hilbert transform to (1.1) and setting v=-Hu. The term “algebraic solitary wave” is explained by the fact that v0(X) has a 1/X decay for large X. More precisely (see [15]),
(3.14)v0(X)~2π·1X+O(1X2)for|X|→∞.(8)
Forα≥0, the functions uα and vα are the elements of L2(ℝ). Moreover, they are orthogonal in the principal value sense, namely, for all α1,α2≥0,(3.15)P.V.∫-∞∞uα1(X)vα2(X)dX=0.
Equations (3.12) and (3.14) imply that v0∈L2(ℝ). The fact that uα and vα with α>0 are the elements of L2(ℝ) follows from their asymptotics obtained in [15], namely,(3.16)uα(X)~4Γ(1+α)π2+αcos[(1+α)arctan(2X/π)][1/4+(X/π)2](1+α)/2+O(1|X|2+α),vα(X)~4Γ(2+α)π2+α2sin[(1+α)arctan(2X/π)](1+α)[1/4+(X/π)2](1+α)/2+O(1|X|2+α).
Orthogonality of uα and vα follows from the fact that all uα(X) with α≥0 are even functions and all vα(X) with α≥0 are odd functions of X (see Property  2 and Figures 1–4).(9)
At the pointX=0,one has for all α>-1(3.17)uα(0)=4Γ(2+α)π2+α(2α+2-1)ζ(2+α)>0,(3.18)vα(0)=0,
where ζ(s) is the Riemann Zeta function (see [15]).


### 3.2. Sturm-Liouville Problem

It is convenient to representwα(X) in the exponential form, namely,(3.19)wα(X)=Rα(X)exp[iΘα(X)],
where(3.20)Rα(X)=[uα(X)]2+[vα(X)]2,Θα(X)={Φα(X),uα(X)>0,Φα(X)+π,uα(X)<0,vα(X)≥0,Φα(X)-π,uα(X)<0,vα(X)<0,Φα(X)=arctanvα(X)uα(X).
Exponential representation (3.19) allows one to deduce the boundary value problem for the functions wα(X). It turns out that for all α>-1 the functions wα(X) solve the equation(3.21)ddX(Pα(X)dwdX)-Qα(X)w+ρα(X)w=0,X∈ℝ,
where(3.22)Pα(X)=CW[uα(X),vα(X)],C=const,Qα(X)=[Pα(X)Rα′(X)]′Rα(X),ρα(X)=[Θα′(X)]2,(3.23)Θα′(X)=W[uα(X),vα(X)]Rα2(X).
Here W[uα(X),vα(X)] is the Wronskian of uα(X) and vα(X). Below we shall use a shorter notation Wα(X)=W[uα(X),vα(X)].Taking into account the behavior ofwα(X) for large X (see [15]), we can restate the obtained results in another form. Indeed, wα(X) are solutions of the Sturm-Liouville problem(3.24)-ddX(Pα(X)dwdX)+Qα(X)w=λρα(X)w,X∈ℝ,(3.25)limX→±∞w(X)=0,
corresponding to λ=1. Without loss of generality, we can choose the constant in (3.22) to be positive. In the next subsection, we shall prove that Wα(X)>0. It implies that Pα(X)>0 and ρα(X)>0 in (3.22).It follows from (3.7) that for α>0 the functions wα(X) also satisfy the zero mean condition(3.26)∫-∞∞wα(X)dX=0.
This reflects the oscillatory behavior of wα(X) for α>0 (see Figures 3 and 4).The graph of the arctangent functionΦα(X) is shown in Figure 5. Φα(X) conveniently serves as a zero counter for both functions: uα(X) and vα(X). It possesses zeros at the points where vα(X) has zeros and has jumps at the points where uα(X) has zeros.Figure 5
Graph of the arctangent functionΦ3.8(X).Remark 3.3.
Observe that a general solution of (3.24) can be written in the form
(3.27)w(g)(X)=C1uα(X)+C2uα(X)∫0XWα(y)uα2(y)dy,
where C1, C2=const.Remark 3.4.
We would like to point out that the differential equation given by (3.21) can be factored in the following way (see [26, page 269]):
(3.28)ddX(Pα(X)dwdX)-Qα(X)w+ρα(X)w=Wα(X)uα(X)ddX[uα2(X)Wα(X)ddX(w(X)uα(X))].


### 3.3. Wronskian ofuα and vα

Lemma 3.5.
The following properties hold for the WronskianWα(X) for α>-1 and all x∈ℝ:
(3.29)Wα(-X)=Wα(X),Wα′(0)=0,(3.30)Wα(X)>0.Proof.
We start with (3.29). Taking into account relations in (3.3) and the fact that ∂X=H∘D, we can write
(3.31)Wα(X)=uα(X)vα′(X)-vα(X)uα′(X)=uα(X)uα+1(X)+vα(X)vα+1(X).
Since the functions uα(X) are even and vα(X) are odd with respect to X∈ℝ, Wα(X) is even for X∈ℝ. Differentiation of (3.31) with the help of (3.6) and (3.18) yields Wα′(0)=0. Next, we turn to the proof of (3.30). Since the functions uα(X) and vα(X) are linearly independent solutions of the equation (3.24), Wα(X)≠0 for all α>-1 and x∈ℝ. It remains to establish the sign of the Wronskian. In view of (3.17) and (3.18),
(3.32)Wα(0)=uα(0)uα+1(0)=16Γ(2+α)Γ(3+α)π5+2α(22+α-1)(23+α-1)ζ(2+α)ζ(3+α)>0.
By Abel's formula, for all X∈ℝ,
(3.33)Wα(X)=Wα(0)exp[-∫0XPα′(η)Pα(η)dη],
where the function
(3.34)Pα′(X)Pα(X)=-i(s+1)π·ζ-(s,z)ζ+(s+2,z)-ζ+(s,z)ζ-(s+2,z)ζ+(s,z)ζ+(s+1,z)-ζ-(s,z)ζ-(s+1,z),s=2+α,z=1/2+iX/π,
is continuous on ℝ. After some simplification, we have that
(3.35)Wα(X)=Wα(0)exp{-s+1πℑ[ζ(s,z)]ℜ[ζ(s+2,z)]-ℜ[ζ(s,z)]ℑ[ζ(s+2,z)]ℜ[ζ(s,z)]ℜ[ζ(s+1,z)]+ℑ[ζ(s,z)]ℑ[ζ(s+1,z)]}.
This representation yields (3.30). The lemma is proved.Three-dimensional graph ofWα(X) is given in Figure 6.Figure 6
Three-dimensional graph of the WronskianWα(X).Remark 3.6.
What does the positivity of the Wronskian yield for the soliton and its conjugate? Forα=0, (3.30) simplifies to read
(3.36)W0(X)=2sech2(X)[v0′(X)+tanhX·v0(X)]>0.
We notice that v0(X)>0 for X>0, v0(-X)=-v0(X) for X∈ℝ, and v0(0)=0 (see Figure 2). Integrating the inequality v0′(X)/v0(X)+tanhX>0 over the interval [ε,X] for X≥ε>0 and the inequality v0′(X)/v0(X)+tanhX<0 over [-ε,X] for X≤-ε<0 yields the estimate
(3.37)|v0(X)|>|v0(ε)|coshεcoshXfor|X|≥ε>0.


### 3.4. Zeros of the Functionsuα(X) and vα(X)

This subsection is devoted to the estimates of the number of zeros for the functions in question. By a strictly monotone change of variable(3.38)y=∫bXdηPα(η),b=const∈ℝ, Equation (3.21) can be reduced to the equation(3.39)d2𝒲dy2+Q̃α(y)𝒲=0,
where 𝒲(y)=w(X(y)) and Q̃α(y)=Pα(X(y))Qα(X(y)). Therefore, any nontrivial solution of (3.39) can have not more than a finite number of zeros on any bounded interval ([24], page 323).Theorem 3.7.
LetNα be the number of zeros of the function vα(X) on the interval [0,X0], where X0<∞. Then the following inequality holds:
(3.40)Iα-1≤Nα≤Iα+1,
where
(3.41)Iα=arctaniζ-(2+α,1/2+iX0/π)ζ+(2+α,1/2+iX0/π).Proof.
Since we choseC=1 in (3.22),
(3.42)Pα(X)W[uα(X),vα(X)]=1.
Therefore,
(3.43)Θα′(X)=vα′uα-uα′vαvα2>0.
Therefore, by Theorem 2.1 of Section 2, for the interval [0,X0] we have the estimate
(3.44)-1+Iα≤Nα≤1+Iα,
where
(3.45)Iα=∫0X0Wα(η)dηvα2(η)+uα2(η)=∫0X0Θα′(η)dη=arctaniζ-(2+α,1/2+iX0/π)ζ+(2+α,1/2+iX0/π).
Here we have used the fact that ζ-(s,1/2)=0 for s>1.Theorem 3.8.
The zeros ofuα(X) separate and are separated by those of vα(X).Proof.
This follows from Sturm's Separation Theorem (see [24], page 335).


## 3.1. Properties of the Functionsuα and vα

In this subsection, we collect the properties of the functionsuα and vα. Some of them were established in [15] and some are given for the first time as follows.Properties of the Functionsuα and vα
(1)
The functionsuα(X) and vα(X) satisfy the functional differential equations(3.6)uα′(X)=-vα+1(X),vα′(X)=uα+1(X),
where α>-1, X∈ℝ, and the prime denotes differentiation with respect to X. This follows from (3.3) and the relation d/dX=H∘D.(2)
The functionsuα(X) are even and the functions vα(X) are odd on ℝ (see [15] and Figures 1–4).(3)
The functionuα(X) is periodic with the period 𝒳 = iπ. It follows from the periodicity of ζ+(s,a) with the unit period.(4)
For allα>0 (see [15]),(3.7)∫-∞∞uα(X)dX=0,∫-∞∞vα(X)dX=0.
These properties are reflected on the graphs (see Figures 3 and 4).(5)
For allα>0 and n∈ℕ,(3.8)∂αn∫-∞∞uα(X)dX=0,∂αn∫-∞∞vα(X)dX=0.
These relations follow from the differentiation of the identities in (3.7).(6)
For allα>0 and c∈ℝ,(3.9)P.V.∫-∞∞uα(X-c2X)dX=0,P.V.∫-∞∞vα(X-c2X)dX=0.
Moreover,(3.10)P.V.∫-∞∞uα(X-∑j=1∞bjX-cj)dX=0,P.V.∫-∞∞vα(X-∑j=1∞bjX-cj)dX=0,
where bj is any sequence of positive constants, cj are any real constants, and the series converges. These relations follow from (3.7) and Theorem 2.2.(7)
The functional sequence of Riesz potentials{uα(X)} converges pointwise to the soliton u0(X), and the functional sequence {vα(X)} converges pointwise to the conjugate soliton v0(X) for α→0+ (see [15]). Notice that(3.11)∫-∞∞u0(X)dX=4,P.V.∫-∞∞v0(X)dX=0
(see Figures 1 and 2). Here the conjugate soliton v0(X) is given by
(3.12)v0(X)=2iπ2[ψ′(12+iXπ)-ψ′(12-iXπ)].
Equation (3.12) can be recovered from (3.5) with α=0 thanks to (2.13).Figure 2
Graph of the conjugate solitonv0(X).Figure 3
Graph of the fractional derivativeu3.8(X).Figure 4
Graph of the conjugate fractional derivativev3.8(X).Remark 3.2.
The conjugate soliton (3.12) is an algebraic solitary wave for extended KdV:
(3.13)vt+vxxx+3[2v·Hv-H(v2)]x=0,
obtained by applying the Hilbert transform to (1.1) and setting v=-Hu. The term “algebraic solitary wave” is explained by the fact that v0(X) has a 1/X decay for large X. More precisely (see [15]),
(3.14)v0(X)~2π·1X+O(1X2)for|X|→∞.(8)
Forα≥0, the functions uα and vα are the elements of L2(ℝ). Moreover, they are orthogonal in the principal value sense, namely, for all α1,α2≥0,(3.15)P.V.∫-∞∞uα1(X)vα2(X)dX=0.
Equations (3.12) and (3.14) imply that v0∈L2(ℝ). The fact that uα and vα with α>0 are the elements of L2(ℝ) follows from their asymptotics obtained in [15], namely,(3.16)uα(X)~4Γ(1+α)π2+αcos[(1+α)arctan(2X/π)][1/4+(X/π)2](1+α)/2+O(1|X|2+α),vα(X)~4Γ(2+α)π2+α2sin[(1+α)arctan(2X/π)](1+α)[1/4+(X/π)2](1+α)/2+O(1|X|2+α).
Orthogonality of uα and vα follows from the fact that all uα(X) with α≥0 are even functions and all vα(X) with α≥0 are odd functions of X (see Property  2 and Figures 1–4).(9)
At the pointX=0,one has for all α>-1(3.17)uα(0)=4Γ(2+α)π2+α(2α+2-1)ζ(2+α)>0,(3.18)vα(0)=0,
where ζ(s) is the Riemann Zeta function (see [15]).


## 3.2. Sturm-Liouville Problem

It is convenient to representwα(X) in the exponential form, namely,(3.19)wα(X)=Rα(X)exp[iΘα(X)],
where(3.20)Rα(X)=[uα(X)]2+[vα(X)]2,Θα(X)={Φα(X),uα(X)>0,Φα(X)+π,uα(X)<0,vα(X)≥0,Φα(X)-π,uα(X)<0,vα(X)<0,Φα(X)=arctanvα(X)uα(X).
Exponential representation (3.19) allows one to deduce the boundary value problem for the functions wα(X). It turns out that for all α>-1 the functions wα(X) solve the equation(3.21)ddX(Pα(X)dwdX)-Qα(X)w+ρα(X)w=0,X∈ℝ,
where(3.22)Pα(X)=CW[uα(X),vα(X)],C=const,Qα(X)=[Pα(X)Rα′(X)]′Rα(X),ρα(X)=[Θα′(X)]2,(3.23)Θα′(X)=W[uα(X),vα(X)]Rα2(X).
Here W[uα(X),vα(X)] is the Wronskian of uα(X) and vα(X). Below we shall use a shorter notation Wα(X)=W[uα(X),vα(X)].Taking into account the behavior ofwα(X) for large X (see [15]), we can restate the obtained results in another form. Indeed, wα(X) are solutions of the Sturm-Liouville problem(3.24)-ddX(Pα(X)dwdX)+Qα(X)w=λρα(X)w,X∈ℝ,(3.25)limX→±∞w(X)=0,
corresponding to λ=1. Without loss of generality, we can choose the constant in (3.22) to be positive. In the next subsection, we shall prove that Wα(X)>0. It implies that Pα(X)>0 and ρα(X)>0 in (3.22).It follows from (3.7) that for α>0 the functions wα(X) also satisfy the zero mean condition(3.26)∫-∞∞wα(X)dX=0.
This reflects the oscillatory behavior of wα(X) for α>0 (see Figures 3 and 4).The graph of the arctangent functionΦα(X) is shown in Figure 5. Φα(X) conveniently serves as a zero counter for both functions: uα(X) and vα(X). It possesses zeros at the points where vα(X) has zeros and has jumps at the points where uα(X) has zeros.Figure 5
Graph of the arctangent functionΦ3.8(X).Remark 3.3.
Observe that a general solution of (3.24) can be written in the form
(3.27)w(g)(X)=C1uα(X)+C2uα(X)∫0XWα(y)uα2(y)dy,
where C1, C2=const.Remark 3.4.
We would like to point out that the differential equation given by (3.21) can be factored in the following way (see [26, page 269]):
(3.28)ddX(Pα(X)dwdX)-Qα(X)w+ρα(X)w=Wα(X)uα(X)ddX[uα2(X)Wα(X)ddX(w(X)uα(X))].


## 3.3. Wronskian ofuα and vα

Lemma 3.5.
The following properties hold for the WronskianWα(X) for α>-1 and all x∈ℝ:
(3.29)Wα(-X)=Wα(X),Wα′(0)=0,(3.30)Wα(X)>0.Proof.
We start with (3.29). Taking into account relations in (3.3) and the fact that ∂X=H∘D, we can write
(3.31)Wα(X)=uα(X)vα′(X)-vα(X)uα′(X)=uα(X)uα+1(X)+vα(X)vα+1(X).
Since the functions uα(X) are even and vα(X) are odd with respect to X∈ℝ, Wα(X) is even for X∈ℝ. Differentiation of (3.31) with the help of (3.6) and (3.18) yields Wα′(0)=0. Next, we turn to the proof of (3.30). Since the functions uα(X) and vα(X) are linearly independent solutions of the equation (3.24), Wα(X)≠0 for all α>-1 and x∈ℝ. It remains to establish the sign of the Wronskian. In view of (3.17) and (3.18),
(3.32)Wα(0)=uα(0)uα+1(0)=16Γ(2+α)Γ(3+α)π5+2α(22+α-1)(23+α-1)ζ(2+α)ζ(3+α)>0.
By Abel's formula, for all X∈ℝ,
(3.33)Wα(X)=Wα(0)exp[-∫0XPα′(η)Pα(η)dη],
where the function
(3.34)Pα′(X)Pα(X)=-i(s+1)π·ζ-(s,z)ζ+(s+2,z)-ζ+(s,z)ζ-(s+2,z)ζ+(s,z)ζ+(s+1,z)-ζ-(s,z)ζ-(s+1,z),s=2+α,z=1/2+iX/π,
is continuous on ℝ. After some simplification, we have that
(3.35)Wα(X)=Wα(0)exp{-s+1πℑ[ζ(s,z)]ℜ[ζ(s+2,z)]-ℜ[ζ(s,z)]ℑ[ζ(s+2,z)]ℜ[ζ(s,z)]ℜ[ζ(s+1,z)]+ℑ[ζ(s,z)]ℑ[ζ(s+1,z)]}.
This representation yields (3.30). The lemma is proved.Three-dimensional graph ofWα(X) is given in Figure 6.Figure 6
Three-dimensional graph of the WronskianWα(X).Remark 3.6.
What does the positivity of the Wronskian yield for the soliton and its conjugate? Forα=0, (3.30) simplifies to read
(3.36)W0(X)=2sech2(X)[v0′(X)+tanhX·v0(X)]>0.
We notice that v0(X)>0 for X>0, v0(-X)=-v0(X) for X∈ℝ, and v0(0)=0 (see Figure 2). Integrating the inequality v0′(X)/v0(X)+tanhX>0 over the interval [ε,X] for X≥ε>0 and the inequality v0′(X)/v0(X)+tanhX<0 over [-ε,X] for X≤-ε<0 yields the estimate
(3.37)|v0(X)|>|v0(ε)|coshεcoshXfor|X|≥ε>0.


## 3.4. Zeros of the Functionsuα(X) and vα(X)

This subsection is devoted to the estimates of the number of zeros for the functions in question. By a strictly monotone change of variable(3.38)y=∫bXdηPα(η),b=const∈ℝ, Equation (3.21) can be reduced to the equation(3.39)d2𝒲dy2+Q̃α(y)𝒲=0,
where 𝒲(y)=w(X(y)) and Q̃α(y)=Pα(X(y))Qα(X(y)). Therefore, any nontrivial solution of (3.39) can have not more than a finite number of zeros on any bounded interval ([24], page 323).Theorem 3.7.
LetNα be the number of zeros of the function vα(X) on the interval [0,X0], where X0<∞. Then the following inequality holds:
(3.40)Iα-1≤Nα≤Iα+1,
where
(3.41)Iα=arctaniζ-(2+α,1/2+iX0/π)ζ+(2+α,1/2+iX0/π).Proof.
Since we choseC=1 in (3.22),
(3.42)Pα(X)W[uα(X),vα(X)]=1.
Therefore,
(3.43)Θα′(X)=vα′uα-uα′vαvα2>0.
Therefore, by Theorem 2.1 of Section 2, for the interval [0,X0] we have the estimate
(3.44)-1+Iα≤Nα≤1+Iα,
where
(3.45)Iα=∫0X0Wα(η)dηvα2(η)+uα2(η)=∫0X0Θα′(η)dη=arctaniζ-(2+α,1/2+iX0/π)ζ+(2+α,1/2+iX0/π).
Here we have used the fact that ζ-(s,1/2)=0 for s>1.Theorem 3.8.
The zeros ofuα(X) separate and are separated by those of vα(X).Proof.
This follows from Sturm's Separation Theorem (see [24], page 335).


## 4. Inequality for Hurwitz Zeta Functions

Here we discuss a new inequality for the Hurwitz Zeta functions which follows from Lemma3.5. The next statement is a corollary of this lemma.Corollary 4.1.
Fors>1 and z=1/2+iX/π with X∈ℝ, the following inequality holds:
(4.1)𝒦(s,z)=ζ(s,z̅)ζ(s+1,z)+ζ(s,z)ζ(s+1,z̅)>0.
This inequality can also be written in another form:
(4.2)𝒦(s,z)=ℜ{ζ(s,z)}ℜ{ζ(s+1,z)}+ℑ{ζ(s,z)}ℑ{ζ(s+1,z)}>0.Proof.
Dropping positive terms in front of the full-range Hurwitz Zeta functions in (3.4) and (3.5) and using (3.30) lead to
(4.3)ζ+(s,z̅)ζ+(s+1,z)-ζ-(s,z)ζ-(s+1,z̅)=2[ζ(s,z̅)ζ(s+1,z)+ζ(s,z)ζ(s+1,z̅)]=4[ℜ{ζ(s,z)}ℜ{ζ(s+1,z)}+ℑ{ζ(s,z)}ℑ{ζ(s+1,z)}]>0.Remark 4.2.
Settingζ(s,z)=rs(z)eiφs(z) and ζ(s+1,z)=rs+1eiφs+1(z), we can rewrite (4.1) in the form
(4.4)𝒦=2rsrs+1{exp[i(φs+1-φs)]+exp[-i(φs+1-φs)]}=4rsrs+1cos(φs+1-φs).
It shows that, for z=1/2+iX/π, X∈ℝ,
(4.5)cos(φs+1(z)-φs(z))>0.
Introduce the scalar product for the complex-valued functions f=f1+if2 and g=g1+ig2 by the formula
(4.6)〈f·g〉=f·g̅=f1f2+f2g2+i(f2g1-f1g2).
Then for s>1 and z=1/2+iX/π,
(4.7)𝒦(s,z)=2ℜ{〈ζ(s,z)·ζ(s+1,z)〉}>0.Remark 4.3.
The proof of (4.1) becomes quite difficult when one approaches it from the point of view of special functions. For example, the use of integral representations (2.21) and (2.22) yields
(4.8)𝒦=12Γ(s)Γ(s+1)∬0∞ts-1τscos[X/π(t-τ)]sinh(t/2)sinh(τ/2)dtdτ.
The change of variables ξ=(t-τ)/2,η=(t+τ)/2 leads to
(4.9)𝒦=2Γ(s)Γ(s+1)∫0∞cos(2Xπξ)dξ∫-ξξ(ξ+η)s-1(η-ξ)scoshη-coshξdη.
It is not clear at all that the integral (4.9) is positive for all s>1 and X∈ℝ. However, (4.1) shows that it is. Multiplication of the series representations (2.16) and (2.17) does not make the proof any easier.



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