Coexistence of the solitary and periodic waves in convecting shallow water fluid

Nonlinear Analysis: Real World Applications 53 (2020) 103067

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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa

Coexistence of the solitary and periodic waves in convecting shallow water fluid Xianbo Sun a,c , Wentao Huang b ,∗, Junning Cai a a

Department of Applied Mathematics, Guangxi University of Finance and Economics, Nanning 530003, China b Guangxi Colleges and Universities Key Laboratory of Unmanned Aerial Vehicle Telemetry, Guilin University of Aerospace Technology, Guilin 541004, China c Department of Applied Mathematics, Western University, London, Ontario, Canada N6A 5B7

article

info

abstract

Article history: Received 12 May 2019 Received in revised form 5 November 2019 Accepted 6 November 2019 Available online xxxx Keywords: Periodic traveling wave Solitary wave Coexistence

The existence of a solitary wave for the shallow water model in convecting circumstance was established in previous works. It is still unknown that whether there exist periodic waves. In this paper, we prove that the models possess periodic waves with a fixed range of wave speed. The amplitude and wave speed are explicitly given. Moreover, the coexistence of the solitary wave and one periodic wave is established. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction The KdV equation plays a central role in modeling the motion of shallow water. In real world problem, certain relatively weak influences or perturbations are unavoidable. The following KdV equation with additional three dissipative terms was derived in [1] to model the convecting shallow water fluid, ut + λ1 uux + λ3 uxxx + ϵ(λ2 uxx + λ4 uxxxx + λ5 (uux )x ) = 0,

(1)

where ϵ is very small and appears with λ2 , λ4 and λ5 in the original model, λi s are different constants in different circumstances. Eq. (1) describes the water motion in a wide range of dissipative circumstances. We have known that Eq. (1) can govern the wave evolution of a shallow liquid layer when being exposed to open air and heated from air side [2–5], or involving a transfer of the surface-active material across the open surface [6,7], or being heated from below and experiencing shear to model internal waves [8,9]. The parameters λi s have a wide range choices, from negative values to positive values and can almost be treated as free parameters, see [10,11]. For Marangoni–B´enard convection in shallow fluid layers as ∗ Corresponding author. E-mail addresses: [email protected] (X. Sun), [email protected] (W. Huang), [email protected] (J. Cai).

https://doi.org/10.1016/j.nonrwa.2019.103067 1468-1218/© 2019 Elsevier Ltd. All rights reserved.

2

X. Sun, W. Huang and J. Cai / Nonlinear Analysis: Real World Applications 53 (2020) 103067

Boussinesq description, λ4 and λ5 are strictly positive in the absence of buoyancy λ1 . λ3 is positive or negative depending on the value of the air–liquid surface tension, and λ2 is negative when the diffusion is smaller than the instability threshold and otherwise it is positive. When buoyancy effect is added, all of the coefficients may change signs, see [10,11] and references therein for details. When the layer is relatively long and the surface tension is relatively weak, Topper and Kawahara [12] derived Eq. (1) with the coefficients approximated as λ1 = λ2 = λ3 = λ4 = 1 and λ5 = 0, and the existence and uniqueness of the periodic wave were studied in [13,14]. In Aspe and Depassier’s study [15] the parameters were approximated as follows: 3 (10 + σG), λ1 = 2σG 682σ 3 G + 717σ , λ4 = 2079

√ σR2 σ G 1 34 λ2 = , λ3 = ( + σ), 15 2 3 21 8 λ5 = √ , G

(2)

ϵ is a small parameter such that the excess of the Rayleigh number above its critical values is given by εR2 , σ is the P randtl number, G is the Galileo number. Eq. (1) includes dissipative-perturbation of KdV–Burger equation, KS equation with one or more parameters vanishes, and they were studied in many works such as [16–19]. Another interesting problem is that how a strongly nonlinear buoyancy affects the wave motion. That is the case when the buoyancy is u2 ux , see [20,21], ut + λ1 u2 ux + λ3 uxxx + ϵ(λ2 uxx + λ4 uxxxx + λ5 (uux )x ) = 0.

(3)

For Eq. (1), the existence of the solitary wave was rigorously proved in [10,22] via dynamical system method. In [10], the PDE model was reduced into a three-dimensional ODE system and then to analyze the intersection of the related stable manifold and unstable manifold via an auxiliary model, while the PDE was reduced into a two-dimensional ODE system via singular perturbation theory in [22], and the existence of the solitary wave was established by computing the Melnikov integral along the homoclinic loop. However, the existence of periodic wave train is still unverified in both papers. A numerical evidence of periodic wave was presented in [11] as a subsequent work of [10]. For Eq. (3), the existence of the solitary wave was established in [20,21] by using the same method as that of [22]. In this paper, we mainly prove there exist periodic waves for both equations (1) and (3). In particular, we show their difference on the number of periodic waves and coexistence with the solitary wave. Moreover, we finally apply our results to Aspe’s coefficients given in (2) to estimate the wave speed and amplitude. We organize the paper as follows: the reduction and construction of Melnikov function are given in Section 2. We mainly pay attention to Eq. (3), and the analysis for Eq. (1) is similar as that for (3). We explain why the problem on the existence of periodic waves is reduced to the zeros of the Melnikov functions; In Section 3, we mainly prove that the Melnikov function for (3) has two zeros and a unique zero for (1). A further analysis of Melnikov function for Eq. (3) is conducted to prove the coexistence of a solitary wave and a periodic wave in Section 4. In Section 5, we apply our results to the Apse’s case, to detect the periodic wave and estimate the wave speed and amplitude. Finally, a conclusion is drawn. 2. System reduction In this section, we reduce the system and construct the Melnikov functions on the Hamiltonian structure. First, substituting the wave profile z = x − ct into Eq. (3) and direct integration yields − cu +

λ1 3 d2 u du du d3 u u + λ3 2 + ϵ(λ2 + λ5 u + λ4 3 ) = 0, 3 dz dz dz dz

(4)

X. Sun, W. Huang and J. Cai / Nonlinear Analysis: Real World Applications 53 (2020) 103067

where we assume c > 0 and the boundary conditions are imposed as can be written as a system of first order equations,

du d2 u d3 u dz , dz 2 , dz 3

du = v, dz dv = w, dz dw λ1 ϵλ4 = cu − u3 − λ3 w − ϵλ5 uv − ϵλ2 v. dz 3

3

→ 0 as z → ∞. Eq. (4)

(5)

Introducing the new scale z = ϵζ, the system becomes du = ϵv, dζ dv = ϵw, dζ dw λ1 λ4 = cu − u3 − λ3 w − ϵλ5 uv − ϵλ2 v. dζ 3

(6)

Systems (5) and (6) are equivalent for ϵ > 0. When ϵ → 0 in (5) and (6), we have the following systems: du = v, dz dv = w, dz 0 = cu − and

(7) λ1 3 u − λ3 w, 3

du = 0, dζ dv = 0, dζ dw λ1 = cu − u3 − λ3 w. dζ 3

(8)

Thus, the flow of system (7) is confined to the set M0 = {(u, v, w) ∈ R3 : cu −

λ1 3 u − λ3 w = 0} 3

which is the equilibrium set of (8). System (7) is referred to as the slow system, since the time-scale z is slow, and (8) is referred to as the fast system, since the time-scale ζ is fast and we call M0 the slow critical manifold. If each point on M0 is the hyperbolic equilibrium of the fast system (8), then we call M0 is normally hyperbolic. In classical geometric-singular perturbation theory, Fenichel’s criteria [23] assures the persistence of normally hyperbolic M0 as Mϵ for any sufficiently small ϵ−perturbation, while Mϵ approaches M0 in Hausdorf distance as ϵ approaches 0. Then, the flows of (5) can be restricted to Mϵ , and one can detect the solitary wave of (3) by tracking the homoclinic loop of the ODEs on the invariant manifold Mϵ . One direct method to determine whether M0 is a normally hyperbolic manifold is to check if the linearization of the fast system (6), restricted to M0 , has exactly dim M0 eigenvalues with zero real part, see [23]. The linearization of the fast system (6), restricted to M0 , is given by ⎡ ⎤ 0 0 0 ⎣ 0 0 0 ⎦, c − λ1 u 0 −λ3

4

X. Sun, W. Huang and J. Cai / Nonlinear Analysis: Real World Applications 53 (2020) 103067

the eigenvalues of which are 0, 0, −λ3 . Thus, M0 is normally hyperbolic. Therefore, the critical manifold M0 is persistent as a two-dimensional manifold Mϵ for any sufficiently small ϵ. We assume that Mϵ is governed by λ1 1 (cu − u3 ) + ϵΘ(u, v, ϵ), (9) w= λ3 3 and 1 λ1 Mϵ = {(u, v, w) ∈ R3 : w = (cu − u3 ) + ϵΘ(u, v, ϵ)}, λ3 3 where the function Θ(u, v, ϵ) has the form, Θ(u, v, ϵ) = Θ0 (u, v) +

∞ ∑

Θi ϵi .

(10)

i=1

Substituting (10) into the slow system (5), we have ∂Θ 1 1 λ1 ∂Θ v+ϵ ( (cu − u3 ) + ϵΘ) + (c − λ1 u2 )v] ∂u ∂v λ3 3 λ3 1 λ1 λ1 =cu − u3 − λ3 ( (cu − u3 ) + ϵΘ(u, v, ϵ)) − ϵλ5 uv − ϵλ2 v. 3 λ3 3 ϵλ4 [ϵ

Then, comparing the coefficient of ϵ on both sides gives Θ0 (u, v) =

1 (−(λ3 λ2 + λ4 c) − λ3 λ5 u + λ4 λ1 u2 )v. λ23

(11)

Then, system (5) reduced to Mϵ has the form, du = v, dz (12) dv 1 ϵ λ1 = (cu − u3 ) + 2 (−(λ2 λ3 + λ4 c) − λ3 λ5 u + λ1 λ4 u2 )v + O(ϵ2 ). dz λ3 3 λ3 √ √ √ λ3 3c 3 4k , β = c and k = Introducing u = β1 u ˜, v = β2 v˜, z = kξ and ˜ ϵ = ϵ3cλ with β = 2 1 2 λ1 λ1 λ3 c , we have λ3 the new system by dropping the tildes of the symbols u ˜, v˜ and ˜ ϵ, du = v, dz dv = u(1 − u2 ) + ϵ(α0 + α1 u + u2 )v, dz where

λ2 λ3 1 λ3 λ5 √ + ), α1 = − 3cλ1 . 3cλ4 3 3cλ1 λ4 is a Hamiltonian system with the Hamiltonian function α0 = −(

System (13)ϵ=0

H(u, v) =

(13)

(14)

v2 u2 u4 − + . 2 2 4

Therefore, system (13) is a near-Hamiltonian system or a perturbed Hamiltonian system. The portraits of H(u, v) is depicted in Fig. 1. H(u, v) = 0 defines a double homoclinic loop (left and right branches) corresponding to the dark and light solitary wave of the unperturbed mKdV equation (3). In this paper, we only consider the right side because of the shallow water layer. (1, 0) is a center satisfying H(1, 0) = − 14 . (13)ϵ=0 has a family of periodic orbits and they correspond to the bounded periodic waves of the shallow water model (3)ϵ=0 . For any h ∈ (− 41 , 0), H(u, v) = h defines a periodic orbit Γh of (13)ϵ=0 . Let (ρ(h), 0) denote the intersection point of Γh and the positive u-axis, T the period of Γh . Further, let Γh,ϵ be the positive orbit of

X. Sun, W. Huang and J. Cai / Nonlinear Analysis: Real World Applications 53 (2020) 103067

5

Fig. 1. The portraits of H(u, v).

(13) starting from the point (ρ(h), 0) at time z = 0, and (π(h, ϵ), 0) the first intersection point of Γh,ϵ with the positive u-axis at time z = z(ϵ). Then, the difference between the two points is given by ∫ dH = H((π(h, ϵ), 0)) − H((ρ(h), 0)) Γh,ϵ ∫ = (u − u3 )du + vdv Γh,ϵ ∫ z(ϵ) = (u − u3 )v + v(u − u3 ) + ϵ(α0 + α1 u + α2 u2 )v 2 dz 0 ∫ z(ϵ) = ϵ(α0 + α1 u + α2 u2 )v 2 dz 0 ∫ z(ϵ) = ϵ (α0 + α1 u + α2 u2 )v 2 dz 0

≜ ϵF (h, ϵ). By continuousness theorem, we have lim Γh,ϵ = Γh ,

lim π(h, ϵ) = ρ(h),

ϵ→0

and thus,

ϵ→0

lim z(ϵ) = T,

ϵ→0

T

∫

ϵ(α0 + α1 u + α2 u2 )v 2 dτ + O(ϵ),

F (h, ϵ) = ∮0 =

(α0 + α1 u + α2 u2 )vdu + O(ϵ)

(15)

Γh

≜ M(h) + O(ϵ), where M(h) is called Melnikov function, given by ∮ M(h) = (α0 + α1 u + α2 u2 )vdu.

(16)

Γh

Similarly, for the perturbed KdV equation (1), we take u′ (τ ) = v and follow the above procedure to obtain the following regular perturbation problem du = v, dτ dv = u − u2 + ϵ(a0 + u)v, dτ where a0 =

λ1 (λ2 λ3 + λ4 c) . 2c(λ3 λ5 − λ1 λ4 )

(17)

(18)

6

X. Sun, W. Huang and J. Cai / Nonlinear Analysis: Real World Applications 53 (2020) 103067

Fig. 2. The portraits of H ∗ (u, v).

System (17) is a near-Hamiltonian system with the Hamiltonian v2 u2 u3 − − . 2 2 3 The portraits of H ∗ (u, v) is shown as in Fig. 2. H ∗ (u, v) = 0 defines the homoclinic loop Γ0∗ , and a family of periodic orbits Γh∗ : {(u, v)|H ∗ (u, v) = h ∈ (− 16 , 0)} surrounding the center (1, 0) with H ∗ (1, 0) = − 61 . ∗ Let (ρ∗ (h), 0) be the intersection point of Γh∗ and the positive u-axis, T the period of Γh∗ , Γh,ϵ the positive ∗ ∗ orbit of (17) starting from the point (ρ (h), 0) at time τ = 0, and (π (h, ϵ), 0) the first intersection point of ∗ Γh,ϵ with the positive u-axis at time τ = τ ∗ (ϵ). Then, the difference between the two points (ρ∗ (h), 0) and ∗ (π (h, ϵ), 0) can be expressed as ∫ ∗ ∗ ∗ ∗ H (π (h, ϵ), 0) − H (ρ (h), 0) = dH ∗ ∗ Γh,ϵ ∫ (19) = ϵ (a0 + u)vdu + O(ϵ) ≜ ϵM∗ (h) + O(ϵ2 ), H ∗ (u, v) =

Γh∗

where the Melnikov function M∗ (h) is given by ∫

∗

M (h) = Γh∗

(a0 + u)vdu.

(20)

We note that Mansour [20–22] obtained the slow manifolds Mϵ for the models (1) and (3) and obtained twodimensional ODE systems on the Manifolds without re-scaling. They only computed the Melnikov integral along the homoclinic loop to verify the existence of the solitary wave. In our work, we re-scale the ODE systems to obtain much simpler ones, and then extend the Melnikov integral on a larger range of Hamiltonian structures. We get the Melnikov functions (16) and (20). To investigate the existence of periodic waves for the two perturbation problems, we need study the zeros of the functions H(π(h, ϵ), 0)−H(ρ(h), 0) and H ∗ (π ∗ (h, ϵ), 0)−H ∗ (ρ∗ (h), 0) and their distributions. It follows from (15) and (19) that it is suffice to study the zeros of the Melnikov functions M(h) and M∗ (h). 3. Existence of periodic waves of the models (1) and (3) In the section, we study the periodic waves of perturbed mKdV (3). Based on the discussion in the previous sections, we need only study the Melnikov function M(h). Let, ∮ Jn (h) = un vdu, for n = 0, 1, 2. (21) Γh

Then, M(h) = α0 J0 (h) + α1 J1 (h) + J2 (h).

X. Sun, W. Huang and J. Cai / Nonlinear Analysis: Real World Applications 53 (2020) 103067

7

By results in [24], we have For 0 < −h ≪ 1,

Lemma 1.

M(h) = c0 + c1 h ln(−h) + c2 h + O(h2 ln(h)), and

√ 1 (80 α0 + 15 2πα1 + 64), 60 c1 = −α0 , √ c2 = 2πα1 + 4. c0 =

(22)

At another endpoint of the annulus, we have the following result. 1 4

For 0 < h +

Lemma 2.

≪ 1, M(h) = d0 h + d1 h2 + d2 h3 + O(h4 )

with the coefficients:

√

2π (α0 + α1 + 1) , √ 2π d1 = (3 α1 − 1) , 8 √ 5 π 2 (7 α0 − 1) . d2 = 64 d0 =

(23)

√

Proof . Let u =

2 2 r cos θ

+ 1, v = r sin θ. Then, H − h = 0 becomes √ 3 3 4 2r (cos (θ)) r2 r4 (cos (θ)) F(r, ρ) ≜ + + − ρ2 = 0, 2 16 4

1

where ρ = (h + 41 ) 2 . Applying the implicit function theorem to F(r, ρ) at (r, ρ) = (0, 0), we can show that there exist a smooth function r = χ(ρ) and a small positive number δ, 0 < ρ < δ ≪ 1 such that F(χ(ρ), ρ) = 0, and χ(ρ) can be expanded as ( ) √ 4 2 √ 3 2 (cos (θ)) 5 (cos (θ)) − 1 ρ3 √ 2 (cos (θ)) ρ2 χ(ρ) = 2ρ − + 2 8 (24) ( ) √ 7 2 4 2 (cos (θ)) 8 (cos (θ)) − 3 ρ − + O(ρ5 ). 8 Therefore, ∮ ∫∫ ∫ 2π ∫ χ(ρ) ( √ )n+1 2r n n Jn (h) = u vdu = u dudv = dθ (cos θ + 1)n dr. (25) 2 Γh intΓh 0 0 1

Noticing ρ = (h + 41 ) 2 and substituting (24) into (25) yields √ √ √ 1 3 2π 1 2 35 2π 1 1 J0 (h) = 2π(h + ) + (h + ) + (h + )3 + O((h + )4 ), 4 8 4 64 4 4 √ √ 1 23008029 2π 1 4 1 5 J1 (h) = 2π(h + ) − (h + ) + O((h + ) ), 4 33554432 4 4 √ √ √ 1 2π 1 2 5 2 1 3 1 J2 (h) = 2π(h + ) − (h + ) − (h + ) + O((h + )4 ), 4 8 4 64 4 4 for 0 < h +

1 4

≪ 1. This completes the proof of Lemma 2.

□

(26)

X. Sun, W. Huang and J. Cai / Nonlinear Analysis: Real World Applications 53 (2020) 103067

8

Similarly, we have the asymptotic expansions of M∗ (h). Lemma 3.

M∗ (h) has the following forms, M∗ (h) =

for 0 < −h ≪ 1, and 0 < h +

1 6

36 + 42a0 − a0 h ln(−h) + O(h2 ln(−h)) 35

≪ 1,

1 (5a0 − 1)π 1 1 (h + )2 + O((h + )3 ). M∗ (h) = 2(a0 + 1)π(h + ) + 6 6 6 6 Using the expansions of M(h), we have the following two theorems. Theorem 4. √

2 (A) There exist some (α0 , α1 ) near (0, − 32 15π ) such that the perturbed model (3) has two periodic waves emerging in an sufficiently small neighborhood of the homoclinic loop Γ0 . (B) There exist some (α0 , α1 ) near (− 34 , 13 ) such that the perturbed model (3) has two small amplitude periodic waves emerging in an sufficiently small neighborhood of the center Γ− 1 . 4

Proof . It is only to prove that M(h) has two zeros near h = 0 and h = − 41 to prove assertions A and B, respectively. We only prove the first assertion and the second one can√be proved similarly. 2 4 ∗ ∗ The equation {c0 = c1 = 0} has the root (α0∗ , α1∗ ) = (0, − 32 15π ) and c2 (α0 , α1 ) = − 15 . Further, ∂(c0 ,c1 ) 16 ∗ ∗ ∂(α0 ,α1 ) (α0 , α1 ) = − 15 . Then, c0 and c1 can be treated as free parameters in a sufficiently small neighborhood U of (α0∗ , α1∗ ). We suppose the sufficiently small radius is ε. We take a subset U ∗ ⊂ U satisfying U ∗ = {(α0 , α1 )|0 < −c0 ≪ c1 ≪

4 }. 15

4 Choosing (α0 , α1 ) in U ∗ , c2 = − 15 +O(ε) and M(h) has two zeros near h = 0. This completes the proof.

□

Similarly, we have Theorem 5. The perturbed model (1) has a larger amplitude periodic wave emerging in an sufficiently small neighborhood of the homoclinic loop when taking a0 = − 76 − δ1 with positive δ1 sufficiently small; and there exist some a0 = −1 + δ2 with positive δ2 sufficiently small such that the perturbed model (1) has a very small amplitude periodic wave emerging in an sufficiently small neighborhood of the center. Remark. When choosing δ1 = 0, M∗ (0) = 0, implying the Melnikov integral along the homoclinic ∗ loop vanishes, which indicating the existence of the solitary wave Γ0,ϵ , a very little deformation from Γ0∗ . System (17) is the reduced near-Hamiltonian system in studying limit cycles of the classical Bogdanov– Takens system x˙ = y, y˙ = β1 + β2 x + x2 + σxy, and the global existence of zeros of M∗ (h) has been proved in [25], see the introduction of Bogdanov–Takens bifurcation on the Scholarpedia website. The uniqueness of the zero of M∗ (h) has been recently proved in a new paper [26]. Therefore, we have the following result, while we provide a new and simpler proof by another technique. Theorem 6. The shallow water model (1) possesses a unique periodic wave if and only if a0 is located in the interval (−1, − 67 ). The amplitude of the persistent periodic wave is increasing with respect to a0 .

X. Sun, W. Huang and J. Cai / Nonlinear Analysis: Real World Applications 53 (2020) 103067

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Proof . We write

m1 (h) + a0 ), (27) m0 (h) ∮ ∮ ∮ ∮ where m0 (h) = Γ ∗ vdv, m1 (h) = Γ ∗ uvdv. First, m0 (h) = Γ ∗ vdv = intΓ ∗ dudv > 0 by Green formula. M∗ (h) = m0 (h)(

Then,

m1 (h) m0 (h)

h

h

h

h

is well defined. In the remainder of the proof, we will show

Let A(u) = H ∗ (u, v) −

2

v 2

2

= − u2 +

3

u 3

m1 (h) m0 (h)

is monotone for h ∈ (− 61 , 0).

, and

li (u, z) =

ui zi − , for i = 0, 1, A′ (u) A′ (z)

where z is defined by q(u, z) = 2u2 + 2uz + 2z 2 − 3u − 3z satisfying A(u) − A(z) = d ∂ ∂u q(u, z) ∂ li (x, z) = li (u, z) − × li (u, z), i = 0, du ∂u ∂z q(u, z) ∂z ⏐ ⏐ l (u, z) l (u, z) 1 ⏐ 0 Straightforward computation shows that both l0 (u, z) and ⏐ ′ ⏐ l0 (u, z) l1′ (u, z) < u < 1, 1 < z <

3 2 }.

m1 (h) m0 (h)

Therefore, m1 (h) m0 (h)

u−z 6 q(u, z).

We have

1. ⏐ ⏐ ⏐ ⏐ do not vanish on {(u, z)|0 ⏐

is monotone for h ∈ (− 61 , 0) by the Chebyshev criterion [27].

decreases from (− 16 , 1) to (0, 67 ) monotonically. By (27), M∗ (h) has Combining Lemma 3, we have a zero at some h when choosing any a0 ∈ (−1, − 67 ), indicating a periodic wave of the model (1). Further, m1 (h) a0 = a0 (h) = − m increases monotonically, as well as the amplitudes of the periodic wave with respect 0 (h) 1 (h) −1 1 (h) to a0 , h(a0 ) = (− m , we have simulated − m m0 (h) ) m0 (h) in our case study, see Fig. 3(a) in Section 5. □

4. Coexistence of a solitary wave and a periodic wave of model (3) Lemma 7.

For h ∈ (− 41 , 0), J0′ (h) > 0 and J0 (h) > 0.

Proof . It is easy to obtain du = v

∮

1 J0 (− ) = lim 4 h→− 1 4

∮

J0′ (h) =

∮ Γh

Γh

vdτ = v

∫

T (h)

dτ = T (h) > 0, 0

where T (h) denotes the period of Γh . When h → − 14 , v → 0. Therefore, ∫

T

vdu = lim Γh

1 h→− 4

v 2 dτ = 0.

0

which, together with J ′ (0) > 0, implies J0 (h) > 0 for h ∈ (− 14 , 0).

□

It follows from Lemma 7 that the following ratio is well defined, X (h) =

α1 J1 (h) + J2 (h) . J0 (h)

(28)

Then, M(h) = J0 (h)(α0 + X (h)). We have the following result. Lemma 8.

The following hold,

(29)

X. Sun, W. Huang and J. Cai / Nonlinear Analysis: Real World Applications 53 (2020) 103067

10

m (h)

Fig. 3. (a) The simulation of the ratio − m1 (h) , showing it is monotonic; (b) The simulation of α0 (c), showing it is monotone and 0

insects the curves

m (0) − m1 (0) 0

m1 (− 1 )

and − m

6 1 0 (− 6

)

at c = 46.06859050 and c = 32.90613608, respectively; For any fixed h ∈ (− 61 , 0), the

m (h)

curve α0 (c) insects − m1 (h) at a unique point, indicating the existence of a unique zero of M∗ (h) for some c. (c) The wave speed 0

of the persistent periodic waves depends the amplitude, c(h) increases for h ∈ (− 16 , 0); (d) c(h(u)) decreases for u ∈ (0, 1), where h(u) = H ∗ (u, 0).

(i) lim X (h) = α1 + 1,

h→− 1 4

√ 3 2πα1 4 lim X (h) = + h→0 16 5

and (ii) lim X ′ (h) = −

1 h→− 4

3α1 1 − , 3 2

lim X ′ (h) = Sign(−

h→0

√ 3 2π α1 4 − )∞. 16 5

Proof . First, we have X ′ (h) =

(α1 J1′ (h) + J2′ (h))J0 (h) − (α1 J1 (h) + J2 (h))J0′ (h) . J02 (h)

X. Sun, W. Huang and J. Cai / Nonlinear Analysis: Real World Applications 53 (2020) 103067

11

The asymptotic expansions of Ji (h) and Ji′ (h) near h = 0 and h = − 41 for i = 0, 1, 2 can be easily derived from the asymptotic expansions of M(h) in Lemmas 1 and 2. Then, the limits of X (h) and X ′ (h) can be derived via straightforward computation. □ Further, we have the following results. Lemma 9. For h ∈ (− 41 , 0), the following hold. ∗ (i) X (h) decreases from (− 14 , α1 +1) to a minimum point (h , X (h)) and then increases to the right endpoint √ √ 3 2π α1 4 4 32 2 (0, 16 + 5 ) when α1 is located in the interval (− 3 , − 15π ). √ α1 (ii) X (h) increases monotonically from (− 41 , α1 + 1) to the right endpoint (0, 3 2π + 54 ) when α1 is 16 4 located in the interval (−∞, − 3 ). √ 3 2π α1 1 (iii) X (h) decreases monotonically from (− , α + 1) to the right endpoint (0, + 45 ) when α1 is 1 4 16 √ 32 2 located in the interval (− 15π , +∞). Proof . By (ii) of Lemma 8, we have 1 X ′ (− )X ′ (0) < 0, 4

√

2 ′ for α1 ∈ (− 34 , − 32 15π ) which implies X (h) has 2n + 1 zeros accounting multiplicities with n ≥ 0. If n ≥ 1, then there must exist a suitable α0 such that α0 +X (h) has three zeros on (− 41 , 0), which contradicts with the partial results that M(h) has at most 2 zeros on (− 14 , 0) in [28]. Hence, X ′ (h) has only one zero. Determining the signs of X ′ (h) on the two endpoints completes the proof of (i). When α1 ∈ (−∞, − 34 ), 1 X ′ (− ) > 0, X ′ (0) > 0, 4 ′ which implies X (h) has 2n zeros accounting multiplicities with n ≥ 0. If n ≥ 1, consider

1 X (− ) < X (0), 4 then there must exist a suitable α0 such that α0 + X (h) has three zeros on (− 41 , 0). It contradicts with the partial results that M(h) has at most 2 zeros on (− 41 , 0) in [28]. Therefore, X ′ (h) has none zeros and X (h) is monotone. This completes the proof (ii) and (iii) can be proved similarly. □ The following Theorem follows from the results (ii) and (iii) of Lemma 9. Theorem 10. Consider the dissipative KdV model (3), there exists a unique periodic wave when α1 is √ √ ⋃ 3 2π α1 2 located in the interval (−∞, − 34 ) √(− 32 , +∞) and choosing α ∈ (−(α + 1), −( + 45 )), and there 0 1 15π 16 3 2π α1 exists a solitary wave if α0 = −( 16 + 45 ). Theorem 11. For√the dissipative √model (3), the solitary wave √ can coexist with one periodic wave when 3 2π α1 16 32 2 4 32 2 √ α1 ∈ ( 15 2π−80 , − 15π ) ⊂ (− 3 , − 15π ) and choosing α0 = −( 16 + 45 ). Proof . We have X (− 14 ) > X (0) and 1 X ′ (− ) < 0, X ′ (0) > 0 4 √

2 1 , − 32 if and only if α1 ∈ ( 15 √16 15π ). Then, X (h) decreases from left endpoint (− 4 , α1 + 1) to a minimum 2π−80

point and then increases to the right endpoint (0, 3

√

2π α1 16

+ 54 ) which is lower than the left endpoint. Then,

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the line m(h) = α0 = −( 3 Therefore,

√

2π α1 16

+ 45 ) intersects the curve X (h) at two points h = 0 and some h = h∗ . M(h) = (α0 + X (h))J0 (h)

has two zeros at h = 0 and h = h∗ . This completes the proof.

□

5. A case study: estimate the wave speed As we mentioned before, the parameter λi s can be treated as almost free parameters. Now, we fix the parameters according to the formulas (2) that describing the dissipative circumstance in [15], and then apply our results obtained before to investigate the wave speed of the solitary wave and periodic waves of the models (1) and (3). First, we consider the model (1). To do this, we choose Rayleigh number R = 104 and fix the layer length as L = 1000m. Assuming the temperature of the water is near 20◦ C, then the Prandtl number σ ≈ 7.56 and the viscosity of water ν ≈ 1.0016, see the website for Prandtl number and viscosity of 3 ≈ 9.768715103 × 109 , g is the gravitative acceleration. We water in W ikipedia. The Galilei number G = gL ν2 use equals signs instead of almost equal signs for convenience in this section. First, we consider the model (1). Substituting these values into (2) and (18), we obtain the function a0 (c) = −

1.337437587 × (0.3738492209c + 12.30193334) , c

which increases monotonically for c > 0. By Theorem 6, we solve the inequalities 6 −1 < a0 (c) ≤ − , 7 and obtain the wave speed of the uniquely persistent periodic wave satisfies 32.90613608 < c < 46.06859050, where c = 46.06859050 is the wave speed of the persistent solitary wave. For each c ∈ (32.90613608, 46.06859050), Theorem 6 implies that m1 (h) a0 (c) = − m0 (h) determines a unique periodic wave with the amplitude h, and c(h) increases as h increases on (− 61 , 0). We m1 (h) have simulated − m and a0 (c) and numerically solved c(h) and c(h(u)), where h(u) = H(u, 0), see Fig. 3. 0 (h) We choose c = 44 and c = 36, to simulate M∗ (h), M∗ (h(u)) and the oscillation of the model (1), respectively, see Figs. 4 and 5. Finally, we consider the model (3) with the same parameters. Inserting evaluated λi s (i = 1, 2, 3, 4) into (14), we obtain 10.96871203 1 4.794347170 × 10−12 √ α0 (c) = − − , α1 (c) = − . c 3 c The condition in Theorem 11 does not hold. Therefore, there is no coexistence of a solitary wave and a periodic wave. Inserting α1 (c) into X (− 14 ) and X (0), respectively, we have √ 4.794347170 × 10−12 1 −3.993885607 × 10−12 + 0.8 c √ √ X (− , c) = − + 1, X (0, c) = . 4 c c Numerical solving the inequality

1 −X (0, c) < α0 (c) < −X (− , c) 4

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Fig. 4. We take c = 44 to simulate M∗ and the periodic wave. (a) indicating M∗ (h) has a zero at h = −0.006944211; (b) M∗ (h(u)) has a zero u = 0.12; (c) a time series shows that an orbit with initial value u = 0.9 approaches a periodic wave which passes through u = 0.123; (d) shows a periodic wave passes through u = 0.123.

gives the range of the wave speed of the persistent periodic wave, 16.45306805 < c < 23.50438292 and the wave speed of the persistent solitary wave is 23.50438292. α0 (c) increases for c ∈ (16.45306805, 23.50438292]. The range of α1 (c) implies that X (h) is monotone by Lemma 9(iii). Therefore, for any c there exists a unique h such that M(h) = 0, determining a periodic wave with amplitude h, see Fig. 6. Further, the wave speed c must increase as h increases in (− 41 , 0), see Fig. 7 for the graphs of c(h) and c(h(u)) obtained by numerically analysis. In order to illustrate the accuracy of our analysis, we choose c = 21 and c = 18 to simulate the Melnikov function M(h), M(h(u)) (h(u) = H(u, 0)) and the oscillation of the model (3), respectively, see Figs. 8 and 9. 6. Conclusion This paper mainly solves the existence of periodic waves for the dissipative KdV models (1) and (3). In detail, we prove that the model (3) can possesses two periodic waves of different amplitudes and the model (1) has only a unique periodic wave. The ranges of parameters are also given explicitly. Particularly, the

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Fig. 5. We take c = 36 to simulate M∗ and the periodic wave. (a) indicating M∗ (h) has a zero at h = −0.09082433333; (b) M∗ (h(u)) has a zero u = 0.53; (c) a time series shows that an orbit with initial value u = 0.2 approaches a periodic wave which passes through u = 0.53; (d) shows a periodic wave passes through u = 0.53.

Fig. 6. The blue curve is α0 (c), the red curves are −χ(− 14 , c) and −χ(0, c), showing the curve α0 (c) insects the curves −χ(− 14 , c) and −χ(0, c) at c = 16.45306805 and 23.50438292, respectively; For any fixed h, the curve α0 (c) insects −χ(h, c) at a unique point, indicating the existence of a unique zero of M(h) for some c. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 7. The wave speed of the persistent periodic waves is the function of the amplitude. (a) c(h) increases for h ∈ (− 14 , 0), and (b) c(h(u)) decreases for u ∈ (0, 1), where h(u) = H(u, 0).

Fig. 8. We take c = 21 to simulate M and the periodic wave. (a) indicating M(h) has a zero at h = −0.0353846607287624; (b) M(h(u)) has a zero u = 0.27105; (c) a time series shows that an orbit with initial value u = 1.01 approaches a periodic wave which passes through u = 0.27105; (d) shows a periodic wave passes through u = 0.27105.

coexistence of the solitary wave and one periodic wave of the model (3) is established, while the model (1) could not possess this phenomenon. Finally, we apply our results to Aspe’s approximation for the parameters (2) to conduct a case study. We shows that the model (3) could not possess the coexistence phenomenon under (2). The ranges of wave speeds of the persistent periodic waves are estimated for both models.

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Fig. 9. We take c = 18 to simulate M and the periodic wave. (a) indicating M(h) has a zero at h = −0.1437565975; (b) indicating that M(h(u)) has a zero u = 0.59; (c) a time series shows that an orbit with initial value u = 0.27105 approaches a periodic wave which passes through u = 0.59; (d) shows a periodic wave passes through u = 0.59.

Acknowledgments Any comments and suggestions from referees and editors are highly appreciated. This work is supported by National Natural Science Foundation of China (No. 11861009), Natural Science Foundation of Guangxi Province (2016GXNSFDA380031, 2018GXNSFAA138198), Program for Innovative Team of GUFE (2018–2021). References [1] M.G. Velarde, Physicochemical Hydrodynamics: Interfacial Phenomena, Plenum, New York, 1987. [2] X.L. Chu, M.G. Velarde, Sustained transverse and longitudinal-waves at the open surface of a liquid, Physicochem. Hydrodyn. 10 (1988) 727–737. [3] X.L. Chu, M.G. Velarde, Transverse and longitudinal waves induced and sustained by surfactant gradients at liquid-liquid interfaces, J. Colloid Interface Sci. 131 (1989) 471–484. [4] X.L. Chu, M.G. Velarde, Korteweg–de Vries soliton excitation in B´ enard-Marangoni convection, Phys. Rev. A 43 (1991) 1094. [5] A. N.Garazo, M.G. Velarde, Dissipative Korteweg–de Vries description of Marangoni–B´ enard oscillatory convection, Phys. Fluids A 3 (1991) 2295–2300. [6] M.G. Velarde, X.L. Chu, The harmonic oscillator approach to sustained gravity-capillary (Laplace) waves at liquid interfaces, Phys. Rev. A 131 (1988) 430–432.

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