J. Math. Fluid Mech. 18 (2016), 89–102 c 2016 Springer International Publishing  1422-6928/16/010089-14 DOI 10.1007/s00021-015-0240-7

Journal of Mathematical Fluid Mechanics

Dynamic Transitions of Generalized Burgers Equation Limei Li and Kiah Wah Ong Communicated by D. Chae.

Abstract. In this article, we study the dynamic transition for the one dimensional generalized Burgers equation with periodic boundary condition. The types of transition are dictated by the sign of an explicitly given parameter b, which is derived using the dynamic transition theory developed by Ma and Wang (Phase transition dynamics. Springer, New York, 2014). The rigorous result demonstrates clearly the types of dynamics transition in terms of length scale l, dispersive parameter δ and viscosity ν. Mathematics Subject Classification. Primary 35Q35; Secondary 35Q53. Keywords. Generalized Burgers equation, centre manifold reduction, phase transition dynamics, spatial-temporal patterns.

1. Introduction By making some assumptions about the flow field and pressure gradients in a particular class of fluid flow, Burgers [2] derived a simple equation to model some aspects of turbulence. The equation he derived is now known as Burgers equation and is given by ∂u ∂2u ∂u = −u + ν 2. (1.1) ∂t ∂x ∂x This evolutionary partial differential equation is a simple model that couples the nonlinear convective behaviour of fluids with the dissipative viscous behavior. It is well known that Burgers’ equation can arise in a wide variety of physical models where dissipation plays a role, such as modelling of traffic flow, shallow water waves, and gas dynamics [7,8,18,19]. Ma and Wang [12–15] first gave the following classification scheme for dissipative system. It says that dynamic transitions of all dissipative systems can be classified into three categories, namely, Type-I (Continuous), Type-II (Jump) or Type-III (Random). For Type-I transition, the transition states are represented by a local attractor Σλ , which attracts a neighborhood of the basic solution. This says that as the control parameter crosses a threshold, the transition states stay in the close neighborhood of the basic state. There are many physical systems which can undergo a continuous transition. For classical example, see [9] regarding the bifurcation and stability of the solutions of the Boussinesq equations, and the onset of the Rayleigh-Benard convection. For Type-II transition, the transition states are represented by some local attractors away from the basic solution. Intuitively speaking, the system undergoes a more drastic change. For Type-III transition, the transition states are represented by two local attractors, with one as in Type-I transition and the other as in Type-II transition. In this case the fluctuations of the basic state can be divided into two regions such that fluctuations in one of the regions lead to continuous transitions and those in the other region lead to jump transitions. For example of Type-II and Type-III The research of Limei Li is supported by National Science Foundation China Grant 11271271 and Sichuan Education Foundation Grant 12ZB108, and by National Study Abroad Foundation.

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transitions, the reader is refer to [16] which study the dynamic phase transitions associated with the spatial-temporal oscillation of the Belousov–Zhabotinsky reactions. In this paper we study the generalized Burgers equation (GBE) in one-dimensional space by introducing a dispersive term in the Burgers–Sivashinsky type equations [3,6]. The equation is supplemented with periodic boundary condition and is given in the form of ∂2u ∂u ∂u ∂u = ν 2 + λu + δ − γu , ∂t ∂x ∂x ∂x u(x, t) = u(x + 2l, t), t ≥ 0, u(x, 0) = u0 (x), x ∈ Ω = (−l, l),

(1.2)

where ν > 0 is the viscosity of a given fluid, γ and λ are positive parameters while δ is a dispersive parameter which take values in R. We also assume that the average flow vanishes  l u(x, t) dx = 0. (1.3) −l

The dynamics of generalized Burgers equation for the case where ν = 1, γ = λ and δ = 0 was studied by Hsia and Wang [5]. Using attractor bifurcation method developed by Ma and Wang [10,11], Hsia and Wang shown that the equation bifurcated from trivial solution to a S 1 attractor consisting of steady states when the control parameter λ crosses some critical value, that critical value has also been explicitly identified. The objective of this article is to study the dynamic transition of GBE using dynamic transition theory developed by Ma and Wang [10,17]. One crucial component of dynamic transition theory is center manifold reduction. Under the assumption (2.7) which is known as Principle of Exchange of Stability (PES), we reduce (2.2) to a center manifold, then the type of transitions for (2.2) at (0, λ0 ) is completely dictated by its reduced equation (3.13) and (3.14) near λ = λ0 . Using first-order approximation formula (See Theorem A.1.1. in [17]) we obtain the reduced equations (3.55) and (3.66). Analysis of these reduced equations give the dynamic transition theorem in Sect. 4 which provides a precise criterion for the transition type. In particular, for domain Ω = (−2π, 2π) and taking viscosity ν = 1, we show that there exists a positive real number η, such that whenever the dispersive parameter δ ∈ (−η, η), then the phase transition of (1.2) at λ = λ0 = 14 is Type-I (Continuous) and the bifurcated periodic orbit is an attractor as the control parameter λ crosses the critical value λ0 . If the dispersive parameter δ lies outside of the interval (excluding the endpoints of the interval as well), then the phase transition is Type-II (Jump) and the trivial solution bifurcates to a unique unstable periodic orbit as the control parameter λ crosses the critical value λ0 . The value of η is also calculated as a root of some rational function. Besides this special case, more general observation can be made about the relation between length scale l, viscosity ν, and the dispersive parameter δ in determining the types of phase transition which occur as the control parameter λ crosses the critical value λ0 . These observations are summarize as Physical Conclusion 4.1 and 4.2 respectively. The paper is organized as follows. Linear eigenvalue problem for system (2.2) is studied in Sect. 2. Center manifold reduction and its first-order approximation is addressed in Sect. 3. The main theorems, its proof and physical conclusions are given in Sect. 4.

2. Mathematical Setting and Linear Problem For the mathematical setting of (1.2) we let 1 H1 = H 2 (Ω) ∩ Hper (Ω),  

H=

u∈

L2per (Ω)

l

: −l

 u dx = 0 ,

(2.1)

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  where L2per = u ∈ L2 (Ω) : u(x) = u(x + 2l) . Notice that H1 and H are two Hilbert spaces and H1 → H. We then write (1.2) as du = Lλ u + G(u), dt u(x, 0) = φ(x),

(2.2)

where Lλ = −A + Bλ , with ∂u ∂2u , and Bλ u = λu + δ . ∂x2 ∂x Since A : H1 → H is a linear homeomorphism and Bλ : H1 → H is a linear compact operator, we see that the operators Lλ = −A + Bλ are parameterized linear completely continuous fields and is continuously depending on λ ∈ R. Also notice that G(u) = −γu ∂u ∂x is a bounded mapping satisfying G(u) = o(uH1 ). Next we consider the eigenvalue problem Au = −ν

Lλ u = β(λ)u.

(2.3)

This eigenvalue problem is equivalent to ν

∂u ∂2u = β(λ)u, + λu + δ ∂x2 ∂x

(2.4)

and we obtain β2k−1 (λ) = σk (λ) + iρk , β2k (λ) = σk (λ) − iρk , as the eigenvalues where

 σk (λ) =

λ−

νk 2 π 2 l2

 and ρk =

kπ δ. l

(2.5)

The corresponding eigenfunctions are

  kπ 1 kπ e2k−1 (x) = √ x − sin x , cos l l 2l   kπ 1 kπ x + sin x , e2k (x) = √ cos l l 2l

and these eigenfunctions are orthonormal, that is

 0 < ei , ej >= 1

With these eigenvalues, we have (with λ0 =

if i = j, if i = j.

(2.6)

νπ 2 l2 ),

⎧ ⎪ ⎨> 0 if λ > Reβ1 (λ) = Reβ2 (λ) = = 0 if λ = ⎪ ⎩ < 0 if λ <

νπ 2 l2 , νπ 2 l2 , νπ 2 l2 ,

Imβ1 (λ0 ) = −Imβ2 (λ0 ) = 0, Reβj (λ0 ) < 0,

∀j ≥ 3.

(2.7)

2

∗ Also for Lλ u = ν ∂∂xu2 + λu + δ ∂u ∂x , its adjoint can be calculated. By writing (Lλ u, ϕ) = (u, Lλ ϕ), we have

L∗λ ϕ = ν

∂2ϕ ∂ϕ . + λϕ − δ 2 ∂x ∂x

(2.8)

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Calculations show

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 νπ 2 π e1 − δe2 = σ1 (λ)e1 − ρ1 e2 , l2 l   2 π νπ Lλ e2 = δe1 + λ − 2 e2 = ρ1 e1 + σ1 (λ)e2 , l l   2 νπ π L∗λ e∗1 = λ − 2 e∗1 + δe∗2 = σ1 (λ)e∗1 + ρ1 e∗2 , l l   π ∗ νπ 2 ∗ ∗ Lλ e2 = − δe1 + λ − 2 e∗2 = −ρ1 e∗1 + σ1 (λ)e∗2 . l l Lλ e1 =

λ−

(2.9) (2.10) (2.11) (2.12)

With this, we say that e1 = e∗1 ,

e2 = e∗2 ,

(2.13)

and in general, we have e∗n = en , ∀n ∈ N.

3. Center Manifold Reduction By the spectral theorem, see Theorem 3.4 [10], we have the following decomposition H = E 1 ⊕ E2 , H1 = E1 ⊕ E2 , E1 = span{e1 , e2 }, E2 = span{en : n ≥ 3}.

(3.1)

Let u ∈ H, we write u = v + y, with v = x1 e1 + x2 e2 , where x1 ∈ R, x2 ∈ R and y ∈ E2 . We can then write du = Lλ u + G(u) dt as dv = Jλ v + P1 G(v + y), dt dy = Lλ y + P2 G(v + y), dt where Jλ = Lλ |E1 : E1 → E1 and Lλ = Lλ |E2 : E2 → E2 . Since we write v = x1 e1 + x2 e2 , we dx1 dx2 dv dt = dt e1 + dt e2 and by (2.9), (2.10) and (3.4) we have

(3.2)

(3.3)

(3.4) (3.5) have

dx1 dx2 e1 + e2 = Jλ (x1 e1 + x2 e2 ) + P1 G(v + y) dt dt = x1 Jλ e1 + x2 Jλ e2 + P1 G(v + y) = x1 [σ1 (λ)e1 − ρ1 e2 ] + x2 [ρ1 e1 + σ1 (λ)e2 ] + P1 G(v + y) = (σ1 (λ)x1 + ρ1 x2 )e1 + (−ρ1 x1 + σ1 (λ)x2 )e2 + P1 G(v + y).

(3.6)

Suppose P1 G(v + y) = ae1 + be2 , a, b ∈ R, then (3.6) can be written as

Hence

dx1 dx2 e1 + e2 = (σ1 (λ)x1 + ρ1 x2 )e1 + (−ρ1 x1 + σ1 (λ)x2 )e2 + ae1 + be2 . dt dt

(3.7)

   dx1 dx2 − (σ1 (λ)x1 + ρ1 x2 ) − a e1 = − + (−ρ1 x1 + σ1 (λ)x2 ) + b e2 . dt dt

(3.8)



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Using the fact that e1 and e2 are independent, we obtain dx1 = σ1 (λ)x1 + ρ1 x2 + a, (3.9) dt dx2 = −ρ1 x1 + σ1 (λ)x2 + b. (3.10) dt

l

l

l

l Notice that −l P1 G(v + y)e1 dx = a and −l P1 G(v + y)e1 dx = −l G(v + y)P1 e1 dx = −l G(v + y)e1 dx, therefore (3.9) can be written as  l dx1 = [σ1 (λ)x1 + ρ1 x2 ] + G(v + y)e1 dx. (3.11) dt −l Similarly, we have dx2 = [−ρ1 x1 + σ1 (λ)x2 ] + dt



l

−l

G(v + y)e2 dx.

(3.12)

By the center manifold theorem, there exist a function, Φ(·, λ) : Bδ (0) ∩ E1 → E2 , see [4], such that the bifurcation of (3.11), (3.12) and (3.5) is equivalent to  l dx1 = [σ1 (λ)x1 + ρ1 x2 ] + G(v + Φ(x))e1 dx, (3.13) dt −l  l dx2 = [−ρ1 x1 + σ1 (λ)x2 ] + G(v + Φ(x))e2 dx. (3.14) dt −l The key is to find a good approximation of the center manifold function Φ so that (3.13) and (3.14) with Φ replaced by the approximation provides a complete bifurcation information. Note that the higher order term in this case is G(u) = −γu ∂u ∂x . Let us define G(u, v) = −γu

∂v , ∂x

(3.15)

then G(u, v) is bilinear as G(αu1 + βu2 , v) = αG(u1 , v) + βG(u2 , v), G(u, αv1 + βv2 ) = αG(u, v1 ) + βG(u, v2 ).

(3.16)

The approximate center manifold function Φ can then be expressed as (See Theorem A.1.1. in [17]) Φ = Φ1 + Φ2 + Φ3 + o(2), where

(3.17)

  o(k) = o(xk ) + O |Reβ(λ)|xk .

Functions Φ1 , Φ2 , Φ3 are calculated by − Lλ Φ1 = x21 G11 + x22 G22 + x1 x2 (G12 + G21 ) ,     2 2 (−Lλ ) + 4ρ1 (−Lλ )Φ2 = 2ρ21 (x21 − x22 ) (G22 − G11 ) − 2x1 x2 (G12 + G21 ) , and

    (−Lλ )2 + 4ρ21 Φ3 = ρ1 (−Lλ ) (x22 − x21 ) (G12 + G21 ) + 2x1 x2 (G11 − G22 ) ,

where Gij = P2 G(ei , ej , λ) for 1 ≤ i, j ≤ 2, and Lλ = Lλ |E2 : E2 → E2 . Let ∞ ∞   Φ1 (x1 , x2 ) = Φ1 = yn (x1 , x2 )en = yn en , n=3

n=3

(3.18) (3.19)

(3.20)

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and from the following calculations, 

−l



0

otherwise,

0 

otherwise,

 G(e2 , e2 )e2n dx =

 −2γπ 

G(e2 , e1 )e2n dx =



G(e2 , e2 )e2n−1 dx =

(3.22)

G(e2 , e1 )e2n−1 dx =

l

−l

(3.21)

l

−l



l

−l

if n = 2 if n = 2

G(e1 , e2 )e2n−1 dx + G(e1 , e2 )e2n dx +

√γπ ( 2l)3

√γπ ( 2l)3

l

−l



G(e1 , e1 )e2n dx =

l

−l





l

−l



G(e1 , e1 )e2n−1 dx =

l

−l





l

−γπ √ ( 2l)3

if n = 2

0

otherwise,

−γπ √ ( 2l)3

if n = 2

0

otherwise.

√ ( 2l)3

if n = 2

0

otherwise,

2γπ √ ( 2l)3

if n = 2

0

otherwise,

(3.23)

(3.24)

(3.25)

(3.26)

and (3.18), we obtain the nonzero terms as   γπ √ (ρ2 − σ2 )x21 + (σ2 − ρ2 )x22 + 2(σ2 + ρ2 )x1 x2 , y3 (x1 , x2 ) = 2 2 (σ2 + ρ2 )( 2l)3

(3.27)

and y4 (x1 , x2 ) =

  −γπ 2 2 √ (ρ . + σ )x − (ρ + σ )x + 2(σ − ρ )x x 2 2 2 2 2 2 1 2 1 2 (σ22 + ρ22 )( 2l)3

(3.28)

Hence, we obtain all the coefficients of Φ1 (x1 , x2 ). Next, we calculate Φ2 (x1 , x2 ) = Φ2 =

∞ 

pn (x1 , x2 )en =

n=3

∞ 

pn en .

n=3

By using (3.19) and (3.21)–(3.26), we obtain the nonzero terms as p3 (x1 , x2 ) =

  2ρ21 γπ √ A1 [4x1 x2 − 2(x21 − x22 )] + B1 [4x1 x2 + 2(x21 − x22 )] , 2 + B1 )( 2l)3

(3.29)

  2ρ21 γπ √ B1 [4x1 x2 − 2(x21 − x22 )] − A1 [4x1 x2 + 2(x21 − x22 )] , 2 2 3 (A1 + B1 )( 2l)

(3.30)

(A21

and p4 (x1 , x2 ) = where A1 = −σ23 + 3ρ22 σ2 − 4ρ21 σ2 ,

B1 = −3ρ2 σ22 + ρ32 − 4ρ21 ρ2 . Hence, we obtain all the coefficients of Φ2 (x1 , x2 ). Finally let Φ3 (x1 , x2 ) = Φ3 =

∞  n=3

qn (x1 , x2 )en =

∞  n=3

qn en .

(3.31)

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Using (3.20) and (3.21)–(3.26), we obtain the nonzero terms of Φ3 as q3 (x1 , x2 ) =

 2πγ √ ρ1 (x22 − x21 ) (σ2 [A2 + B2 ] + ρ2 [B2 − A2 ]) (A22 + B22 )( 2l)3  + 2ρ1 x1 x2 (σ2 [B2 − A2 ] − ρ2 [A2 + B2 ]) ,

(3.32)

 2πγ √ ρ1 (x22 − x21 ) (σ2 [B2 − A2 ] − ρ2 [B2 + A2 ]) (A22 + B22 )( 2l)3  − 2ρ1 x1 x2 (σ2 [B2 + A2 ] + ρ2 [B2 − A2 ]) ,

(3.33)

and q4 (x1 , x2 ) =

where A2 = σ22 − ρ22 + 4ρ21 , B2 = 2ρ2 σ2 . Hence the coefficients of Φ3 (x1 , x2 ) are fully determined. Let  l ∂ej ek dx, −γei (G(ei , ej ), ek ) = ∂x −l we then compute (G(x1 e1 + x2 e2 + y), ei ) for i = 1, 2. With  l ∂e1 e1 −γe1 ∂x −l  l ∂e2 e1 −γe1 ∂x −l  l ∂e1 e1 −γe2 ∂x −l  l ∂e2 e1 −γe2 ∂x −l

(3.34)

dx = 0,

(3.35)

dx = 0,

(3.36)

dx = 0,

(3.37)

dx = 0,

(3.38)

we obtain 

 l ∂y ∂y (G(x1 e1 + x2 e2 + y), e1 ) = −γx1 e1 e1 dx + −γx2 e2 e1 dx ∂x ∂x −l −l  l  l  l ∂e1 ∂e2 ∂y e1 dx + e1 dx + −γyx1 −γyx2 −γy e1 dx. + ∂x ∂x ∂x −l −l −l l

(3.39)

Next, by writing y = Φ(x1 , x2 ) we have  (G(ei , Φ), ej ) =

l

−l l

−γei

∂Φ ej dx, ∂x

−γΦ

∂ei ej dx ∂x

 (G(Φ, ei ), ej ) =

−l

for i = 1, 2.

(3.40)

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To avoid fourth-order term in the later calculations, we then drop the last term We then obtain the approximate equation of (3.11) as

JMFM

l −l

∂y −γy ∂x e1 dx in (3.39).

 l dx1 = [σ1 (λ)x1 + ρ1 x2 ] + G(v + Φ)e1 dx dt −l = σ1 (λ)x1 + ρ1 x2 + x1 [(G(e1 , Φ), e1 ) + (G(Φ, e1 ), e1 )] +x2 [(G(e2 , Φ), e1 ) + (G(Φ, e2 ), e1 )] .

(3.41)

 l dx2 = [−ρ1 x1 + σ1 (λ)x2 ] + G(v + Φ)e2 dx dt −l = −ρ1 x1 + σ1 (λ)x2 + x1 [(G(e1 , Φ), e2 ) + (G(Φ, e1 ), e2 )] +x2 [(G(e2 , Φ), e2 ) + (G(Φ, e2 ), e2 )] .

(3.42)

Similarly, we have

From (3.27)–(3.30) and (3.32)–(3.33), we know that Φ can be written as Φ(x1 , x2 ) = ϕ3 (x1 , x2 )e3 + ϕ4 (x1 , x2 )e4 ,

(3.43)

where ϕ3 (x1 , x2 ) = y3 (x1 , x2 ) + p3 (x1 , x2 ) + q3 (x1 , x2 ), ϕ4 (x1 , x2 ) = y4 (x1 , x2 ) + p4 (x1 , x2 ) + q4 (x1 , x2 ).

(3.44)

With these notations, we have (G(e1 , Φ), e1 ) = ϕ3 (G(e1 , e3 ), e1 ) + ϕ4 (G(e1 , e4 ), e1 ), (G(Φ, e1 ), e1 ) = ϕ3 (G(e3 , e1 ), e1 ) + ϕ4 (G(e4 , e1 ), e1 ),

(3.45) (3.46)

(G(e2 , Φ), e1 ) = ϕ3 (G(e2 , e3 ), e1 ) + ϕ4 (G(e2 , e4 ), e1 ), (G(Φ, e2 ), e1 ) = ϕ3 (G(e3 , e2 ), e1 ) + ϕ4 (G(e4 , e2 ), e1 ).

(3.47) (3.48)

Since −2πγ (G(e1 , e3 ), e1 ) = √ , ( 2l)3 πγ (G(e3 , e1 ), e1 ) = √ , ( 2l)3 2πγ (G(e2 , e3 ), e1 ) = √ , ( 2l)3 −πγ (G(e3 , e2 ), e1 ) = √ , ( 2l)3

−2πγ (G(e1 , e4 ), e1 ) = √ , ( 2l)3 πγ (G(e4 , e1 ), e1 ) = √ , ( 2l)3 −2πγ (G(e2 , e4 ), e1 ) = √ , ( 2l)3 πγ (G(e4 , e2 ), e1 ) = √ , ( 2l)3

(3.49) (3.50) (3.51) (3.52)

we have from (3.49)–(3.52) that −2πγ (ϕ3 + ϕ4 ) , (G(e1 , Φ), e1 ) = √ ( 2l)3 2πγ (G(e2 , Φ), e1 ) = √ (ϕ3 − ϕ4 ) , ( 2l)3

πγ (G(Φ, e1 ), e1 ) = √ (ϕ3 + ϕ4 ) , ( 2l)3 πγ (G(Φ, e2 ), e1 ) = √ (ϕ4 − ϕ3 ) . ( 2l)3

(3.53) (3.54)

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Putting all these together, (3.41) can be written as dx1 = σ1 (λ)x1 + ρ1 x2 dt  2σ2 π2 γ 2 8ρ2 A1 + + 21 2+ 2 2 3 8l σ + ρ2 A1 + B1  2 2 2 −2ρ2 π γ 8ρ2 B1 + − 21 2+ 2 2 3 8l σ + ρ2 A1 + B1  2 2 2 2σ2 π γ 8ρ2 A1 + + 21 2+ 2 2 3 8l σ + ρ2 A1 + B1  2 2 2 −2ρ2 π γ 8ρ2 B1 + − 21 2+ 2 2 3 8l σ2 + ρ2 A1 + B1   3 + o(x ) + O |Reβ(λ)|x3 .

 4 3 [ρ σ B − ρ ρ A ] 1 2 2 1 2 2 x1 A22 + B22  4 2 [ρ σ A + ρ ρ B ] 1 2 2 1 2 2 x1 x2 A22 + B22  4 2 [ρ σ B − ρ ρ A ] 1 2 2 1 2 2 x1 x2 A22 + B22  4 3 [ρ σ A + ρ ρ B ] 1 2 2 1 2 2 x2 A22 + B22 (3.55)

To work on (3.42), we need the following 2πγ (G(e1 , e3 ), e2 ) = √ , ( 2l)3 −πγ (G(e3 , e1 ), e2 ) = √ , ( 2l)3 2πγ (G(e2 , e3 ), e2 ) = √ , ( 2l)3 −πγ (G(e3 , e2 ), e2 ) = √ , ( 2l)3

−2πγ (G(e1 , e4 ), e2 ) = √ , ( 2l)3 πγ (G(e4 , e1 ), e2 ) = √ , ( 2l)3 2πγ (G(e2 , e4 ), e2 ) = √ , ( 2l)3 −πγ (G(e4 , e2 ), e2 ) = √ , ( 2l)3

(3.56) (3.57) (3.58) (3.59)

and from (3.56)–(3.59), we have 2πγ (ϕ3 − ϕ4 ) , (G(e1 , Φ), e2 ) = √ ( 2l)3 2πγ (G(e2 , Φ), e2 ) = √ (ϕ3 + ϕ4 ) , ( 2l)3

πγ (G(Φ, e1 ), e2 ) = √ (ϕ4 − ϕ3 ) , ( 2l)3 −πγ (G(Φ, e2 ), e2 ) = √ (ϕ3 + ϕ4 ) , ( 2l)3

(3.60) (3.61)

therefore, from (G(e1 , Φ), e2 ) = ϕ3 (G(e1 , e3 ), e2 ) + ϕ4 (G(e1 , e4 ), e2 ),

(3.62)

(G(Φ, e1 ), e2 ) = ϕ3 (G(e3 , e1 ), e2 ) + ϕ4 (G(e4 , e1 ), e2 ), (G(e2 , Φ), e2 ) = ϕ3 (G(e2 , e3 ), e2 ) + ϕ4 (G(e2 , e4 ), e2 ), (G(Φ, e2 ), e2 ) = ϕ3 (G(e3 , e2 ), e2 ) + ϕ4 (G(e4 , e2 ), e2 ),

(3.63) (3.64) (3.65)

we obtain the second reduced equation, dx2 = −ρ1 x1 + σ1 (λ)x2 dt   2ρ2 π2 γ 2 8ρ21 B1 4 + + 2 − 2 [ρ1 σ2 A2 + ρ1 ρ2 B2 ] x31 8l3 σ22 + ρ22 A1 + B12 A2 + B22   2σ2 π2 γ 2 8ρ21 A1 4 + + 2 + 2 [ρ1 σ2 B2 − ρ1 ρ2 A2 ] x21 x2 8l3 σ22 + ρ22 A1 + B12 A2 + B22   2ρ2 π2 γ 2 8ρ21 B1 4 + + 2 − 2 [ρ1 σ2 A2 + ρ1 ρ2 B2 ] x1 x22 8l3 σ22 + ρ22 A1 + B12 A2 + B22

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  2σ2 π2 γ 2 8ρ21 A1 4 + + 2 + 2 [ρ1 σ2 B2 − ρ1 ρ2 A2 ] x32 8l3 σ22 + ρ22 A1 + B12 A2 + B22   + o(x3 ) + O |Reβ(λ)|x3 .

JMFM

(3.66)

4. Main Theorem and Physical Conclusions By generalizing a theorem of Andronov et al. [1] in finite-dimensional case to the infinite-dimensional case, Ma and Wang gave a transition criteria based on the sign of bifurcation number. Let Φ(x, λ) be the center manifold function of (2.2) near λ = λ0 and ei = ei (λ0 ) for i = 1, 2. Also assume that  aipq xp1 xq2 + o(x3 ), i = 1, 2. (4.1) (G(x + Φ(x, λ0 ), λ0 ), ei ) = 2≤p+q≤3

For (4.1) the bifurcation number is given by b=

 π 1  3π  1 a30 + a203 + a + a221 4 4 12  π  1 2 + a02 a02 − a120 a220 2ρ1  π  1 1 + a11 a20 + a111 a102 − a211 a220 − a211 a202 . 4ρ1

(4.2)

Theorem 2.3.7 [17] says that if the bifurcation number b is negative, then the transition of (1.2) is Type-I (Continuous), and the bifurcated periodic orbit is an attractor. For the case where b is positive, then the transition is a Type-II (Jump) transition, and (1.2) bifurcates on λ < λ0 to a unique unstable periodic orbit. The following theorem is a direct application of the Theorem 2.3.7 cited above. Theorem 4.1. Let us define f (ν, γ, δ, l) =

2πlδ 2 γ 2 −3νπ 3 γ 2 + 2 2 2 2 2 2 2l(9ν π + 4δ l ) 3ν(4δ l + 9π 2 ν 2 )(16δ 2 l2 + 9π 2 ν 2 ) 2δ 2 π 3 γ 2 + 2 2 , 9ν π l + 16δ 2 l3

(4.3)

where ν, γ, l are positive parameters, while δ is a dispersive parameter which take values in R. Then the following assertions hold true: (1) If f (ν, γ, δ, l) < 0, then the transition of (1.2) is Type-I (Continuous), and the bifurcated periodic orbit is an attractor. (2) If f (ν, γ, δ, l) > 0, then the transition is a Type-II (Jump) transition, and (1.2) bifurcates on λ < λ0 to a unique unstable periodic orbit. Proof. Notice that in our case, we do not have second order terms, hence b=

 π 1  3π  1 a30 + a203 + a12 + a221 . 4 4

(4.4)

From calculation, we obtain a130 = a112 = a221 = a203 , a121

=

a103

=

−a230

=

−a212 ,

(4.5) (4.6)

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where lδ 2 γ 2 −3νπ 2 γ 2 + 4l[9ν 2 π 2 + 4δ 2 l2 ] 3ν(4δ 2 l2 + 9π 2 ν 2 )(16δ 2 l2 + 9π 2 ν 2 ) δ2 π2 γ 2 + 2 2 , 9ν π l + 16δ 2 l3 6πγ 2 δ 3 l2 −πγ 2 δ + = 2(9ν 2 π 2 + 4δ 2 l2 ) (4δ 2 l2 + 9π 2 ν 2 )(16δ 2 l2 + 9π 2 ν 2 ) πδγ 2 (9π 2 ν 2 + 8δ 2 l2 ) . − 2 6νl (9π 2 ν 2 + 16δ 2 l2 )

a130 =

a103

(4.7)

(4.8)

This gives b=

−3νπ 3 γ 2 2πlδ 2 γ 2 + 2l(9ν 2 π 2 + 4δ 2 l2 ) 3ν(4δ 2 l2 + 9π 2 ν 2 )(16δ 2 l2 + 9π 2 ν 2 ) 2δ 2 π 3 γ 2 + 2 2 , 9ν π l + 16δ 2 l3

and the results follow from Theorem 2.3.7 [17].

(4.9) 

We next consider the special case where the length scale is 2π and viscosity ν = 1. By normalization, we assume γ = 1 throughout this section. Theorem 4.2. Let Ω = (−2π, 2π) and viscosity ν = 1. Then the following assertions hold true: (1) If the dispersive parameter satisfies −η < δ < η, then the phase transition of (1.2) at λ = λ0 = 14 is Type-I (Continuous) and the bifurcated periodic orbit is an attractor Σλ = S 1 as the control parameter λ crosses the critical value λ0 . Furthermore, the bifurcated periodic solution can be expressed as u = x1 e1 + x2 e2 + o(|σ1 (λ)|),  1/2 σ1 (λ) cos ρ1 λt + o(|σ1 (λ)|), x1 (t) = |b|  1/2 σ1 (λ) x2 (t) = sin ρ1 λt + o(|σ1 (λ)|). |b|

(4.10)

where σ1 (λ) and ρ1 are as in (2.9)–(2.12), see Figure 1. (2) If the dispersive parameter satisfies δ < −η or δ > η, then the phase transition is Type-II (Jump) and the trivial solution bifurcates to a unique unstable periodic orbit as the control parameter λ crosses the critical value λ0 . Here η is a root of 1 1 + 256π 2 x4 1 + 48π 2 x2 + − 768x2 2304x2 + 16384x4 576x2 + 1024x4 2 2 2 2 16x − 81π − 468π x + 192π 2 x4 , = 12(9 + 16x2 )(9 + 64x2 )

ζ(x) =

which is η 1.6099.

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Fig. 1. Bifurcated periodic orbit from λ = λ0

Proof. From (4.3) and with l = 2π, ν = 1 and γ = 1, we obtain  π 1  3π  1 a30 + a203 + a12 + a221 b= 4 4 3π  1  π  1  = 2a30 + 2a30 4 4 = 2πa130   1 1 1 + 256π 2 δ 4 1 + 48π 2 δ 2 = 2 + − π 768δ 2 2304δ 2 + 16384δ 4 576δ 2 + 1024δ 4   1 16δ 2 − 81π 2 − 468π 2 δ 2 + 192π 2 δ 4 = 2 . π 12(9 + 16δ 2 )(9 + 64δ 2 ) Suppose we write 1 b(δ) = 2 ζ(δ). π Numerical methods show that, ζ(η) = 0 for η ±1.6099 and

(4.11)

(4.12)

ζ(δ)

is positive for

δ ∈ (−∞, η) ∪ (η, ∞),

(4.13)

ζ(δ)

is negative for

δ ∈ (−η, η).

(4.14)

Hence b(δ) is positive for δ ∈ (−∞, η) ∪ (η, ∞) and is negative for δ ∈ (−η, η). The result then follow from Theorem 2.3.7 [17], and proof of the theorem is complete.  Based on Theorems 4.1 and 4.2, one see that the key phase transition criteria is the sign of bifurcation number. Hence instead of investigating the special case in Theorem 4.2, one can say more about the relation between length scale l, viscosity ν, and the dispersive parameter δ in determining the types of phase transition which occur as the control parameter λ crosses the critical value λ0 . We next deduce from (4.9) the following physical conclusion. Physical Conclusion 4.1. Let domain Ω be an interval centered at the origin with fix radius l = L, then for every ν > 0, there are values ±Dν such that the phase transition of (1.2) at λ = λ0 is Type-I (Continuous), whenever the dispersive parameter satisfies −Dν < δ < Dν . The transition will be of Type-II (Jump), whenever the dispersive parameter satisfies δ < −Dν or δ > Dν . Figure 2 show the special case where Ω = (−1, 1). Similar conclusion can be made if we fix the viscosity and let the dispersive parameter δ ∈ R and length scale l > 0 to vary. Physical Conclusion 4.2. Let viscosity ν > 0 be fix, then for every domain Ω = (−l, l) with l > 0, there exist values ±Dl such that the phase transition of (1.2) at λ = λ0 is Type-I (Continuous), whenever the dispersive parameter satisfies −Dl < δ < Dl . The transition will be of Type-II (Jump), whenever the dispersive parameter satisfies δ < −Dl or δ > Dl . Figure 3 show the special case where viscosity ν = 1.

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Fig. 2. Type-I and Type-II transitions as given by Physical Conclusion 4.1

Fig. 3. Type-I and Type-II transitions as given by Physical Conclusion 4.2

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[11] Ma, T., Wang, S.: Dynamic bifurcation of nonlinear evolution equations and applications. Chin. Ann. Math. 26(2), 185– 206 (2005) [12] Ma, T., Wang, S.: Stability and Bifurcation of Nonlinear Evolutions Equations. Science Press, Beijing (2007) [13] Ma, T., Wang, S.: Dynamic model and phase transitions for liquid helium. J. Math. Phys. 49:073304:1–18 (2008) [14] Ma, T., Wang, S.: Cahn–Hilliard equations and phase transition dynamics for binary system. Dist. Cont. Dyn. Syst. Ser. B 11(3), 741–784 (2009) [15] Ma, T., Wang, S.: Phase separation of binary systems. Phys. A Stat. Mech. Appl. 388(23), 4811–4817 (2009) [16] Ma, T., Wang, S.: Phase transitions for Belousov–Zhabotinsky reactions. Math. Methods Appl. Sci. 34(11), 1381– 1397 (2011) [17] Ma, T., Wang, S.: Phase Transition Dynamics. Springer, New York (2014) [18] Richards, P.I.: Shock waves on the highway. Oper. Res. 4, 42–51 (1956) [19] Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1999) Limei Li Department of Mathematics and Software Sichuan Normal University, Chengdu 610066 People’s Republic of China e-mail: [email protected]

Kiah Wah Ong Department of Mathematical and Actuarial Sciences, Universiti Tunku Abdul Rahman, Cheras, 43000 Kajang, Selangor, Malaysia e-mail: [email protected]

Limei Li and Kiah Wah Ong Department of Mathematics, Indiana University Bloomington, IN 47405, USA (accepted: September 9, 2015; published online: January 4, 2016)