REDUCTION AND BIFURCATION OF TRAVELING WAVES OF THE KDV-BURGERS-KURAMOTO EQUATION

. In this paper, the Lie symmetry analysis is performed on the KBK equation. By constructing its one-dimensional optimal system, we obtain four classes of reduced equations and corresponding group-invariant solutions. Par-ticularly, the traveling wave equation, as an important reduced equation, is investigated in detail. Treating it as a singular perturbation system in R 3 , we study the phase space geometry of its reduced system on a two-dimensional invariant manifold by using the dynamical system methods such as tracking the unstable manifold of the saddle, studying the equilibria at inﬁnity and dis- cussing the homoclinic bifurcation and Poincar´e bifurcation. Correspongding wavespeed conditions are determined to guarantee the existence of various bounded traveling waves of the KBK equation.


(Communicated by Shigui Ruan)
Abstract. In this paper, the Lie symmetry analysis is performed on the KBK equation. By constructing its one-dimensional optimal system, we obtain four classes of reduced equations and corresponding group-invariant solutions. Particularly, the traveling wave equation, as an important reduced equation, is investigated in detail. Treating it as a singular perturbation system in R 3 , we study the phase space geometry of its reduced system on a two-dimensional invariant manifold by using the dynamical system methods such as tracking the unstable manifold of the saddle, studying the equilibria at infinity and discussing the homoclinic bifurcation and Poincaré bifurcation. Correspongding wavespeed conditions are determined to guarantee the existence of various bounded traveling waves of the KBK equation.

1.
Introduction. This paper considers the KdV-Burgers-Kuramoto (KBK) equation [14] ∂u ∂t + u ∂u ∂x +ᾱ ∂ 2 u ∂x 2 +β which is an appropriate model to describe phenomena that are simultaneously involved in nonlinearity, dissipation, dispersion and instability, whereᾱ,β,γ are nonzero real constants. Equation (1) plays important roles in describing physical processes in motion of turbulence and unstable systems [30,17,18] and is also known as the generalized Kuramoto-Sivashinsky equation [17] or Benney equation [25]. The KBK equation has been investigated widely and various direct methods have been proposed to obtain exact traveling wave solutions of it, such as the Weiss-Tabor-Carnevale transformation method [17], trial-function method [25], tanh-function method and extended tanh-function method [19,28,4]. In recent decade, more methods are applied to obtain new exact solutions of it, including the trigonometric function expansion method [6], generalized F-expansion method [34], unified ansätze approach [15], a combination method [32] and Exp-function equation so that all possible bounded traveling waves and corresponding existence conditions can be identified clearly. We first prove that it has a two-dimensional invariant submanifold M in R 3 by the geometric singular perturbation theory [5,12]. Restricted on M , the singular perturbation system is reduced to a dynamical system with perturbation in R 2 . By using the dynamical system methods and some techniques such as tracking unstable manifold of the saddle and studying equilibria at infinity, we investigate the phase space geometry of the corresponding unperturbed system in detail. The result clearly shows that under appropriate wavespeed conditions there exist the saddlespiral shock waves besides the traveling fronts mentioned in [7] for the unperturbed system. Subsequently, by using the Fredholm theorem, we prove that the two types of traveling waves can persist in the KBK equation. Furthermore, Mel'nikov type computation is carried out to study the homoclinic bifurcation and Poincaré bifurcation of the reduced system. Various wavespeed conditions are determined to guarantee the existence of solitary waves and periodic waves of the KBK equation. Not only that, based on the bifurcation results, we also discuss the existence of two types of oscillatory bounded traveling waves for the KBK equation.

Reduction and series solutions of the KBK equation.
To seek a symmetry σ(x, t, u) of the KBK equation, we set where u = u(x, t) satisfies equation (1) and a(x, t), b(x, t), d(x, t), e(x, t) are functions to be determined later. Based on Lie group theory [26], σ satisfies the following equation Noting that u t = −uu x −ᾱu xx −βu xxx −γu xxxx , we can calculate the expressions of σ t , σ x , σ xx , σ xxx and σ xxxx (see Appendix 1 ). Substituting them into (2), we get the differential equations with respect to a(x, t), b(x, t), d(x, t) and e(x, t). Solving them yields where c 1 , c 2 , c 3 are arbitrary real constants. Thus, the symmetry of the KBK equation can be written as It follows that the vector field of the KBK equation is spanned by the vectors Next, we construct the one-dimensional optimal system for the KBK equation. Firstly, one can check that the following commutation relations hold It means that the vector field is closed under the Lie bracket. From the commutation relations (3), we calculate adjoin representations of the vector field as follows for any ε ∈ R. Given a nonzero vector we expect to simplify as many of the coefficients a i as possible through suitable applications of adjoint maps to V . Firstly, suppose that a 3 = 0. Without loss of generality, we can set a 3 = 1. If we act on the vector V by Ad(exp(a 2 V 1 )), the coefficient of the V 2 can be eliminated: It is easy to see thatV can not be reduced further by above adjoint maps. So, every one-dimensional subalgebra generated by V with a 3 = 0 is equivalent to the subalgebra spanned by The remaining one-dimensional subalgebras are spanned by vectors V with a 3 = 0 i.e., V = a 1 V 1 + a 2 V 2 . Similarly, suppose that a 1 = 1. If we act on the vector V by Ad(exp(−a 2 V 3 )), the coefficient of the V 2 can be eliminated: So, every one-dimensional subalgebra generated by V with a 3 = 0 is equivalent to the subalgebra spanned by V 1 (a 1 = 0) or V 2 (a 1 = 0). Thus , we obtain an optimal system of one-dimensional subalgebras of the KBK equation as follows: where a 1 = 0 is an arbitrary constant.
According to the optimal system, we will reduce the KBK equation and construct its group-invariant solutions.
Substituting (4) into (1), we reduce the KBK equation to the form: Solving it, we get the trivial solution of the KBK equation u(x, t) = C, where C is an arbitrary constant.
Substituting (5) into (1), we reduce the KBK equation to the form: where C is an arbitrary constant.
Case 3. For the generator V 1 , we have Substituting (6) into (1), we reduce the KBK equation to the form: where f 3 = df3 dx . Integrating (7) once, we havē where e is an integral constant.
We assume a solution of Eq.(8) in a power series of the from Substituting (9) into (8) leads to From (10), we have and for n ≥ 1 Thus, the power series solution of the KBK equation can be written as follows where c i (i = 0, 1, 2) are arbitrary constants, the other coefficients c n (n ≥ 3) are determined by (11) and (12). By a simple application of the Implicit Function Theorem, it is easy to check that power series (13) is convergent.
where ξ = x − t 2 2a1 . Substituting (14) into (1), we reduce the KBK equation to the form:γ where f 4 = df4 dξ . Similarly, we can obtain the corresponding series solution of Eq. (15) as follows where c i (i = 0, 1, 2) are arbitrary constants, , (n ≥ 2). Especially, the traveling wave solutions correspond to the symmetry group generated by which convert the KBK equation into its traveling wave system where denotes d/dξ and constantc > 0 is wavespeed.
3. Existence of invariant submanifold M in R 3 and the flow on it. Integrating (17) once will yield where we set the integration constant e = 0 (Otherwise, we can always make a suitable homeomorphic transformation to eliminate it and convert the equation into the same form as equation (18) ). Firstly, we rescale the parameterγ = γ for the small > 0. Then, the homeomorphic transformation f (ξ) = γU (ξ) convert (18) into the form With ξ = τ , the 'fast system' associated with (19) has the form Noting that d∆(x) dx | x=0 = 0 , we can see that J(E 0 ) and J(E 1 ) have at least a pair nonzero real eigenvalues with opposite signs. We want to prove that system (19) has two manifolds intersecting along an one-dimensional curve in R 3 . This curve just corresponds to a bounded traveling wave solution of the KBK equation.
If is set to zero in (19), then U and V are governed by where Y lies on the set which is a two-dimensional submanifold in R 3 . From [5], the manifold M 0 is said to be normally hyperbolic if the linearization of the fast system, restricted to M 0 , has exactly dimM 0 eigenvalues on the imaginary axis, with the remainder of the system hyperbolic. The 'fast system' (20), restricted to the manifold M 0 , has the Jacobian matrix which has the eigenvalues 0, 0, −β. It means that the manifold M 0 is normally hyperbolic. So, Fenichels invariant manifold theory [5] guarantees that there exists a two-dimensional submanifold M diffeomorphic to M 0 in R 3 , which is within the distance ε of M 0 and is invariant for the flow (19). Next, we assume that the manifold M can be written as In order to obtain the approximation of manifold M , we expand function h(U, V, ) in Taylor series in the variable Substituting (22) into (19), we get YUQIAN ZHOU AND QIAN LIU by power of . This allows one to write (19) as the following system which determines the dynamics on the 'slow' manifold M . Letting = 0 in (23), we have which has two equilibria E 0 : (0, 0) and E 1 : (2c, 0), at which the Jacobian matrices are respectively.
Proof. Note that the expression ∂P (U, V )/∂U + ∂Q(U, V )/∂V = − α β has a fixed sign. By the Bendixon Theorem, system (24) has no closed orbit in phase plane (U-V plane). When α > 0, β > 0 and c ≤ α 2 4β , we consider the problem in the triangle region D which is enclosed by three lines U = 0, V = 0 and V = k(U − 2c) in phase plane of (24), where k < 0 is a constant to be determined. In region D, there is no equilibrium of (24). Firstly, by [3], there exists an unstable manifold Γ of the saddle E 0 in the first quadrant, which intersects neither the U -axis nor the V -axis in an enough small neighborhood of the origin E 0 .
The vector field defined by (24) guarantees that orbits confined to the first quadrant move to the right as ξ increases. It means that Γ can not intersect the boundary ∂D 1 := {(U, V ) ∈ R 2 : U = 0, 0 ≤ V ≤ −2kc}. On the boundary ∂D 2 := {(U, V ) ∈ R 2 : 0 < U < 2c, V = 0}, dV dξ | ∂D2 = cU β − U 2 2β > 0, which means that Γ can not intersect the boundary ∂D 2 . On the boundary To be more specific, we can choose the constant k in the interval ( ) freely to guarantee Γ not to intersect the boundary ∂D 3 .
From the facts above, it concludes that Γ can not go out of the region D and E 1 is exactly the ω-limit set of it. Thus, we prove the existence of saddle-node heteroclinic orbit connecting E 0 and E 1 . Moreover, the fact dU/dξ = V > 0 means that the bounded kink wave solution corresponding to Γ is monotone increasing with respect to ξ.
By implicit function theorem, we can solve the equation y + p 2 (x, y) = 0 in an enough small neighborhood of the origin (0, 0) and obtain By Theorem 7.2 and its corollary in [35], the degenerate equilibrium (0, 0) is an unstable degenerate node. So, we can give the global phase portrait of system (24) in figure 1. It shows that there exists a saddle-focus heteroclinic orbit connecting E 0 and E 1 , which corresponds to the saddle-spiral shock wave. Figure 1. Global phase portrait of (24) for α > 0, β > 0 and α 2 < 4βc.
Assume that the heteroclinic connection of (24) can be expressed as (U 0 , V 0 ). The one thing left is to prove there exists analogous heteroclinic connection for (23) when is sufficiently small.
In order to seek such connection in (23), we set Substituting (25) into (23) and taking the lowest order in , we obtain the approximate system which governsŨ andṼ , where the linear differential operator is defined by Next, we want to prove system (26) has a solution satisfying U ,Ṽ → 0 as ξ → ±∞.
By Fredholm theory [33], system (26) has a square-integrable solution if and only if the following compatibility condition holds for all functions X(ξ) in the kernel of the adjoint operator L * , where ·, · is the inner product on R 2 . The adjoint system for (26) can be expressed as Noting that ξ → +∞, U 0 → 2c, we can see that (28) has a constant matrix with two eigenvalues λ 1,2 = α± √ α 2 −4 β c 2β . Obviously, both eigenvalues λ 1 and λ 2 have positive real parts. Any solutions of (28), other than the zero solution, must grow exponentially. The only solution in L 2 is therefore a zero solution X(ξ) = 0, and consequently the Fredholm orthogonality condition (27) trivially holds. Thus, we prove the existence of analogous heteroclinic orbits for (23) when is sufficiently small, which implies the desired existence of kink wave and saddle-spiral shock wave for the KBK equation.

4.
Solitary wave and periodic wave solutions. In this section, under the condition that α > 0 and β > 0, we will prove the existence of homoclinic and periodic orbits of (23), which corresponds to the existence of solitary wave and periodic wave solutions of the KBK equation respectively.

Proof. Consider the Mel'nikov function
where I k := Γ U k V dU , k = 0, 1, R(h) := I 1 /I 0 and Γ corresponds to homoclinic loop Υ 0 ∪ {Ẽ 0 } or periodic orbits ΓẼ 1 (h). First, we claim that In fact, assume that (a(h), 0) and (b(h), 0) (a(h) < b(h)) are the points where periodic orbit ΓẼ implying (32). Note the facts that homoclinic orbit Υ 0 intersects U -axis at the point (3c, 0) and periodic orbits ΓẼ 1 (h) approach the homoclinic orbit Υ 0 as h → 0. Thus lim , U ∈ (0, 3c) is the explicit expression of the curve Υ 0 . Direct computation gives the result I * 1 /I * 0 = 12 7 c and thus proves (33). From the fact that I 0 > 0 is the area of the region enclosed by Γ, the Mel'nikov function has a simple zero c = 7 5α β. By [8], we know that for every sufficiently small , there exists a corresponding c = c( ) (c(0) = 7 5α β) to guarantee that the homoclinic orbit survive from homoclinic bifurcation and therefore prove the existence of homoclinic orbit.
The monotonicity of R(h) implies that ifαβ < c < 7 5α β, there exists a h * ∈ (− 2c 3 3β , 0) satisfies M (h * ,α, β, c) = 0. Furthermore, h * is a simple zero of the Mel'nikov function M (h,α, β, c), since M (h * ,α, β, c) = β 2 R (h * ) < 0. Therefore, by the Poincaré bifurcation theory [3], whenαβ < c < 7 5α β and is small enough, there exists a limit cycle for system (29), which is close to the periodic orbit ΓẼ 1 (h * ). 5. Existence of other bounded traveling waves. In section 3 and 4, we obtain different wavespeed conditions to guarantee the existence of three basic types of bounded traveling waves of the KBK equation. From these bifurcation results, existence of more bounded traveling waves of the KBK equation can be identified. In fact, one can note that the region enclosed by a periodic orbit or a homoclinic loop is compact. It means that orbits in the compact region are bounded, and therefore correspond to the bounded traveling waves of the KBK equation. In addition, from properties of the dynamical system, it is well known that the limit cycle and the homoclinic loop are limit sets. So, when a limit cycle appears from the Poincare bifurcation, there will exist some connections between the equilibrium and the limit cycle, which correspond to a kind of oscillatory bounded traveling wave. Similarly, if a homoclinic orbit persists from the homoclinic bifurcation, there will exist some connections between the equilibrium and the homoclinic loop, which corresponds to another kind of oscillatory bounded traveling wave.