Saddle-Node Heteroclinic Orbit and Exact Nontraveling Wave Solutions for (2+1)D KdV-Burgers Equation

Abstract

We have undertaken the fact that the periodic solution of (2+1)D KdV-Burgers equation does not exist. The Saddle-node heteroclinic orbit has been obtained. Using the Lie group method, we get two-(1+1)-dimensional PDE, through symmetric reduction; and by the direct integral method, spread F-expansion method, and (𝐺^′/𝐺)-expansion method, we obtain exact nontraveling wave solutions, for the (2+1)D KdV Burgers equation, and find out some new strange phenomenons of sympathetic vibration to evolution of nontraveling wave.

1. Introduction

We consider the (2+1)-dimensional Korteweg-de Vries Burgers ((2+1)D KdV Burgers) equation 𝑢_𝑡+𝑢𝑢_𝑥−𝛽𝑢_𝑥𝑥+𝛼𝑢_{𝑥𝑥𝑥𝑥}+𝛾𝑢_𝑦𝑦=0, (1) where 𝑢∶𝑅_𝑥×𝑅_𝑦×𝑅⁺_𝑡→𝑅, 𝛼, 𝛽, and 𝛾 are real parameters. Equation (1) is model equation for wide class of nonlinear wave models in an elastic tube, liquid with small bubbles, and turbulence [1–3]. Much attention has been put on the study of their exact solutions by some methods [4], such as, a complex line soliton by extended tanh method with symbolic computation [5], exact traveling wave solutions including solitary wave solutions, periodic wave and shock wave solutions by extended mapping method, and homotopy perturbation method [6, 7].

It is well known that the investigation of exact solutions of nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. Many effective methods have been presented [7–22], such as functional variable separation method [8, 9], homotopy perturbation method [12], F-expansion method [7, 13], Lie group method [14, 15], variational iteration method [16], homoclinic test method [17–19], Exp-function method [20, 21], and homogeneous balance method [22]. Practically, there is no unified method that can be used to handle all types of nonlinearity.

In this paper, we will discuss the existence of periodic traveling wave solution and seek the Saddle-Node heteroclinic orbit, and further use the Lie group method with the aid of the symbolic computation system Maple to construct the non-traveling wave solutions for (1).

2. Existence of Periodic Traveling Wave Solution of (1)

Introducing traveling wave transformation in this form 𝑢𝑥,𝑦,𝑡=𝑢(𝜉),𝜉=𝑝𝑥+𝑞𝑦−𝑐𝑡 (2) permits us to convert (1) into an ODE for 𝑢=𝑢(𝜉)𝑝𝑝𝑢𝑢_𝜉−𝛽𝑝²𝑢_𝜉𝜉+𝛼𝑝³𝑢_{𝜉𝜉𝜉𝜉}−𝑟𝑢_𝜉𝜉=0, (3) where 𝑟=𝑝𝑐−𝑞²𝛾, Integrating (3) with respect to 𝜉 twice and taking integration constant to 𝐴 yields 2𝛼𝑝⁴𝑢_𝜉𝜉−2𝛽𝑝³𝑢_𝜉+𝑝²𝑢²−2𝑟𝑢=𝐴. (4) Letting 𝑢_𝜉=𝑣, thus nonlinear ordinary differential equation (4) is equivalent to the autonomous dynamic system as follows: 𝑑𝑢𝑑𝜉=𝑣,𝑑𝑣=1𝑑𝜉2𝛼𝑝⁴2𝛽𝑝³𝑣−𝑝²𝑢²+2𝑟𝑢+𝐴. (5)(6) The dynamic system (5) has two balance points: 𝑃₁𝑢₁,𝑣₁=𝑟+𝑟²+𝑝²𝐴𝑝²,𝑃,0₂𝑢₂,𝑣₂=𝑟−𝑟²+𝑝²𝐴𝑝²,0. (7) The Jacobi matrixes at the balance points for the right-hand side of (5) are obtained as follows, respectively: 𝐽₁=−01𝑟²+𝑝²𝐴𝑝⁴𝛼𝛽,𝐽𝑝𝛼₂=01𝑟²+𝑝²𝐴𝑝⁴𝛼𝛽.𝑝𝛼 (8) Their latent equations are expressed, respectively, as, 𝑝³𝜆𝑝𝛼𝜆−𝛽+𝑟²+𝑝²𝑝𝐴=0,₃𝜆𝑝𝛼𝜆−𝛽−𝑟²+𝑝²𝐴=0. (9) Relevant latent roots are as follows respectively: 𝜆₁=𝑝𝛽±𝑝²𝛽²−4𝛼𝑟²+𝑝²𝐴2𝑝²𝛼,𝜆₂=𝑝𝛽±𝑝²𝛽²+4𝛼𝑟²+𝑝²𝐴2𝑝²𝛼. (10) Obviously, if 𝑝²𝛽²>4𝛼𝑟²+𝑝²𝐴, then 𝜆₁ are two positive real roots, therefore 𝑃₁ is a nonsteady node point. If 0<𝑝²𝛽²<4𝛼𝑟²+𝑝²𝐴, then 𝜆₁ are conjugate complex roots and real part is positive, so 𝑃₁ is a nonsteady focus point. And 𝜆₂ is a positive and minus real root, thus 𝑃₂ is a saddle point. From (5), we know the phase trajectory on the phase plane satisfies 𝑑𝑣=𝑑𝑢2𝛽𝑝³𝑣−𝑝²𝑢²+2𝑟𝑢+𝐴2𝛼𝑝⁴𝑣. (11) Integrating (11), we can obtain 𝐻(𝑢,𝑣)=𝐴𝑢+𝑟𝑢²−13𝑝²𝑢³+2𝛽𝑝³𝑢𝑣−𝛼𝑝⁴𝑣², (12) where 𝐻(𝑢,𝑣) is a total energy or Hamiliton function of system (4). Apparently 𝑢_𝜉≠−𝜕𝐻𝜕𝑣,𝑣_𝜉≠𝜕𝐻.𝜕𝑢 (13) Consequently, the system expressed in (12) is not a conservative one, then periodic traveling wave solution of (1) does not exist.

We conclude the above analysis in the following theorem.

Theorem 1. Under the traveling wave transformation, the periodic solution of (2+1)-dimensional KdV-Burgers equation does not exist.
But, saddle-node heteroclinic orbits and nontraveling periodic solution do exist, which will be discussed later in this paper.

3. Saddle-Node Heteroclinic Orbits of KdV-Burgers Equation

First, we assume the solutions of (4) in the form 𝑢(𝜉)=𝑟+𝑟²+𝑝²𝐴𝑝²+𝑏1+𝑒^𝑎𝜉2. (14) Substituting (14) into (4) yields 24𝛼𝑝⁴𝑎²+𝑟²+𝑝²𝐴+2𝛽𝑝³𝑎𝑒^2𝑎𝜉−4𝛼𝑝⁴𝑎²−𝑟²+𝑝²𝐴−𝛽𝑝³𝑎𝑒^𝑎𝜉+2𝑟²+𝑝²𝐴+𝑝²𝑏=0. (15) Then we get 4𝛼𝑝⁴𝑎²+𝑟²+𝑝²𝐴+2𝛽𝑝³𝑎=0,𝛼𝑝⁴𝑎²−𝑟²+𝑝²𝐴−𝛽𝑝³2𝑎=0,𝑟²+𝑝²𝐴+𝑝²𝑏=0. (16) Solving the system (16) gets 𝛽𝑎=−5𝛼𝑝,𝑏=−12𝛽²,25𝛼𝑟²+𝑝²𝐴=6𝑝²𝛽².25𝛼 (17) Substituting (17) into (14) obtains 𝑢(𝜉)=𝑟+𝑟²+𝑝²𝐴𝑝²−12𝛽²125𝛼1+𝑒^{−(𝛽/5𝛼𝑝)𝜉2}=𝑢₁−3𝛽²𝛽25𝛼1+tanh𝜉20𝛼². (18) Evidently, 𝜉→−∞⇒𝑢(𝜉)→𝑢₁, 𝜉→+∞⇒𝑢(𝜉)→𝑢₁−(6𝛽²/25𝛼)=𝑢₂. Thus (18) is a saddle-node heteroclinic orbit through nonsteady node point 𝑃₁ and saddle point 𝑃₂ [23].

Ecumenic, taking the Hamiliton function 𝐻(𝑢,𝑣)=𝐵, we obtain 𝑑𝑢=𝑑𝜉=𝑣3𝑝𝛽𝑢±3𝑢3𝐴𝛼+3𝑝²𝛽²+𝑟𝛼𝑢−𝑝²𝛼𝑢²−9𝐵𝛼3𝛼𝑝², (19) where 𝐵 is an arbitrary constant. Integrating (19) with respect to 𝜉 we have ^𝑢(𝜉)3𝛼𝑝²3𝑝𝛽𝑠±3𝑠3𝐴𝛼+3𝑝²𝛽²+𝑟𝛼𝑠−𝑝²𝛼𝑠²−9𝐵𝛼𝑑𝑠=𝜉+𝜉₀, (20) where 𝜉₀ is an arbitrary constant. We can see that (4) has the general solution (20) and all partial cases as include above result can be found from the general solution of (20). Example, take 𝛼𝑟²+𝑝²𝐴−𝑝²𝛽²=0, 3𝐵𝛼+𝐴𝛽²=0, 𝑟𝛼+𝑝²𝛽²=0 in (20), we find a solution of (4) as follows: 𝑢(𝜉)=−3𝛽²𝛽4𝛼1+tanh4𝑝𝛼𝜉+𝜉₀². (21) It is a heteroclinic orbit too.

4. Li Symmetry of (1)

This section devotes to Li symmetry of (1) [14, 15]. Let 𝜎=𝜎𝑥,𝑦,𝑡,𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑦,…. (22) be the Li symmetry of (1). From Lie group theory, 𝜎 satisfies the following equation 𝜎_𝑥𝑡+2𝑢_𝑥𝜎_𝑥+𝑢𝜎_𝑥𝑥+𝜎𝑢_𝑥𝑥−𝛽𝜎_𝑥3+𝛼𝜎_𝑥4+𝛾𝜎_𝑦𝑦=0. (23) We take the function 𝜎 in the form 𝜎=𝑎₁𝑢_𝑥+𝑎₂𝑢_𝑦+𝑎₃𝑢_𝑡+𝑎₄𝑢+𝑎₅, (24) where 𝑎_𝑖=𝑎_𝑖(𝑥,𝑦,𝑡)∶𝑅_𝑥×𝑅_𝑦×𝑅⁺_𝑡→𝑅  (𝑖=1,…,5) are functions to be determined later. Substituting (3) into (2) yields 𝑎₁1=−𝑘2𝛾^′2(𝑡)𝑦+𝑘₁(𝑡),𝑎₂=𝑘₂(𝑡),𝑎₃=𝑐,𝑎₄=0,𝑎₅=1𝑘2𝛾^2′′(𝑡)𝑦−𝑘^′₁(𝑡), (25) where 𝑘_𝑗(𝑡)  (𝑗=1,2) are arbitrary functions of 𝑡, 𝑐 is an arbitrary constant. Substituting (25) into (24), we obtain the Li symmetries of (1) as follows: −1𝜎=𝑘2𝛾^′2(𝑡)𝑦+𝑘₁(𝑡)𝑢_𝑥+𝑘₂(𝑡)𝑢_𝑦+𝑐𝑢_𝑡+1𝑘2𝛾^2′′(𝑡)𝑦−𝑘^′₁(𝑡). (26)

5. Symmetry Reduction and Solutions of (1)

Based on the integrability of reduced equation of symmetry (26), we are to consider the following three cases.

Case 1. Taking 𝑘₂(𝑡)=0 and 𝑐=0 in (26) yields 𝜎=𝑘₁(𝑡)𝑢_𝑥−𝑘^′₁(𝑡). (27) The solution of the differential equation 𝜎=0 is 𝑘𝑢=^′1(𝑡)𝑘₁(𝑡)𝑥+𝐹𝑦,𝑡,𝐹𝑦,𝑡∶𝑅_𝑦×𝑅⁺_𝑡→𝑅. (28) Substituting (28) into (1) yields the function 𝐹(𝑦,𝑡) which satisfies the following linear PDE: 𝑘₁^′′𝑘₁𝜕+𝛾²𝐹𝜕𝑦²=0. (29) By integrating both sides, we find out the following result: 𝐹𝑘𝑦,𝑡=−^1′′2𝛾𝑘₁𝑦²+𝑘₃(𝑡)𝑦+𝑘₄(𝑡), (30) where 𝑘₃(𝑡), 𝑘₄(𝑡) are new arbitrary functions of 𝑡. Substituting (30) into (28), we can get the solutions of (1) as follows: 𝑢₁=𝑘𝑥,𝑦,𝑡^′1(𝑡)𝑘₁(𝑡)𝑘𝑥−^1′′2𝛾𝑘₁𝑦²+𝑘₃(𝑡)𝑦+𝑘₄(𝑡). (31) (1) Given 𝑘_𝑖(𝑡)=cn(𝑡,0.95)  (𝑖=1,3,4), 𝑥=1, 𝛾=0.6 in (31), the local structure of 𝑢₁ is obtained (Figure 1). Where cn(𝑡,0.95) is an Jacobian elliptic cosine function.
(2) Given 𝑘₁(𝑡)=sech(𝑡), 𝑘₃(𝑡)=sin(𝑡), 𝑘₄(𝑡)=cn(𝑡,0.1), 𝑦=1, 𝛾=0.6 in (31), the local structure of 𝑢₁ is obtained (Figure 2).

Figure 1 

The strange phenomenon which is a sympathetic vibration of periodicity on the 𝑡-axis and paraboloid on 𝑦-axis for 𝑢1(𝑥,𝑦,𝑡) as 𝑥=1.

Figure 2 

The periodic solution which is a periodic nontraveling wave traveling on the 𝑡-axis for 𝑢1(𝑥,𝑦,𝑡) as 𝑦=1.

Case 2. Take 𝑘₁(𝑡)=𝑡, 𝑘₂(𝑡)=1 and 𝑐=0 in (26), then 𝜎=𝑡𝑢_𝑥+𝑢_𝑦−1. (32) Solving the differential equation 𝜎=0, we can get 𝑢=𝑦+𝐹(𝑡,𝜉),𝜉=𝑥−𝑡𝑦. (33) Substituting (33) into (1) and integrating once with respect to 𝜉 yield 𝐹_𝑡+𝐹𝐹_𝜉+𝛾𝑡²𝐹_𝜉−𝛽𝐹_𝜉𝜉+𝛼𝐹_𝜉𝜉𝜉=0. (34) Again, further using the transformation of dependent variable to (34), 𝐹()(𝜃)1𝑡,𝜉=𝐹,𝜃=𝑘𝑡−3𝛾𝑡³+𝜉. (35) Substituting (35) into (34) and integrating once with respect to 𝜃 yield 2𝑘²𝛼𝐹^′′−2𝑘𝛽𝐹^′+𝐹²+2𝐹+𝐴=0, (36) where 𝐴 is an integration constant, 𝐹^′=𝑑𝐹(𝜃)/𝑑𝜃. We assume that the solution of (36) can be expressed in the form 𝐹(𝜃)=𝑎₀+𝑎₁𝑤(𝜃)+𝑎₂𝑤(𝜃)², (37) where 𝑎_𝑖  (𝑖=0,1,2) are constants to be determined later, 𝑤(𝜃) satisfies the following auxiliary equation 𝑤^′=𝑝+𝑞𝑤². (38) Substituting (37) and (38) into (36) and equating the coefficients of all powers of 𝑤 to zero yield a set of algebra equations for 𝑎₀, 𝑎₁, 𝑎₂, and 𝐴 as follows. 𝑤⁴:𝑎₂𝑎₂+12𝛼𝑘²𝑞²𝑤=0,₃:−4𝛽𝑘𝑎₂𝑞+2𝑎₁𝑎₂+4𝛼𝑘²𝑎₁𝑞²𝑤=0,₂:𝑎²₁+16𝛼𝑘²𝑎₂𝑞𝑝−2𝛽𝑘𝑎₁𝑞+2𝑎₂+2𝑎₂𝑎₀𝑤=0,₁:2𝑎₁𝑎₀−4𝛽𝑘𝑎₂𝑝+2𝑎₁+4𝛼𝑘²𝑎₁𝑤𝑞𝑝=0,₀:2𝑎₀+𝐴+4𝛼𝑘²𝑎₂𝑝²+𝑎²₀−2𝛽𝑘𝑎₁𝑝=0. (39) Solving the system of function equations with the aid of Maple, we obtain 𝑎₀=3𝛽²−25𝛼25𝛼,𝑎₁=6𝛽²𝑞25𝑠𝛼,𝑎₂=3𝛽²𝑞.25𝛼𝑝 (40) when 𝑘=𝛽/10𝑠𝛼, 𝑝𝑞<0, 𝐴=(625𝛼²−36𝛽⁴)/625𝛼², where 𝑠=√−𝑝𝑞.
It is known that solutions of (38) are as follows [24]: 𝑤(𝜃)=−𝑠tanh(𝑠𝜃),𝑤(𝜃)=−𝑠coth(𝑠𝜃). (41) Substituting (41), (40), (37), and (35) into (33), we obtain solutions of (1) as follows: 𝑢₂=1𝑥,𝑦,𝑡25𝛼3𝛽²−25𝛼−3𝑞𝛽²×𝛽tanh𝛾10𝑠𝛼𝑥−𝑡𝑦+𝑡−3𝑡³−2𝑝𝑞tanh²𝛽𝛾10𝑠𝛼𝑥−𝑡𝑦+𝑡−3𝑡³𝑢+𝑦,₃=1𝑥,𝑦,𝑡25𝛼3𝛽²−25𝛼−3𝑞𝛽²×𝛽coth𝛾10𝑠𝛼𝑥−𝑡𝑦+𝑡−3𝑡³−2𝑝𝑞coth²𝛽𝛾10𝑠𝛼𝑥−𝑡𝑦+𝑡−3𝑡³+𝑦. (42) (see Figures 3 and 4).

Figure 3 

Local structure of 𝑢2(𝑥,𝑦,𝑡) is shown as 𝑥=1,  𝛼=1,  𝛽=10,  𝑝=−1,  𝑞=1,  and 𝛾=6.

Figure 4 

Local structure of 𝑢3(𝑥,𝑦,𝑡) is shown as 𝑥=1,  𝛼=1,  𝛽=10,  𝑝=−1,  𝑞=1,  𝛾=6.

Remark 2. If we direct assume that the solution of (34) can be expressed in the form 𝐹(𝑡,𝜉)=𝑎₀(𝑡)+𝑎₁(𝑡)𝑤(𝜃)+𝑎₂(𝑡)𝑤(𝜃)², (43) where 𝜃=𝑓(𝑡)𝜉+𝑔(𝑡), 𝑓(𝑡), and 𝑔(𝑡) are continuous functions of 𝑡 to be determined later. 𝑤(𝜃) satisfies the auxiliary equation (38). Substituting (43) and (38) into (34), equating the coefficients of all powers of 𝑤 to zero yields a set of function equations for 𝑎₀(𝑡), 𝑎₁(𝑡), 𝑎₂(𝑡), 𝑓(𝑡), and 𝑔(𝑡) as follows: 𝑤⁵:2𝑓𝑎₂𝑞12𝑓²𝑞²𝛼+𝑎₂𝑤=0,₄:−3𝑓𝑞−2𝑎₁𝑞²𝑓²𝛼+2𝑞𝑓𝑎₂𝛽−𝑎₁𝑎₂𝑤=0,₃:−2𝛽𝑎₁𝑓²𝑞²+2𝑎²₂𝑓𝑝+40𝛼𝑎₂𝑓³𝑞²𝑝+2𝑎₂𝑔^′𝑞+2𝑎₀𝑎₂𝑓𝑞+𝑎²₁𝑓𝑞+2𝑎₂𝑓^′𝜉𝑞+2𝛾𝑡²𝑎₂𝑤𝑓𝑞=0,₂:−8𝛽𝑎₂𝑓²𝑝𝑞+𝑎₁𝑔^′𝑞+𝑎^′₂+𝑎₀𝑎₁𝑓𝑞+𝛾𝑡²𝑎₁𝑓𝑞+8𝛼𝑎₁𝑓³𝑝𝑞²+𝑎₁𝑓^′𝜉𝑞+3𝑎₁𝑎₂𝑤𝑓𝑞=0,₁:𝑎²₁𝑓𝑝+16𝛼𝑎₂𝑓³𝑝²𝑞+𝑎^′₁+2𝛾𝑡²𝑎₂𝑓𝑝+2𝑎₀𝑎₂𝑓𝑝+2𝑎₂𝑔^′𝑝+2𝑎₂𝑓^′𝜉𝑝−2𝛽𝑎₁𝑓²𝑤𝑝𝑞=0,₀:𝑎₁𝑔^′𝑝+𝑎₁𝑓^′𝜉𝑝+𝑎₀𝑎₁𝑓𝑝−2𝛽𝑎₂𝑓²𝑝²+𝑎^′₀+2𝛼𝑎₁𝑓³𝑝²𝑞+𝛾𝑡²𝑎₁𝑓𝑝=0. (44) Solving the system of function equations, we obtain 𝑎₀(𝑡)=3𝛽²25𝛼,𝑎₁(𝑡)=±6𝛽²𝑞,𝑎25𝑠𝛼₂(𝑡)=3𝛽²𝑞(𝑡)𝛽25𝛼𝑝,𝑓=±(𝑡)10𝑠𝛼,𝑔=∓𝛽𝛾𝑡30𝑠𝛼³. (45) This result indicate the idea is equivalent to idea of Case 2 above.

Case 3. Take 𝑘₂(𝑡)=0 and 𝑐=1 in (26), then 𝜎=𝑘₁(𝑡)𝑢_𝑥+𝑢_𝑡−𝑘₁(𝑡). (46) Solving the differential equation 𝜎=0, we obtain 𝑢=𝑘₁(𝑡)+𝐹𝜉,𝑦,𝜉=𝑥−𝑘₁(𝑡)𝑑𝑡. (47) Substituting (47) into (1) yield 𝛼𝐹_{𝜉𝜉𝜉𝜉}−𝛽𝐹_𝜉𝜉𝜉+𝐹𝐹_𝜉𝜉+𝐹²_𝜉+𝛾𝐹_𝑦𝑦=0. (48) Using the transformation 𝐹(𝜉,𝑦)=𝐹(𝜂),𝜂=𝑘𝜉−𝑐𝑦 and integrating the resulting equation with respect to 𝜂 we have 𝑘²𝐹²+2𝛾𝑐²𝐹+2𝑘⁴𝛼𝐹^′′−2𝑘³𝛽𝐹^′+𝐴=0, (49) where 𝐴 is an arbitrary constant, 𝐹^′=𝑑𝐹/𝑑𝜂. Suppose that the solution of ODE (49) can be expressed by a polynomial in (𝐺^′/𝐺) as follows: 𝐹𝜂=𝑏_𝑛𝐺^′𝐺^𝑛+⋯, (50) where 𝐺=𝐺(𝜂) satisfies the second-order LODE in the form [25] 𝐺^′′+𝜆𝐺^′+𝜇𝐺=0. (51) Balancing 𝐹^′′ with 𝐹² in (49) gives 𝑛=2. So that 𝐹𝜂=𝑏₂𝐺^′𝐺²+𝑏₁𝐺^′𝐺+𝑏₀,𝑏₂≠0, (52) where 𝑏_𝑖  (𝑖=0,1,2) and 𝜇 are constants to be determined later. Substituting (52) and (51) into (49). Setting these coefficients of the 𝐺^′/𝐺 to zero, yields a set of algebraic equations as follows: 𝑘²𝑏₂12𝛼𝑘²+𝑏₂=0,2𝑘²10𝛼𝑘²𝑏₂𝜆+𝑏₁𝑏₂+2𝛼𝑘²𝑏₁+2𝛽𝑘𝑏₂=0,8𝛼𝑘⁴𝑏₂𝜆²+2𝛽𝑘³𝑏₁+𝑘²𝑏²₁+16𝛼𝑘⁴𝑏₂𝜇+2𝑘²𝑏₂𝑏₀+6𝛼𝑘⁴𝑏₁𝜆+4𝛽𝑘³𝑏₂𝜆+2𝛾𝑐²𝑏₂=0,2𝑘²𝑏₁𝑏₀+4𝛽𝑘³𝑏₂𝜇+4𝛼𝑘⁴𝑏₁𝜇+2𝛾𝑐²𝑏₁+2𝛼𝑘⁴𝑏₁𝜆²+2𝛽𝑘³𝑏₁𝜆+12𝛼𝑘⁴𝑏₂𝜆𝜇=0,2𝛾𝑐²𝑏₀+2𝛼𝑘⁴𝑏₁𝜆𝜇+𝐴+2𝛽𝑘³𝑏₁𝜇+4𝛼𝑘⁴𝑏₂𝜇²+𝑘²𝑏²₀=0. (53) Solving the algebraic equations above yields 𝑏₀=15𝑘³𝜆𝛼5𝑘𝜆𝛼+2𝛽−3𝑘²𝛽²+25𝑐²𝛼𝛾25𝑘²𝛼,𝑏₁=−12𝑘5𝑘𝛼𝜆+𝛽5,𝑏₂=−12𝑘²𝛼. (54) when 25𝑘²𝛼²(4𝜇−𝜆²)+𝛽²=0 and 625𝛼²(𝐴𝑘²−𝑐²𝛾²)+36𝑘⁴𝛽⁴=0. Consequently, we obtain the following solution of (1) for 𝜆²−4𝜇>0: 𝑢₄𝑥,𝑦,𝑡=−12𝑘²𝛼𝜏²×𝐶₁sinh𝜏𝑘𝑥−𝑘₁(𝑡)𝑑𝑡−𝑐𝑦+𝐶₂cosh𝜏𝑘𝑥−𝑘₁(𝑡)×𝐶𝑑𝑡−𝑐𝑦₁cosh𝜏𝑘𝑥−𝑘₁(𝑡)𝑑𝑡−𝑐𝑦+𝐶₂×𝑘𝑘sinh𝜏𝑥−₁(𝑡)𝑑𝑡−𝑐𝑦⁻¹²+12𝑘²𝜆𝛼𝜏−12𝑘5𝑘𝛼𝜆+𝛽5×𝐶₁sinh𝜏𝑘𝑥−𝑘₁(𝑡)𝑑𝑡−𝑐𝑦+𝐶₂cosh𝜏𝑘𝑥−𝑘₁(𝑡)×𝐶𝑑𝑡−𝑐𝑦₁cosh𝜏𝑘𝑥−𝑘₁(𝑡)𝑑𝑡−𝑐𝑦+𝐶₂×𝑘𝑘sinh𝜏𝑥−₁(𝑡)𝑑𝑡−𝑐𝑦⁻¹²+15𝑘³𝜆𝛼5𝑘𝜆𝛼+2𝛽−3𝑘²𝛽²+25𝑐²𝛼𝛾25𝑘²𝛼+𝜆²4, (55) where 𝜏=(1/2)𝜆²−4𝜇.

6. Conclusions

Based on the fact that the periodic solution of (2+1)D KdV-Burgers equation does not exist, we have obtained Saddle-node Heteroclinic Orbits. By applying the Lie group method, we reduce the (2+1)D KdV Burgers equation to (1+1)-dimensional equations including the (1+1)-dimensional linear partial differential equation with constants coefficients (29), (48) and (1+1)-dimensional nonlinear partial differential equation with variable coefficients (34). By solving the equations (29), (34), and (48), we obtain some new exact solutions and discover the strange phenomenon of sympathetic vibration to evolution of nontraveling wave soliton for the (2+1)D KdV Burgers equation. Our results show that the unite of Lie group method with others is effective to search simultaneously exact solutions for nonlinear evolution equations. Other structures of solutions with symmetry (26) are to be further studied.

Acknowledgments

The authors would like to thank professor S. Y. Lou for the helpful discussions. This work was supported by key research projects of Sichuan Provincial Educational Administration no. 10ZA021 and Chinese Natural Science Foundation Grant no. 10971169.