A New Variable-Coefficient Riccati Subequation Method for Solving Nonlinear Lattice Equations

Abstract

We propose a new variable-coefficient Riccati subequation method to establish new exact solutions for nonlinear differential-difference equations. For illustrating the validity of this method, we apply it to the discrete (2 + 1)-dimensional Toda lattice equation. As a result, some new and generalized traveling wave solutions including hyperbolic function solutions, trigonometric function solutions, and rational function solutions are obtained.

1. Introduction

Nonlinear differential-difference equations (NDDEs) play an important role in many branches of applied physical sciences such as condensed matter physics, biophysics, atomic chains, molecular crystals, and quantum physics. Unlike difference equations which are fully discretized, NDDEs are semidiscretized; that is, their time variable is usually kept continuous. Since the work of Fermi in the 1960s [1], research for NDDEs has been paid a lot of attention recently (e.g., see [2–10] and the references therein). Among these research works, the investigation of exact solutions of NDDEs has attracted a wide interest, and many effective methods have been presented and applied for solving NDDEs successfully in the literature. For example, these methods include the known (𝐺^′/𝐺)-expansion method [11–14], the exp-function method [15], the exponential function rational expansion method [16, 17], the Jacobi elliptic function method [18, 19], Hirota’s bilinear method [20], the extended simplest equation method [21], and the tanh-function method [22]. The difference among these methods lies in that different ansatzes are used. However, we notice most of the existing methods are dealing with ansatzes with constant coefficients, while very few methods are concerned with variable coefficients. Recent results on solving NDDEs by variable-coefficient methods only include the presentation of Tang et al. [14], in which a variable-coefficient (𝐺^′/𝐺)-expansion method was developed.

It is well known that the Riccati equation method is very effective in finding exact solutions for nonlinear differential-difference equations [23, 24]. But so far little attention is paid to the application of the Riccati equation method to solve NDDEs. In this paper, by introducing a new ansatz, we propose a new discrete variable-coefficient Riccati subequation method for solving NDDEs. Then we apply this method to establish new exact solutions of NDDEs.

We organize this paper as follows. In Section 2, we give the description of the discrete variable-coefficient Riccati subequation method. Then in Section 3 we apply the method to solve the discrete (2 + 1)-dimensional Toda lattice equation. Comparisons between the proposed method and the known (𝐺^′/𝐺)-expansion method are also made. Some conclusions are presented at the end of the paper.

2. Description of the Variable-Coefficient Riccati Subequation Method

The main steps of the variable-coefficient Riccati sub-ODE method for solving NDDEs are summarized as follows.

Step 1. Consider a system of 𝑀 polynomial NDDEs in the form 𝑃𝑢_𝑛+𝑝1(𝑋),…,𝑢_{𝑛+𝑝𝑘}(𝑋)𝑢,…,_{′𝑛+𝑝1}(𝑋),…,𝑢^′_{𝑛+𝑝𝑘}(𝑋)𝑢,…,_{(𝑟)𝑛+𝑝1}(𝑋),…,𝑢^(𝑟)_{𝑛+𝑝𝑘}(𝑋)=0, (1) where the dependent variable 𝑢 has 𝑀 components 𝑢_𝑖, the continuous variable 𝑋 has 𝑁 components 𝑥_𝑗, the discrete variable 𝑛 has 𝑄 components 𝑛_𝑖, the 𝑘 shift vectors 𝑝_𝑠∈𝑍^𝑄 have 𝑄 components 𝑝_𝑠𝑗, and 𝑢^(𝑟)_{𝑛+𝑝𝑠}(𝑋),𝑠=1,2,…,𝑘, denote the collection of mixed derivative terms of order 𝑟.

Step 2. Using a wave transformation 𝑢_{𝑛+𝑝𝑠}(𝑋)=𝑈_{𝑛+𝑝𝑠}𝜉_{𝑛+𝑝𝑠},𝜉_𝑛=^𝑄_𝑖=1𝑑_𝑖𝑛_𝑖(𝑋),𝜉+𝜁_{𝑛+𝑝𝑠}=^𝑄_𝑖=1𝑑_𝑖𝑛_𝑖+𝑝_𝑠𝑖+𝜁(𝑋), (2) where 𝑑_𝑖,𝑝_𝑠𝑖, 𝑖=1,…,𝑄, are constants and 𝜁(𝑋) are underdetermined differentiable functions of 𝑋, we can rewrite (1) in the following nonlinear ODE: 𝑃𝑈_𝑛+𝑝1𝜉_𝑛+𝑝1𝑈,…,_{𝑛+𝑝𝑘}𝜉_{𝑛+𝑝𝑘},…,𝑈^′_𝑛+𝑝1𝜉_𝑛+𝑝1𝑈,…,_{′𝑛+𝑝𝑘}𝜉_{𝑛+𝑝𝑘}𝑈,…,_{(𝑟)𝑛+𝑝1}𝜉_𝑛+𝑝1,…,𝑈^(𝑟)_{𝑛+𝑝𝑘}𝜉_{𝑛+𝑝𝑘}=0. (3)

Step 3. Suppose the solutions of (3) can be denoted by 𝑈_𝑛𝜉_𝑛=^𝑙_𝑖=0𝑎_𝑖(𝑋)𝜙^𝑖𝜉_𝑛, (4) where 𝑎_𝑖(𝑋) are functions of 𝑋 to be determined later, 𝑙 is a positive integer that can be determined by balancing the highest order linear term with the nonlinear terms in (3), and 𝜙(𝜉_𝑛) satisfies the known Riccati equation: 𝜙^′𝜉_𝑛=𝜎+𝜙²𝜉_𝑛. (5)

Step 4. We present some special solutions 𝜙₁,…𝜙₆ for (5).
When 𝜎<0, 𝜙₁𝜉_𝑛=−√−𝜎tanh√−𝜎𝜉_𝑛+𝑐₀,𝜙₂𝜉_𝑛=−√−𝜎coth√−𝜎𝜉_𝑛+𝑐₀,𝜙_1,2𝜉_{𝑛+𝑝𝑠}=𝜙_1,2𝜉_𝑛−√−𝜎tanh√−𝜎∑^𝑄_𝑖=1𝑑_𝑖𝑝_𝑠𝑖1−𝜙_1,2𝜉_𝑛/√−𝜎tanh√−𝜎∑^𝑄_𝑖=1𝑑_𝑖𝑝_𝑠𝑖, (6) where 𝑐₀ is an arbitrary constant.
When 𝜎>0, 𝜙₃𝜉_𝑛=√𝜎tan√𝜎𝜉_𝑛+𝑐₀,𝜙₄𝜉_𝑛=−√𝜎cot√𝜎𝜉_𝑛+𝑐₀,𝜙_3,4𝜉_{𝑛+𝑝𝑠}=𝜙_3,4𝜉_𝑛+√𝜎tan√𝜎∑^𝑄_𝑖=1𝑑_𝑖𝑝_𝑠𝑖1−𝜙_3,4𝜉_𝑛/√𝜎tan√𝜎∑^𝑄_𝑖=1𝑑_𝑖𝑝_𝑠𝑖,𝜙₅𝜉_𝑛=√𝜎tan2√𝜎𝜉_𝑛+𝑐₀+sec2√𝜎𝜉_𝑛+𝑐₀,𝜙₅𝜉_{𝑛+𝑝𝑠}=𝜙₅⁽¹⁾𝜉_𝑛+√𝜎tan2√𝜎∑^𝑄_𝑖=1𝑑_𝑖𝑝_𝑠𝑖1−𝜙₅⁽¹⁾𝜉_𝑛/√𝜎tan2√𝜎∑^𝑄_𝑖=1𝑑_𝑖𝑝_𝑠𝑖+𝜙₅⁽²⁾𝜉_𝑛sec2√𝜎∑^𝑄_𝑖=1𝑑_𝑖𝑝_𝑠𝑖1−𝜙₅⁽¹⁾𝜉_𝑛/√𝜎tan2√𝜎∑^𝑄_𝑖=1𝑑_𝑖𝑝_𝑠𝑖, (7)(8) where 𝜙₅⁽¹⁾(𝜉_𝑛)=√𝜎tan(2√𝜎𝜉_𝑛+𝑐₀), 𝜙₅⁽²⁾(𝜉_𝑛)=√𝜎|sec(2√𝜎𝜉_𝑛+𝑐₀) and 𝑐₀ is an arbitrary constant.
When 𝜎=0, 𝜙₆𝜉_𝑛1=−𝜉_𝑛+𝑐₀,𝜙₆𝜉_{𝑛+𝑝𝑠}=𝜙₆𝜉_𝑛1−𝜙₆𝜉_𝑛∑^𝑄_𝑖=1𝑑_𝑖𝑝_𝑠𝑖, (9) where 𝑐₀ is an arbitrary constant.

Step 5. Substituting (4) into (3), by use of (5)–(9), the left hand side of (3) can be converted into a polynomial in 𝜙(𝜉_𝑛). Equating each coefficient of 𝜙^𝑖(𝜉_𝑛) to zero, yields a set of algebraic equations. Solving these equations, we can obtain a set of under-determined partial equations, from which 𝑎_𝑖(𝑋), 𝜁(𝑋) can be determined using the aid of mathematical software.

Step 6. Substituting the values of 𝑎_𝑖(𝑋) into (4) and combining with the various solutions of (5), we can obtain a variety of exact solutions for (1).

Remark 1. In [25, equations (40)–(42)], Ma and Fuchssteiner presented a general class of solutions to the following Riccati equation: 𝑣_𝜉=𝑎₀+𝑎₁𝑣+𝑎₂𝑣². (10)

We note that despite the difference in forms and parameters, our solutions 𝜙₁(𝜉_𝑛), 𝜙₂(𝜉_𝑛),  𝜙₃(𝜉_𝑛), 𝜙₄(𝜉_𝑛), and 𝜙₆(𝜉_𝑛) to (5) are in accordance with the presented solutions in [25]. In fact, 𝜙₁(𝜉_𝑛) is essentially equivalent to the tanh-function solution in [25, (41)] with 𝑎₁=0,𝑎₂=1, Δ=−4𝜎, and −𝜀ln𝜀₀/2=𝑐₀, while 𝜙₂(𝜉_𝑛) is essentially equivalent to the coth-function solution in [25, (41)] with 𝑎₁=0, 𝑎₂=1, Δ=−4𝜎, and −𝜀ln(−𝜀₀)/2=𝑐₀. 𝜙₃(𝜉_𝑛) is essentially equivalent to the tan-function solution in [25, (42)] with 𝑎₁=0, 𝑎₂=1, Δ=−4𝜎, and𝜉₀=𝑐₀, while 𝜙₄(𝜉_𝑛) is essentially equivalent to the cot-function solution in [25, (42)] with 𝑎₁=0, 𝑎₂=1, Δ=−4𝜎, and 𝜉₀=𝑐₀. 𝜙₇(𝜉_𝑛) is essentially equivalent to the rational function solution in [25, (40)] with 𝑎₁=0, 𝑎₂=1, and 𝜉₀=𝑐₀. Besides, our solution 𝜙₅(𝜉_𝑛) is not shown in [25].

3. Application of the Variable-Coefficient Riccati Subequation Method to the Discrete (2 + 1)-Dimensional Toda Lattice Equation

In this section, we will apply the described method in Section 2 to the discrete (2 + 1)-dimensional Toda lattice equation [13, 14]: 𝜕²𝑦_𝑛𝜕𝑥𝜕𝑡=𝑒^{𝑦𝑛−1−𝑦𝑛}−𝑒^{𝑦𝑛−𝑦𝑛+1}, (11) where 𝑦_𝑛=𝑦_𝑛(𝑥,𝑡), 𝑛∈𝑍, is the displacement from equilibrium of the 𝑛th unit mass under an exponentially decaying interaction force between nearest neighbors.

In [26], Ma presented a general Casoratian formulation for the 2D Toda lattice equation and obtained various Casoratian-type solutions by a Bäcklund transformation under some linear conditions. In [27], Ma derived molecule solutions in Wronskian form for the finite, semi-infinite, and infinite bilinear 2D Toda molecule equations by combining two pieces of molecule bidirectional Wronskian solutions, and in the cases of finite and semi-infinite lattices, separated-variable boundary conditions are imposed.

As in [13, 14], using the transformation 𝜕𝑢_𝑛𝜕𝑡=𝑒^{𝑦𝑛−1−𝑦𝑛}−1, (12)

(11) can be rewritten as 𝜕²𝑢_𝑛=𝜕𝑥𝜕𝑡𝜕𝑢_𝑛𝑢𝜕𝑡+1_𝑛−1−2𝑢_𝑛+𝑢_𝑛+1. (13)

Using a wave transformation 𝑢_𝑛(𝑥,𝑡)=𝑈_𝑛𝜉_𝑛,𝜉_𝑛=𝑑₁𝑛+𝜁(𝑥,𝑡), (14) where 𝑑₁ is a constant, (13) can be turned into 𝜕𝜁(𝑥,𝑡)()𝜕𝑥𝜕𝜁𝑥,𝑡𝜕𝑡+𝑈_𝑛^′′𝜕²𝜁(𝑥,𝑡)𝑈𝜕𝑥𝜕𝑡^′𝑛=𝜕𝜁(𝑥,𝑡)𝑈𝜕𝑡^′𝑛+1𝑈_𝑛−1−2𝑈_𝑛+𝑈_𝑛+1, (15) where 𝑈_𝑛=𝑈_𝑛(𝜉_𝑛),  𝑈^′_𝑛=𝑈^′_𝑛(𝜉_𝑛). Suppose 𝑈_𝑛(𝜉_𝑛) can be denoted by 𝑈_𝑛𝜉_𝑛=^𝑙_𝑖=0𝑎_𝑖(𝑥,𝑡)𝜙^𝑖𝜉_𝑛, (16) where 𝜙(𝜉_𝑛) satisfies (5). Balancing the order of 𝑈_𝑛^′′ and 𝑈^′_𝑛𝑈_𝑛 in (15), we obtain 𝑙+2=(𝑙+1)+𝑙, and then 𝑙=1. So we have 𝑈_𝑛𝜉_𝑛=𝑎₀(𝑥,𝑡)+𝑎₁(𝑥,𝑡)𝜙𝜉_𝑛. (17)

Next we will proceed to solve (15) and (13) in several cases.

Case 1. If 𝜎<0 and assume (5) and (6) hold, then substituting (17), (5), and (6) into (15), collecting the coefficients of 𝜙^𝑖_1,2(𝜉_𝑛), and equating them to zero, we obtain a series of underdetermined partial equations for 𝑎₀(𝑥,𝑡), 𝑎₁(𝑥,𝑡), and 𝜁(𝑥,𝑡). Solving these equations yields 𝑎₁(𝑥,𝑡)=𝐹₁(𝑥),𝑎₀=−𝑡+𝐹₂(𝑥),𝜁(𝑥,𝑡)=𝐹₃(𝑥), (18) where 𝐹_𝑖(𝑥),  𝑖=1,2,3, are arbitrary functions with respect to the variable 𝑥, or 𝑎₁()𝑎𝑥,𝑡=−𝑘,₀=−𝑡−2𝑘𝜎𝑔₁(𝑡)−1+cosh2√−𝜎𝑑₁+𝑔₂(𝑥),𝜁(𝑥,𝑡)=𝑘𝑥+𝑔₁(𝑡), (19) where 𝑘 is an arbitrary nonzero constant and 𝑔₁(𝑡) and 𝑔₂(𝑥) are two arbitrary functions with respect to the variables 𝑥 and 𝑡, respectively. Then we obtain the following hyperbolic function solitary wave solutions for (13): 𝑢_𝑛(𝑥,𝑡)=−𝐹₁(𝑥)√−𝜎tanh√−𝜎𝑑₁𝑛+𝐹₃(𝑥)+𝑐₀−𝑡+𝐹₂(𝑥),𝑢_𝑛(𝑥,𝑡)=−𝐹₁(𝑥)√−𝜎coth√−𝜎𝑑₁𝑛+𝐹₃(𝑥)+𝑐₀−𝑡+𝐹₂(𝑥),𝑢_𝑛(𝑥,𝑡)=𝑘√−𝜎tanh√−𝜎𝑑₁𝑛+𝑘𝑥+𝑔₁(𝑡)+𝑐₀−𝑡−2𝑘𝜎𝑔₁(𝑡)−1+cosh2√−𝜎𝑑₁+𝑔₂(𝑥),𝑢_𝑛(𝑥,𝑡)=𝑘√−𝜎coth√−𝜎𝑑₁𝑛+𝑘𝑥+𝑔₁(𝑡)+𝑐₀−𝑡−2𝑘𝜎𝑔₁(𝑡)−1+cosh2√−𝜎𝑑₁+𝑔₂(𝑥), (20)(21)(22)(23) where 𝑑₁, 𝑐₀ are arbitrary constants.

Case 2. If 𝜎>0 and assume (5) and (7) hold, then substituting (17), (5), and (7) into (15), collecting the coefficients of 𝜙^𝑖_3,4(𝜉_𝑛), and equating them to zero, we obtain a series of underdetermined partial equations for 𝑎₀(𝑥,𝑡), 𝑎₁(𝑥,𝑡), and 𝜁(𝑥,𝑡). Solving these equations yields 𝑎₁(𝑥,𝑡)=𝐹₁(𝑥),𝑎₀=−𝑡+𝐹₂(𝑥),𝜁(𝑥,𝑡)=𝐹₃(𝑥) (24) or 𝑎₁()𝑎𝑥,𝑡=−𝑘,₀=−𝑡−2𝑘𝜎𝑔₁(𝑡)−1+cos2√𝜎𝑑₁+𝑔₂(𝑥),𝜁(𝑥,𝑡)=𝑘𝑥+𝑔₁(𝑡). (25)

Then we obtain the following trigonometric function solutions for (13): 𝑢_𝑛(𝑥,𝑡)=𝐹₁(𝑥)√𝜎tan√𝜎𝑑₁𝑛+𝐹₃(𝑥)+𝑐₀−𝑡+𝐹₂(𝑥),𝑢_𝑛(𝑥,𝑡)=−𝐹₁(𝑥)√𝜎cot√𝜎𝑑₁𝑛+𝐹₃(𝑥)+𝑐₀−𝑡+𝐹₂(𝑥),𝑢_𝑛(𝑥,𝑡)=−𝑘√𝜎tan√𝜎𝑑₁𝑛+𝑘𝑥+𝑔₁(𝑡)+𝑐₀−𝑡−2𝑘𝜎𝑔₁(𝑡)−1+cos2√𝜎𝑑₁+𝑔₂(𝑥),𝑢_𝑛(𝑥,𝑡)=𝑘√𝜎cot√𝜎𝑑₁𝑛+𝑘𝑥+𝑔₁(𝑡)+𝑐₀−𝑡−2𝑘𝜎𝑔₁(𝑡)−1+cos2√𝜎𝑑₁+𝑔₂(𝑥), (26)(27)(28)(29) where 𝑑₁, 𝑐₀ are arbitrary constants.

Case 3. If 𝜎>0 and assume (5) and (8) hold, then substituting (17), (5), and (8) into (15), using [𝜙₅⁽²⁾(𝜉_𝑛)]²=𝜎+[𝜙₅⁽¹⁾(𝜉_𝑛)]², collecting the coefficients of [𝜙₅⁽¹⁾(𝜉_𝑛)]^𝑖[𝜙₅⁽²⁾(𝜉_𝑛)]^𝑗, and equating them to zero, we obtain a series of partial equations. Solving these equations, we also obtain, as in Case 2, 𝑎₁(𝑥,𝑡)=𝐹₁(𝑥),𝑎₀=−𝑡+𝐹₂(𝑥),𝜁(𝑥,𝑡)=𝐹₃(𝑥) (30) or 𝑎₁()𝑎𝑥,𝑡=−𝑘,₀=−𝑡−2𝑘𝜎𝑔₁(𝑡)−1+cos2√𝜎𝑑₁+𝑔₂(𝑥),𝜁(𝑥,𝑡)=𝑘𝑥+𝑔₁(𝑡). (31) So we have the following trigonometric function solutions for (13): 𝑢_𝑛(𝑥,𝑡)=𝐹₁(𝑥)√𝜎tan2√𝜎𝑑₁𝑛+𝐹₃(𝑥)+𝑐₀+sec2√𝜎𝑑₁𝑛+𝐹₃(𝑥)+𝑐₀−𝑡+𝐹₂(𝑥),𝑢_𝑛(𝑥,𝑡)=−𝑘√𝜎tan2√𝜎𝑑₁𝑛+𝑘𝑥+𝑔₁(𝑡)+𝑐₀+sec2√𝜎𝑑₁𝑛+𝑘𝑥+𝑔₁(𝑡)+𝑐₀−𝑡−2𝑘𝜎𝑔₁(𝑡)−1+cos2√𝜎𝑑₁+𝑔₂(𝑥), (32)(33) where 𝑑₁, 𝑐₀ are arbitrary constants.

Case 4. If 𝜎=0 and assume (5) and (9) hold, then substituting (17), (5), and (9) into (15), collecting the coefficients of 𝜙^𝑖₆(𝜉_𝑛), and equating them to zero, we obtain a series of partial equations. Solving these equations yields 𝑎₁(𝑥,𝑡)=𝐹₁(𝑥),𝑎₀=−𝑡+𝐹₂(𝑥),𝜁(𝑥,𝑡)=𝐹₃(𝑥) (34) or 𝑎₁(𝑥,𝑡)=−𝑘,𝑎₀=𝑔₂(𝑥)+𝑔₁(𝑡),𝜁()𝑑𝑥,𝑡=𝑘𝑥+²¹𝑡+𝑔₁(𝑡)𝑘+𝑚, (35) where 𝑚 is an arbitrary constant, and then we can obtain the following rational function solutions for (13): 𝑢_𝑛()𝐹𝑥,𝑡=−¹(𝑥)𝑑₁𝑛+𝐹₃(𝑥)+𝑐₀−𝑡+𝐹₂(𝑥),𝑢_𝑛()=𝑘𝑥,𝑡𝑑₁𝑛+𝑘𝑥+𝑑²₁𝑡+𝑔₁(𝑡)/𝑘+𝑚+𝑐₀+𝑔₂(𝑥)+𝑔₁(𝑡). (36)(37)

Remark 2. In [14], Tang et al. presented some exact solutions for (11) using the variable-coefficient (𝐺^′/𝐺)-expansion method. We note that our results (22), (28), and (37) are generalizations of [14, equations (17), (23), (29)]. In fact, if we let in (22) 𝑐₀ = arth(𝐶₂/𝐶₁) or arcoth(𝐶₂/𝐶₁), 𝜎=(4𝜇−𝜆²)/4, 𝑔₁(𝑡)=𝑔(𝑡), and 𝑔₂(𝑥)=ℎ(𝑥)−𝑘𝜆/2, then the solution denoted by (22) becomes [14, equation (17)]. If we let in (28) 𝑐₀=arctan(−𝐶₂/𝐶₁) or arccot(−𝐶₂/𝐶₁), 𝜎=(4𝜇−𝜆²)/4, 𝑔₁(𝑡)=𝑔(𝑡), and 𝑔₂(𝑥)=ℎ(𝑥)−𝑘𝜆/2, then the solution denoted by (28) becomes [14, equation (23)]. If we let in (37) 𝑐₀=𝐶₂/𝐶₁, 𝑔₁(𝑡)=𝑘𝑔(𝑡)/𝑑²₁−𝑡, 𝑚=𝑘, and 𝑔₂(𝑥)=ℎ(𝑥)−𝑘𝜆/2, then the solution denoted by (37) becomes [14, equation (29)].

Remark 3. It is worthy to note that the solutions in [14, equations (18), (24), (30)] are obtained under the hypothesis 𝑈_𝑛𝜉_𝑛=^𝑚_𝑙=−𝑚𝑎_𝑖(𝑋)𝐺^′𝜉_𝑛𝐺𝜉_𝑛. (38) If we modify (4) by 𝑈_𝑛𝜉_𝑛=^𝑙_𝑖=−𝑙𝑎_𝑖(𝑋)𝜙^𝑖𝜉_𝑛, (39) then following a similar manner as above we can obtain some exact solutions for (13), which are generalizations of [14, equations (18), (24), (30)]. As the process is almost the same, we omit it here.

Remark 4. Our results (32) and (33) are new exact solutions for (13) and have not been reported by other authors so far to our best knowledge.

Remark 5. From the analysis above, we notice that more general exact solutions for the discrete (2 + 1)-dimensional Toda lattice equation are obtained by the proposed variable-coefficient Riccati subequation method than by the (𝐺^′/𝐺)-expansion method. In fact, in the (𝐺^′/𝐺)-expansion method, the solutions 𝑈_𝑛(𝜉_𝑛) are denoted by a polynomial in (𝐺^′(𝜉_𝑛)/𝐺(𝜉_𝑛)), and 𝐺 satisfies 𝐺^′′+𝜆𝐺^′+𝜇𝐺=0, (40) where 𝜆,𝜇 are constants. If we let in (40) (𝐺^′(𝜉_𝑛)/𝐺(𝜉_𝑛))=−𝜙(𝜉_𝑛)−𝜆/2,(4𝜇−𝜆²)/4=𝜎, then (40) reduces to 𝜙^′(𝜉_𝑛)=𝜎+𝜙²(𝜉_𝑛), which is the Riccati equation (5). So (𝐺^′(𝜉_𝑛)/𝐺(𝜉_𝑛)) can be generalized and explained by 𝜙(𝜉_𝑛).

Remark 6. All of the solutions presented in this paper have been checked with Maple 11 by putting them back into the original equations.

Conclusions

We have proposed a variable-coefficient Riccati sub-equation method for solving nonlinear differential-difference equations and applied it to find exact solutions of the discrete (2 + 1)-dimensional Toda lattice equation. As a result, some generalized and new exact solutions for it have been successfully found. We have also compared this method with the known (𝐺^′/𝐺)-expansion method. Comparison results show that solutions obtained by the (𝐺^′/𝐺)-expansion method are only special cases of the solutions by the proposed method, which is to some extent in accordance with the analysis in [28]. Finally, as the present method is concise and powerful, we note that it can be applied to other nonlinear differential-difference equations.

Acknowledgments

This work is partially supported by National Natural Science Foundation of China (11171178). The author would like to thank the reviewers very much for their valuable suggestions on the paper.