The Study of the Solution to a Generalized KdV-mKdV Equation

Abstract

A mathematical technique based on an auxiliary equation and the symbolic computation system Matlab is employed to investigate a generalized KdV-mKdV equation which possesses high-order nonlinear terms. Some new solutions including the Jacobi elliptic function solutions, the degenerated soliton-like solutions, and the triangle function solutions to the equation are obtained.

1. Introduction

Many powerful techniques have been established during the past decades for the study of the nonlinear dispersive partial differential equations [1–9]. The inverse scattering method, the Backlund transformation, the Darboux transformation, the Painleve’ analysis, the pseudospectral method, the finite differences method, and the sine-cosine ansatz are used to acquire solitary wave solutions and compactons solutions for some nonlinear equations. Wadati [1] employed the potential function to handle the KdV equation, and the resulting equation was solved iteratively by making use of a formal series for the potential function. Wadati [2] developed the trace method to handle the KP equation. The tanh method, developed by Malfliet and Hereman [3], is heavily used in the literature to deal with nonlinear partial differential equations. Fan and Zhang [4] introduced a useful extended tanh method that combined the standard tanh method with the Riccati equation. The extensive method was effectively used by many researchers to investigate exact solutions for a lot of nonlinear models (see [10, 11]).

With the development of the symbolic computation system, the direct methods for constructing travelling wave solutions to differential equations become feasible. With the help of Mathematica, Sirendaoreji and Jiong [12, 13] used the auxiliary equation method to investigate KdV and mKdV equations, Boussinesq equations, sine-Gordon equations, and the nonlinear Klein-Gordon equations, respectively. The Jacobi elliptic function expansion method is confirmed as a powerful technique to solve some nonlinear differential equations (see [14]).

Zhang et al. [15] used a sub-ODE technique to investigate the exact solution of the following nonlinear dispersive KdV-mKdV equation: 𝑢_𝑡+𝛼+𝛽𝑢^𝑝+𝜆𝑢^2𝑝𝑢_𝑥+𝑢_𝑥𝑥𝑥=0, (1) where 𝛼,𝛽,𝜆, and 𝑝 are constants. The bell type solitary wave solution, the kink type solitary wave solution, the algebraic solitary wave solution, and the sinusoidal travelling wave solution of (1) with exponent 𝑝>0 were expressed explicitly in [15]. In fact, (1) is turned into a KdV equation if 𝑝=1,𝛽≠0,𝜆=0, and an mKdV equation if 𝑝=1,𝛽=0,𝜆≠0.

In this paper, we further develop the work in [15] for the study of (1). By using a mathematical technique different from those in previous works [1–10], we obtain the exact travelling wave solutions including Jacobi elliptic function solutions, degenerated soliton solutions, and triangle function solutions for (1) with exponent 𝑝. Many of the solutions obtained are different from those presented in Zhang et al.’s work [15].

2. Brief Description of the Approach

To illustrate the basic concept of the auxiliary differential method, we consider that the nonlinear partial differential equation has the form 𝑃𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑥𝑥,𝑢_𝑥𝑡,𝑢_𝑡𝑡,…=0. (2) Using the transformation 𝜉=𝜇(𝑥−𝑐𝑡)(𝜇≠0 and 𝑐≠0), (2) turns into the following nonlinear ordinary different equation: 𝑄𝑢,𝑢_𝜉,𝑢_𝜉𝜉,𝑢_𝜉𝜉𝜉,…=0. (3)

We seek for the solutions of (3) in the form 𝑢(𝜉)=^𝑁_𝑖=0𝑔_𝑖𝑧^𝑖(𝜉), (4) where 𝑔_𝑖(𝑖=0,1,2,…,𝑁) are constants which will be determined later. The parameter 𝑁 is a positive integer and can be determined by balancing the highest-order derivative terms and the highest power nonlinear terms in (3). The highest degree of 𝜕^𝑝𝑢/𝜕𝜉^𝑝 can be calculated by 𝑂𝜕^𝑝𝑢𝜕𝜉^𝑝𝑂𝑢=𝑁+𝑝,𝑝=0,1,2,…,_𝑞𝜕^𝑝𝑢𝜕𝜉^𝑝=𝑞𝑁+𝑝,𝑞,𝑝=0,1,2,…. (5)

We assume that 𝑧(𝜉) represents the solutions of the following auxiliary differential equation 𝑑𝑧𝑑𝜉²=𝑐₁+𝑐₂𝑧²+𝑐₃2𝑧⁴, (6) where 𝑐_𝑖(𝑖=1,2,3) are real constants.

Substituting (4) and (6) into (3) and equating the coefficients of all powers of 𝑧(𝜉) and 𝑧^𝑗(𝜉)𝑐₁+𝑐₂𝑧²+(𝑐₃/2)𝑧⁴(𝑗=0,1,2,…) to be zero in the resulting equation, several algebraic equations will be obtained. Then, solving these algebraic equations by the symbolic computation system Matlab and combining (4) and the solutions of the auxiliary equation (6), we can get the exact solutions for (2).

3. Exact Travelling Wave Solutions to (1)

Firstly, setting 𝑢^𝑝=𝑤 yields 𝑢_𝑥=1𝑝𝑤^1/𝑝−1𝑤_𝑥,𝑢_𝑡=1𝑝𝑤^1/𝑝−1𝑤_𝑡,𝑢_𝑥𝑥=1𝑝1𝑝−1𝑤^1/𝑝−2𝑤_𝑥²+1𝑝𝑤^1/𝑝−1𝑤_𝑥𝑥,𝑢_𝑥𝑥𝑥=1𝑝1𝑝1−1𝑝−2𝑤^1/𝑝−3𝑤³_𝑥+3𝑝1𝑝−1𝑤^1/𝑝−2𝑤_𝑥𝑤_𝑥𝑥+1𝑝𝑤^1/𝑝−1𝑤_𝑥𝑥𝑥, (7) which turns (1) into 𝑤²𝑤_𝑡+𝛼+𝛽𝑤+𝜆𝑤²𝑤²𝑤_𝑥+1𝑝1−1𝑝−2𝑤³_𝑥1+3𝑝−1𝑤𝑤_𝑥𝑤_𝑥𝑥+𝑤²𝑤_𝑥𝑥𝑥=0. (8)

To find the traveling solutions for (1), we use the wave variable 𝜉=𝜇(𝑥−𝑐𝑡), where 𝑐≠0 and 𝜇≠0. The wave variable 𝜉 transforms (8) into the following ordinary differential equation: 𝛼+𝛽𝑤+𝜆𝑤²𝑤²𝑤^′+1𝑝1−1𝑝−2𝜇²𝑤^′3+3𝜇²1𝑝−1𝑤𝑤^′𝑤^′′−𝑐𝑤²𝑤^′+𝜇²𝑤²𝑤^′′′=0. (9) From (9), we have 𝑁=1. Therefore, we choose the ansatz 𝑤(𝜉)=𝑔₀+𝑔₁𝑧, (10) where 𝑧(𝜉) may be determined by 𝑑𝑧𝑑𝜉²=𝑐₁+𝑐₂𝑧²+𝑐₃2𝑧⁴, (11) which possesses several types of solutions listed in Table 1 (see [16]).

Table 1 

In Table 1, functions sn(𝜉)=sn(𝜉,𝑟), cn(𝜉)=cn(𝜉,𝑟), and dn(𝜉)=dn(𝜉,𝑟), are Jacobian elliptic functions with modulus 𝑟(0<𝑟<1), which have the properties sn(−𝜉)=−sn(𝜉),cn(−𝜉)=cn(𝜉),dn(−𝜉)=dn(𝜉), sn²(𝜉)+cn²(𝜉)=1, dn²(𝜉)+𝑟²sn²(𝜉)=1, (sn(𝜉))^′=cn(𝜉)dn(𝜉), (cn(𝜉))^′=−sn(𝜉)dn(𝜉), and (dn(𝜉))^′=−𝑟²sn(𝜉)cn(𝜉). Setting 𝑟→0 yields sn(𝜉)→sin(𝜉),cn(𝜉)→cos(𝜉), and dn(𝜉)→1. When 𝑟→1, it derives that sn(𝜉)→tanh(𝜉), cn(𝜉)→sech(𝜉), and dn(𝜉)→sech(𝜉).

Substituting (10) and (11) into (9) and letting each coefficient of 𝑧^𝑗𝑐₁+𝑐₂𝑧²+(𝑐₃/2)𝑧⁴(0≤𝑗≤4) be zero, we obtain 𝜆𝑔⁵₁+𝜇²21𝑝1−1𝑝−2𝑔³₁𝑐₃1+3𝑝−1𝜇²𝑔³₁𝑐₃+3𝜇²𝑔³₁𝑐₃=0,𝛽𝑔₁+2𝜆𝑔₀𝑔₁𝑔³₁+2𝜆𝑔₀𝑔⁴₁1+3𝑝−1𝜇²𝑔₀𝑔²₁𝑐₃+6𝜇²𝑔₀𝑔²₁𝑐₃=0,−𝑐𝑔³₁+𝛼+𝛽𝑔₀+𝜆𝑔²₀𝑔³₁+2𝛽𝑔₁+2𝜆𝑔₀𝑔₁𝑔₀𝑔²₁+𝜆𝑔²₀𝑔³₁+𝜇²1𝑝1−1𝑝−2𝑔³₁𝑐₂1+3𝑝−1𝜇²𝑔³₁𝑐₂+3𝑐₃𝜇²𝑔²₀+𝜇²𝑐₂𝑔³₁=0,−2𝑐𝑔₀𝑔²₁+2𝛼+𝛽𝑔₀+𝜆𝑔²₀𝑔²₁𝑔₀+𝛽𝑔₁+2𝜆𝑔₀𝑔₁𝑔²₀𝑔₁1+3𝑝−1𝜇²𝑔²₁𝑔₀𝑐₂+2𝜇²𝑔²₁𝑔₀𝑐₂=0,−𝑐𝑔²₀𝑔₁+𝛼+𝛽𝑔₀+𝜆𝑔²₀𝑔²₀𝑔₁+1𝑝1−1𝑝−2𝜇²𝑔³₁𝑐₁+𝜇²𝑔²₀𝑔₁𝑐₂=0. (12)

Solving (12) with the help of Matlab, we acquire the following solutions: 𝑝=1,𝑔₀𝛽=−2𝜆,𝑔₁𝛽=±2𝜆−𝑐₃𝑐₂,𝛽𝜇=±213𝜆𝑐₂𝛽,𝑐=𝛼−²,16𝜆𝑝=2,𝑔₀=−2𝛽5𝜆,𝑔₁=±2𝛽5𝜆−𝑐₃𝑐₂,𝛽𝜇=±523𝜆𝑐₂,𝑐=𝛼−16𝛽²,75𝜆𝑝=2,𝜇=±5𝑐₂𝛽²+5𝜖𝛽²𝑐²₂−2𝑐₁𝑐₃48𝑐₁𝑐₃𝜆^1/2,𝑔₀=−5𝛽8𝜆,𝑔₁𝜇=±4−30𝑐₃𝜆,𝑐=𝛼−5𝛽²+132𝜆4𝜇²𝑐₂,𝑔₀𝛽=−2𝑝+12𝜆𝑝+2,𝑔₁𝛽=±2𝑝+12𝜆𝑝+2−𝑐₃𝑐₂,2𝑐₁𝑐₃=𝑐²₂,𝜇=±𝛽𝑝𝑝+22𝑝+12𝜆𝑝+1𝑐₂𝛽,𝑐=𝛼−²2𝑝+1𝜆𝑝+2²,𝑝+1 (13)(14)(15)(16) where the exponent 𝑝≠−2,−1, and 𝜖=±1.

3.1. The Jacobi Elliptic Function Solutions to (1) in the Case of 𝑝=1

In order to make that wave number 𝜇 a real-valued number, we have to choose constants 𝜆 and modulus 𝑟 (𝑐₂ depends on 𝑟) to satisfy some restrictions. However, in this section, we allow 𝜇 to take values in complex number domain. From the expression of 𝑔₁,𝑔₂,𝜇, and 𝑐 in (13) and the solutions listed in Table 1, we derive the following Jacobi elliptic function solutions for (1) in the case 𝑝=1: 𝑢_1.1()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽𝑟2𝜆2𝑟²𝛽+1×sn2−13𝜆𝑟²𝛽+1𝑥−𝛼𝑡+²𝑡,𝑢6𝜆_1.2()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽𝑟2𝜆2𝑟²𝛽+1×cd2−13𝜆𝑟²𝛽+1𝑥−𝛼𝑡+²𝑡,𝑢6𝜆_1.3()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽𝑟2𝜆22𝑟²𝛽−1×cn213𝜆2𝑟²𝛽−1𝑥−𝛼𝑡+²𝑡,𝑢6𝜆_1.4()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆22−𝑟²𝛽×dn213𝜆2−𝑟²𝛽𝑥−𝛼𝑡+²𝑡,𝑢6𝜆_1.5()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆2𝑟²−12𝑟²𝛽−1×nc213𝜆2𝑟²𝛽−1𝑥−𝛼𝑡+²𝑡,𝑢6𝜆_1.6()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆2𝑟²𝛽+1×ns2−13𝜆𝑟²𝛽+1𝑥−𝛼𝑡+²𝑡,𝑢6𝜆_1.7()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆2𝑟²𝛽+1×dc2−13𝜆𝑟²𝛽+1𝑥−𝛼𝑡+²𝑡,𝑢6𝜆_1.8()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆2𝑟²−1𝑟²𝛽−2×nd213𝜆2−𝑟²𝛽𝑥−𝛼𝑡+²𝑡,𝑢6𝜆_1.9()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆2𝑟²𝛽−2×cs213𝜆2−𝑟²𝛽𝑥−𝛼𝑡+²𝑡,𝑢6𝜆_1.10()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆21−𝑟²𝑟²𝛽−2×sc213𝜆2−𝑟²𝛽𝑥−𝛼𝑡+²𝑡,𝑢6𝜆_1.11()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽𝑟2𝜆2𝑟²−11−2𝑟²𝛽×sd2−13𝜆1−2𝑟²𝛽𝑥−𝛼𝑡+²𝑡,𝑢6𝜆_1.12()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆21−2𝑟²𝛽×ds2−13𝜆1−2𝑟²𝛽𝑥−𝛼𝑡+²𝑡,𝑢6𝜆_1.13()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆11+𝑟²×𝛽𝑟cn223𝜆𝑟²×𝛽+1𝑥−𝛼𝑡+²𝑡𝛽6𝜆±nd223𝜆𝑟²×𝛽+1𝑥−𝛼𝑡+²𝑡,𝑢6𝜆_1.14()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆12𝑟²×𝛽−11×sn223𝜆1−2𝑟²×𝛽𝑥−𝛼𝑡+²𝑡6𝜆⁻¹±𝛽cn223𝜆1−2𝑟²×𝛽𝑥−𝛼𝑡+²𝑡×𝛽6𝜆sn223𝜆1−2𝑟²×𝛽𝑥−𝛼𝑡+²𝑡6𝜆⁻¹,𝑢_1.15()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆𝑟²−11+𝑟²×𝛽1×cn223𝜆𝑟²×𝛽+1𝑥−𝛼𝑡+²𝑡6𝜆⁻¹±𝛽sn223𝜆𝑟²×𝛽+1𝑥−𝛼𝑡+²𝑡×𝛽6𝜆cn223𝜆𝑟²×𝛽+1𝑥−𝛼𝑡+²𝑡6𝜆⁻¹,𝑢_1.16()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆12−𝑟²×𝛽1×sn223𝜆𝑟²×𝛽−2𝑥−𝛼𝑡+²𝑡6𝜆⁻¹±𝛽dn223𝜆𝑟²×𝛽−2𝑥−𝛼𝑡+²𝑡×𝛽6𝜆sn223𝜆𝑟²×𝛽−2𝑥−𝛼𝑡+²𝑡6𝜆⁻¹,𝑢_1.17()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽𝑟2𝜆12−𝑟²×𝛽sn223𝜆𝑟²𝛽−2𝑥−𝛼𝑡+²𝑡𝛽6𝜆±𝑖cn223𝜆𝑟²𝛽−2𝑥−𝛼𝑡+²𝑡,𝑢6𝜆_1.18()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽𝑟2𝜆12−𝑟²×𝛽dn223𝜆𝑟²𝛽−2𝑥−𝛼𝑡+²𝑡×√6𝜆1−𝑟²𝛽sn223𝜆𝑟²×𝛽−2𝑥−𝛼𝑡+²𝑡𝛽6𝜆±cn223𝜆𝑟²×𝛽−2𝑥−𝛼𝑡+²𝑡6𝜆⁻¹,𝑢_1.19()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆12𝑟²×𝛽−1𝑟sn223𝜆1−2𝑟²𝛽𝑥−𝛼𝑡+²𝑡𝛽6𝜆±𝑖dn223𝜆1−2𝑟²𝛽𝑥−𝛼𝑡+²𝑡,𝑢6𝜆_1.20()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆12𝑟²×𝛽−1sn213𝜆1−2𝑟²𝛽𝑥−𝛼𝑡+²𝑡×𝛽6𝜆1±cn223𝜆1−2𝑟²×𝛽𝑥−𝛼𝑡+²𝑡6𝜆⁻¹,𝑢_1.21()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆𝑟²2−𝑟²×𝛽sn213𝜆𝑟²𝛽−2𝑥−𝛼𝑡+²𝑡×𝛽6𝜆1±dn223𝜆𝑟²×𝛽−2𝑥−𝛼𝑡+²𝑡6𝜆⁻¹,𝑢_1.22()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆1−𝑟²𝑟²×𝛽+1dn223𝜆𝑟²𝛽+1𝑥−𝛼𝑡+²𝑡×𝛽6𝜆1±𝑟sn223𝜆𝑟²×𝛽+1𝑥−𝛼𝑡+²𝑡6𝜆⁻¹,𝑢_1.23()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽2𝜆𝑟²−11+𝑟²×𝛽cn223𝜆𝑟²𝛽+1𝑥−𝛼𝑡+²𝑡×𝛽6𝜆1±sn223𝜆𝑟²×𝛽+1𝑥−𝛼𝑡+²𝑡6𝜆⁻¹,𝑢_1.24()𝛽𝑥,𝑡=−+2𝜆𝜖𝛽1−𝑟²2𝜆−1𝑟²×𝛽+1sn223𝜆𝑟²𝛽+1𝑥−𝛼𝑡+²𝑡×𝛽6𝜆dn223𝜆𝑟²×𝛽+1𝑥−𝛼𝑡+²𝑡𝛽6𝜆±cn223𝜆𝑟²×𝛽+1𝑥−𝛼𝑡+²𝑡6𝜆⁻¹,𝑢_1.25()=𝑥,𝑡−𝛽+2𝜆𝜖𝛽𝑟²2𝜆√2−𝑟²×𝛽cn223𝜆𝑟²𝛽−2𝑥−𝛼𝑡+²𝑡×√6𝜆1−𝑟²𝛽±dn223𝜆𝑟²×𝛽−2𝑥−𝛼𝑡+²𝑡6𝜆⁻¹. (17)

Remark 1. From the properties of Jacobi elliptic functions, we know that when the module 𝑟→0 or 𝑟→1, sn, cn, and dn degenerate into triangular or hyperbolic functions, from which we can obtain soliton and triangular function solutions of (1) in the case 𝑝=1. In other word, when 𝑝=1, we can deduce Jacobi elliptic functions solutions, soliton, and triangular function solutions for (1).

Remark 2. When 𝛼=0, 𝑘=(𝛽/2)2/3𝜆(1−2𝑟²), 𝑎₀=−𝛽/2𝜆, and 𝑎₁=±𝑘−3/2𝜆, 𝑢_1.14(𝑥,𝑡) turns into 𝑢_1.14(𝑥,𝑡)=𝑎₀+𝑎₁ns𝑘𝑥−1−2𝑟²2𝑘²+𝜆𝑎²₀+𝛽𝑎₀𝑡+cs𝑘𝑥−1−2𝑟²2𝑘²+𝜆𝑎²₀+𝛽𝑎₀𝑡, (18) which was obtained by Yomba [17].

Remark 3. Supposing 𝑘=(𝛽/6)−1/𝑞𝜈𝜆, 𝑝=(2+√3/3)−6𝑞𝜈𝜆 (𝑞 and 𝜈 are arbitrary constants). When 𝑟→1 in 𝑢_1.17(𝑥,𝑡), we obtain 𝑢_1.17.1(𝑥,𝑡)=𝑎₀−𝜖𝛽12𝜆𝑞−6×±𝑞𝜈𝜆−2𝑞𝜈𝜆+−6𝑞𝜈𝜆tanh−6𝑞𝜈𝜆𝜉±𝑖sech−6𝑞𝜈𝜆𝜉, (19) where 𝑎₀=−(𝛽𝑝/2𝜆)−6𝑞𝜈𝜆+𝛽/2𝜆,𝜉=𝑘(𝑥−(𝜆𝑎²₀+𝛽𝑎₀)𝑡). Solution (19) is in full agreement with that presented in [17] by Yomba.

3.2. The Jacobi Elliptic Function Solutions to (1) in the Case of 𝑝=1/2

Similarly as in Section 3.1, we allow 𝜇 to take values in complex number field. Making use of (14) and the solutions listed in Table 1, we obtain the following Jacobi elliptic function solutions for (1) in the case 𝑝=1/2: 𝑢_2.1()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽𝑟5𝜆2𝑟²𝛽+1×sn5−23𝜆𝑟²+1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_2.2()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽𝑟5𝜆2𝑟²𝛽+1×cd5−23𝜆𝑟²+1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_2.3()=𝑥,𝑡−2𝛽+5𝜆2𝜖𝛽𝑟5𝜆22𝑟²−1×cn2𝛽²75𝜆2𝑟²−1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_2.4()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽5𝜆22−𝑟²𝛽×dn523𝜆2−𝑟²𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_2.5()=𝑥,𝑡−2𝛽+5𝜆2𝜖𝛽5𝜆2𝑟²−22𝑟²−1×nc2𝛽²75𝜆2𝑟²−1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_2.6()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽5𝜆2𝑟²𝛽+1×ns5−23𝜆𝑟²+1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_2.7()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽5𝜆2𝑟²𝛽+1×dc5−23𝜆𝑟²+1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_2.8()=𝑥,𝑡−2𝛽+5𝜆2𝜖𝛽5𝜆2−2𝑟²2−𝑟²𝛽×nd523𝜆2−𝑟²𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_2.9()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽5𝜆−22−𝑟²𝛽×cs523𝜆2−𝑟²𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_2.10()=𝑥,𝑡−2𝛽+5𝜆2𝜖𝛽5𝜆2𝑟²−22−𝑟²𝛽×sc523𝜆2−𝑟²𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_2.11()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽5𝜆2𝑟²1−𝑟²2𝑟²𝛽−1×sd523𝜆2𝑟²−1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_2.12()=𝑥,𝑡−2𝛽+5𝜆2𝜖𝛽5𝜆21−2𝑟²×ds2𝛽²75𝜆2𝑟²−1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_2.13()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽5𝜆1𝑟²×𝛽+1𝑟cn543𝜆𝑟²+1𝑥−𝛼𝑡+16𝛽²𝑡𝛽75𝜆±dn543𝜆𝑟²+1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_2.14()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽5𝜆−11−2𝑟²×𝛽1×sn543𝜆−2𝑟²×+1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆⁻¹±𝛽cn543𝜆−2𝑟²×+1𝑥−𝛼𝑡+16𝛽²𝑡×𝛽75𝜆sn543𝜆−2𝑟²×+1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆⁻¹²,𝑢_2.15()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽5𝜆−1−𝑟²𝑟²×𝛽+11×cn543𝜆𝑟²×+1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆⁻¹±𝛽sn543𝜆𝑟²×+1𝑥−𝛼𝑡+16𝛽²𝑡×𝛽75𝜆cn543𝜆𝑟²×+1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆⁻¹²,𝑢_2.16()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽5𝜆−1𝑟²×𝛽−21×sn543𝜆𝑟²×−2𝑥−𝛼𝑡+16𝛽²𝑡75𝜆⁻¹±𝛽dn543𝜆𝑟²×−2𝑥−𝛼𝑡+16𝛽²𝑡×𝛽75𝜆sn543𝜆𝑟²×−2𝑥−𝛼𝑡+16𝛽²𝑡75𝜆⁻¹²,𝑢_2.17()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽𝑟5𝜆−1𝑟²×𝛽−2sn543𝜆𝑟²×−2𝑥−𝛼𝑡+16𝛽²𝑡𝛽75𝜆±𝑖cn543𝜆𝑟²×−2𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_2.18()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽𝑟5𝜆−1𝑟²×𝛽−2dn543𝜆𝑟²×−2𝑥−𝛼𝑡+16𝛽²𝑡×√75𝜆1−𝑟²sn4𝛽²75𝜆𝑟²×−2𝑥−𝛼𝑡+16𝛽²𝑡75𝜆±cn4𝛽²75𝜆𝑟²×−2𝑥−𝛼𝑡+16𝛽²𝑡75𝜆⁻¹²,𝑢_2.19()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽5𝜆√2𝑟²×𝛽−1sn543𝜆1−2𝑟²×𝑥−𝛼𝑡+16𝛽²𝑡×𝛽75𝜆1±cn543𝜆1−2𝑟²×𝑥−𝛼𝑡+16𝛽²𝑡75𝜆⁻¹²,𝑢_2.20()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽5𝜆12𝑟²×𝛽−1𝑟sn543𝜆1−2𝑟²×𝑥−𝛼𝑡+16𝛽²𝑡𝛽75𝜆±𝑖dn543𝜆1−2𝑟²×𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_2.21()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽𝑟5𝜆12−𝑟²×𝛽sn543𝜆2−𝑟²×𝑥−𝛼𝑡+16𝛽²𝑡×𝛽75𝜆1±dn543𝜆2−𝑟²×𝑥−𝛼𝑡+16𝛽²𝑡75𝜆⁻¹²,𝑢_2.22()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽5𝜆1−𝑟²𝑟²×𝛽+1dn543𝜆𝑟²×+1𝑥−𝛼𝑡+16𝛽²𝑡×𝛽75𝜆1±𝑟sn543𝜆𝑟²×+1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆⁻¹²,𝑢_2.23()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽5𝜆𝑟²−1𝑟²×𝛽+1cn543𝜆𝑟²×+1𝑥−𝛼𝑡+16𝛽²𝑡×𝛽75𝜆1±sn543𝜆𝑟²×+1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆⁻¹²,𝑢_2.24()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽1−𝑟²5𝜆−1𝑟²×𝛽+1sn543𝜆𝑟²×+1𝑥−𝛼𝑡+16𝛽²𝑡×𝛽75𝜆dn543𝜆𝑟²×+1𝑥−𝛼𝑡+16𝛽²𝑡𝛽75𝜆±cn543𝜆𝑟²×+1𝑥−𝛼𝑡+16𝛽²𝑡75𝜆⁻¹²,𝑢_2.25()=−𝑥,𝑡2𝛽+5𝜆2𝜖𝛽𝑟²5𝜆12−𝑟²×𝛽cn543𝜆𝑟²×−2𝑥−𝛼𝑡+16𝛽²𝑡×√75𝜆1−𝑟²𝛽±dn543𝜆𝑟²×−2𝑥−𝛼𝑡+16𝛽²𝑡75𝜆⁻¹². (20)

When 𝑝=1/2, we can also obtain Jacobi elliptic functions solutions, soliton, and triangular function solutions for (1).

3.3. The Jacobi Elliptic Function Solutions to (1) in the Case of??𝑝=2

From the expressions of (15), we get the following jacobi elliptic function solutions to (1): 𝑢_3.1()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇𝑟2−15𝜆×sn𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡+4𝑟²+1𝜇^21/2,𝛽𝜇=±𝑟4𝑟−5²+1+5𝜖𝑟²−16𝜆^1/2,𝑢_3.2()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇𝑟2−15𝜆×cd𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡+4𝑟²+1𝜇^21/2,𝛽𝜇=±𝑟4𝑟−5²+1+5𝜖𝑟²−16𝜆^1/2,𝑢_3.3()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇𝑟215𝜆×cn𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡−42𝑟²−1𝜇^21/2,𝛽𝜇=±54𝑟2𝑟²−1+5𝜖6𝑟²−1𝜆^1/2,𝑢_3.4()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇215𝜆×dn𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡−42−𝑟²𝜇^21/2,𝛽𝜇=±452−𝑟²+5𝜖𝑟²61−𝑟²𝜆^1/2,𝑢_3.5()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇2−151−𝑟²𝜆×nc𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡−42𝑟²−1𝜇^21/2,𝛽𝜇=±54𝑟2𝑟²−1+5𝜖6𝑟²−1𝜆^1/2,𝑢_3.6()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇2−15𝜆×ns𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡+4𝑟²+1𝜇^21/2,𝛽𝜇=±𝑟4𝑟−5²+1+5𝜖𝑟²−16𝜆^1/2,𝑢_3.7()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇2−15𝜆×dc𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡+4𝑟²+1𝜇^21/2,𝛽𝜇=±𝑟4𝑟−5²+1+5𝜖𝑟²−16𝜆^1/2,𝑢_3.8()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇2−15𝑟²−1𝜆×nd𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡−42−𝑟²𝜇^21/2,𝛽𝜇=±452−𝑟²+5𝜖𝑟²61−𝑟²𝜆^1/2,𝑢_3.9()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇2−15𝜆×cs𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡−42−𝑟²𝜇^21/2,𝛽𝜇=±452−𝑟²+5𝜖𝑟²6𝜆1−𝑟^21/2,𝑢_3.10()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇2−151−𝑟²𝜆×sc𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡−42−𝑟²𝜇^21/2,𝛽𝜇=±452−𝑟²+5𝜖𝑟²6𝜆1−𝑟^21/2,𝑢_3.11()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇𝑟2151−𝑟²𝜆×sd𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡−42𝑟²−1𝜇^21/2,𝛽𝜇=±54𝑟2𝑟²−1+5𝜖6𝑟²−1𝜆^1/2,𝑢_3.12()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇2−15𝜆×ds𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡−42𝑟²−1𝜇^21/2,𝛽𝜇=±54𝑟2𝑟²−1+5𝜖6𝑟²−1𝜆^1/2,𝑢_3.13()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇415𝜆×𝑟cn𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡−8𝑟²+1𝜇²±dn𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡−8𝑟²+1𝜇^21/2,𝛽𝜇=±21−𝑟²5𝑟²+1+10𝜖𝑟3𝜆^1/2,𝑢_3.14()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇4−15𝜆×1×sn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆81−2𝑟²⁻¹±cn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆81−2𝑟²𝜇²×sn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆81−2𝑟²𝜇^2−11/2,𝛽𝜇=±251−2𝑟²+10𝜖𝑟√𝑟²−13𝜆^1/2,𝑢_3.15()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇4−151−𝑟²𝜆×1×cn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆8𝑟²+1𝜇²⁻¹±dn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆8𝑟²+1𝜇²×cn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆8𝑟²+1𝜇^2−11/2,𝛽𝜇=±21−𝑟²5𝑟²+1+10𝜖𝑟3𝜆^1/2,𝑢_3.16()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇4−15𝜆×1×sn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆8𝑟²−2𝜇²⁻¹±dn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆8𝑟²−2𝜇²×sn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆8𝑟²−2𝜇^2−11/2,𝛽𝜇=±2𝑟²5𝑟²−2+10𝜖√1−𝑟²3𝜆^1/2,𝑢_3.17()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇𝑟4−15𝜆×sn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆8𝑟²−2𝜇²±𝑖cn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆8𝑟²−2𝜇^21/2,𝛽𝜇=±2𝑟²5𝑟²−2+10𝜖√1−𝑟²3𝜆^1/2,𝑢_3.18()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇𝑟4−15𝜆×sn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆8𝑟²−2𝜇²×√1−𝑟²sn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−32𝜆𝑡𝜇²𝑟²−28±cn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−32𝜆𝑡𝜇²𝑟²−28^−11/2,𝛽𝜇=±2𝑟²5𝑟²−2+10𝜖√1−𝑟²3𝜆^1/2,𝑢_3.19()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇4−15𝜆×𝑟sn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆81−2𝑟²𝜇²±𝑖dn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆81−2𝑟²𝜇^21/2,𝛽𝜇=±251−2𝑟²+10𝜖𝑟√𝑟²−13𝜆^1/2,𝑢_3.20()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇4−15𝜆×sn𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡−81−2𝑟²𝜇²×1±cn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆81−2𝑟²𝜇^2−11/2,𝛽𝜇=±251−2𝑟²+10𝜖𝑟√𝑟²−13𝜆^1/2,𝑢_3.21()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇𝑟4−15𝜆×sn𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡−8𝑟²−2𝜇²×1±dn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆8𝑟²−2𝜇^2−11/2,𝛽𝜇=±5𝑟2𝑟²−2+5𝜖√𝑟⁴−5𝑟²+43𝜆^1/2,𝑢_3.22()=𝑥,𝑡−5𝛽+8𝜆𝜖𝜇4151−𝑟²𝜆×dn𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡−8𝑟²+1𝜇²×1±𝑟sn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆8𝑟²+1𝜇^2−11/2,𝛽𝜇=±21−𝑟²5𝑟²+1+10𝜖𝑟3𝜆^1/2,𝑢_3.23()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇415𝑟²−1𝜆×cn𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡−8𝑟²+1𝜇²×1±sn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆8𝑟²+1𝜇^2−11/2,𝛽𝜇=±21−𝑟²5𝑟²+1+10𝜖𝑟3𝜆^1/2,𝑢_3.24()=−𝑥,𝑡5𝛽+𝑟8𝜆𝜖𝜇²−14−15𝜆×sn𝜇𝑥−𝛼𝑡+5𝛽²𝑡32𝜆𝑡−8𝑟²+1𝜇²×dn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆8𝑟²+1𝜇²±cn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−𝑡32𝜆8𝑟²+1𝜇^2−11/2,𝛽𝜇=±21−𝑟²5𝑟²+1+10𝜖𝑟3𝜆^1/2,𝑢_3.25()=−𝑥,𝑡5𝛽+8𝜆𝜖𝜇𝑟²4−15𝜆×cn𝜇𝑥−𝛼𝑡+5𝛽²32𝜆𝑡−𝑡𝜇²𝑟²−28×√1−𝑟²±dn𝜇𝑥−𝛼𝑡+5𝛽²𝑡−32𝜆𝑡𝜇²𝑟²−28^−11/2,𝛽𝜇=±2𝑟²5𝑟²−2+10𝜖√1−𝑟²3𝜆^1/2. (21)

3.4. Soliton and Triangular Function Solutions of (1)

From the expression of 2𝑐₁𝑐₃=𝑐²₂, we know that 𝑟→0 or 𝑟→1. According to the properties of Jacobi elliptic functions sn, cn, and dn, the following degenerated soliton and triangular function solutions for (1) can be obtained: 𝑢_4.1()=−𝛽𝑥,𝑡2𝑝+1𝛽2𝜆𝑝+2+𝜖2𝑝+12𝜆𝑝+2×tanh𝛽𝑝𝑝+2−2𝑝+1×𝛽4𝜆𝑝+1𝑥−𝛼𝑡+²2𝑝+1𝑡𝜆𝑝+2²𝑝+1^1/𝑝,𝑢_4.2()=−𝛽𝑥,𝑡2𝑝+1√2𝜆𝑝+2+𝜖2𝛽2𝑝+12𝜆𝑝+2×sech𝛽𝑝𝑝+22𝑝+1×𝛽2𝜆𝑝+1𝑥−𝛼𝑡+²2𝑝+1𝑡𝜆𝑝+2²𝑝+1^1/𝑝,𝑢_4.3()=−𝛽𝑥,𝑡2𝑝+1𝛽2𝜆𝑝+2+𝜖2𝑝+12𝜆𝑝+2×coth𝛽𝑝𝑝+2−2𝑝+1×𝛽4𝜆𝑝+1𝑥−𝛼𝑡+²2𝑝+1𝑡𝜆𝑝+2²𝑝+1^1/𝑝,𝑢_4.4()=−𝛽𝑥,𝑡2𝑝+1√2𝜆𝑝+2+𝑖𝜖2𝛽2𝑝+12𝜆𝑝+2×csch𝛽𝑝𝑝+22𝑝+1×𝛽2𝜆𝑝+1𝑥−𝛼𝑡+²2𝑝+1𝑡𝜆𝑝+2²𝑝+1^1/𝑝,𝑢_4.5()=−𝛽𝑥,𝑡2𝑝+1𝛽2𝜆𝑝+2+𝜖2𝑝+1×2𝜆𝑝+2cosh𝛽𝑝𝑝+2−2𝑝+1𝜆×+𝛽𝑝+1𝑥−𝛼𝑡₂2𝑝+1𝑡𝜆𝑝+2²×𝑝+1±1sinh𝛽𝑝𝑝+2−2𝑝+1𝜆×+𝛽𝑝+1𝑥−𝛼𝑡₂2𝑝+1𝑡𝜆𝑝+2²𝑝+1^−11/𝑝,𝑢_4.6()=−𝛽𝑥,𝑡2𝑝+1𝛽2𝜆𝑝+2+𝜖2𝑝+1×2𝜆𝑝+2sinh𝛽𝑝𝑝+2−2𝑝+1𝜆×+𝛽𝑝+1𝑥−𝛼𝑡₂2𝑝+1𝑡𝜆𝑝+2²×𝑝+1±𝑖cosh𝛽𝑝𝑝+2−2𝑝+1𝜆×+𝛽𝑝+1𝑥−𝛼𝑡₂2𝑝+1𝑡𝜆𝑝+2²𝑝+1^−11/𝑝,𝑢_4.7()=−𝛽𝑥,𝑡2𝑝+1𝛽2𝜆𝑝+2+𝜖2𝑝+1×2𝜆𝑝+2sinh𝛽𝑝𝑝+2−2𝑝+1𝜆×+𝛽𝑝+1𝑥−𝛼𝑡₂2𝑝+1𝑡𝜆𝑝+2²×𝑝+11±cosh𝛽𝑝𝑝+2−2𝑝+1𝜆×+𝛽𝑝+1𝑥−𝛼𝑡₂2𝑝+1𝑡𝜆𝑝+2²𝑝+1^−11/𝑝,𝑢_4.8()=−𝛽𝑥,𝑡2𝑝+1√2𝜆𝑝+2+𝜖2𝛽2𝑝+12𝜆𝑝+2×sec𝛽𝑝𝑝+2−2𝑝+1×𝛽2𝜆𝑝+1𝑥−𝛼𝑡+²2𝑝+1𝑡𝜆𝑝+2²𝑝+1^1/𝑝,𝑢_4.9()=−𝛽𝑥,𝑡2𝑝+1√2𝜆𝑝+2+𝜖2𝛽2𝑝+12𝜆𝑝+2×csc𝛽𝑝𝑝+2−2𝑝+1×𝛽2𝜆𝑝+1𝑥−𝛼𝑡+²2𝑝+1𝑡𝜆𝑝+2²𝑝+1^1/𝑝,𝑢_4.10()=−𝛽𝑥,𝑡2𝑝+1𝛽2𝜆𝑝+2+𝑖𝜖2𝑝+12𝜆𝑝+2×cot𝛽𝑝𝑝+22𝑝+1×𝛽4𝜆𝑝+1𝑥−𝛼𝑡+²2𝑝+1𝑡𝜆𝑝+2²𝑝+1^1/𝑝,𝑢_4.11()=−𝛽𝑥,𝑡2𝑝+1𝛽2𝜆𝑝+2+𝑖𝜖2𝑝+12𝜆𝑝+2×tan𝛽𝑝𝑝+22𝑝+1×𝛽4𝜆𝑝+1𝑥−𝛼𝑡+²2𝑝+1𝑡𝜆𝑝+2²𝑝+1^1/𝑝,𝑢_4.12()=−𝛽𝑥,𝑡2𝑝+1𝛽2𝜆𝑝+2+𝑖𝜖2𝑝+1×2𝜆𝑝+21±cos𝛽𝑝𝑝+22𝑝+1𝜆×+𝛽𝑝+1𝑥−𝛼𝑡₂2𝑝+1𝑡𝜆𝑝+2²×𝑝+1sin𝛽𝑝𝑝+22𝑝+1𝜆×+𝛽𝑝+1𝑥−𝛼𝑡₂2𝑝+1𝑡𝜆𝑝+2²𝑝+1^−11/𝑝,𝑢_4.13()=−𝛽𝑥,𝑡2𝑝+1𝛽2𝜆𝑝+2+𝑖𝜖2𝑝+1×2𝜆𝑝+21±sin𝛽𝑝𝑝+22𝑝+1𝜆×+𝛽𝑝+1𝑥−𝛼𝑡₂2𝑝+1𝑡𝜆𝑝+2²×𝑝+1cos𝛽𝑝𝑝+22𝑝+1𝜆×+𝛽𝑝+1𝑥−𝛼𝑡₂2𝑝+1𝑡𝜆𝑝+2²𝑝+1^−11/𝑝,𝑢_4.14()=−𝛽𝑥,𝑡2𝑝+1𝛽2𝜆𝑝+2+𝑖𝜖2𝑝+1×2𝜆𝑝+2sin𝛽𝑝𝑝+22𝑝+1𝜆×+𝛽𝑝+1𝑥−𝛼𝑡₂2𝑝+1𝑡𝜆𝑝+2²×𝑝+11±cos𝛽𝑝𝑝+22𝑝+1𝜆×+𝛽𝑝+1𝑥−𝛼𝑡₂2𝑝+1𝑡𝜆𝑝+2²𝑝+1^−11/𝑝,𝑢_4.15()=−𝛽𝑥,𝑡2𝑝+1𝛽2𝜆𝑝+2+𝑖𝜖2𝑝+1×2𝜆𝑝+2cos𝛽𝑝𝑝+22𝑝+1𝜆×+𝛽𝑝+1𝑥−𝛼𝑡₂2𝑝+1𝑡𝜆𝑝+2²×𝑝+11±sin𝛽𝑝𝑝+22𝑝+1×+𝛽2𝜆𝑝+1𝑥−𝛼𝑡₂2𝑝+1𝑡𝜆𝑝+2²𝑝+1^−11/𝑝. (22)

Remark 4. When 𝑟→0 and 𝑟→1 in the solutions 𝑢_1.1(𝑥,𝑡) to 𝑢_3.25(𝑥,𝑡), the soliton and triangular function solutions we obtain can be found in 𝑢_4.1(𝑥,𝑡) to 𝑢_4.15(𝑥,𝑡) while 𝑝=1,2,1/2, respectively. For example, when 𝑟→1, 𝑢_2.17 and 𝑢_3.17 can be expressed by 𝑢_2.17.1(𝑥,𝑡)=−2𝛽+5𝜆2𝜖𝛽×𝛽5𝜆tanh5−43𝜆𝑥−𝛼𝑡+16𝛽²𝑡𝛽75𝜆±𝑖sech5−43𝜆𝑥−𝛼𝑡+16𝛽²𝑡75𝜆²,𝑢_3.17.1(𝑥,𝑡)=−5𝛽+8𝜆5𝜖𝛽×𝛽8𝜆tanh2−53𝜆𝑥−𝛼𝑡+5𝛽²𝑡𝛽48𝜆±𝑖sech2−53𝜆𝑥−𝛼𝑡+5𝛽²𝑡48𝜆^1/2, (24) which are the same as 𝑢_4.6 in the case of 𝑝=1/2,2, respectively.

Remark 5. In these expressions of solutions, 𝑢_4.1(𝑥,𝑡) and 𝑢_4.2(𝑥,𝑡) are in full agreement with the result which was obtained by Zhang et al. [15].

4. Conclusion

In this paper, by using the auxiliary differential equation method, we obtain the Jacobi function solutions, the degenerated solitons, and the triangular function solutions to the generalized dispersive KdV-mKdV equation (1) in the case of 𝑝=1, 𝑝=2, and 𝑝=1/2 with the help of symbolic computation system Matlab. However, for arbitrary exponent 𝑝, we only obtain the degenerated solitons and triangle function solutions to (1). It may need other technique to deal with the Jacobi elliptic function solutions for the generalized dispersive KdV-mKdV equation with arbitrary exponent 𝑝.

Acknowledgments

The authors are grateful to the referees for their helpful and useful comments. This research is partially supported by the Scientific Research Fund of Chongqing Technology and Business University (no. 1355006).