A New Method with a Different Auxiliary Equation to Obtain Solitary Wave Solutions for Nonlinear Partial Differential Equations

Abstract

A new method with a different auxiliary equation from the Riccati equation is used for constructing exact travelling wave solutions of nonlinear partial differential equations. The main idea of this method is to take full advantage of a different auxilliary equation from the Riccati equation which has more new solutions. More new solitary solutions are obtained for the RLW Burgers and Hirota Satsuma coupled equations.

1. Introduction

In the recent years, remarkable progress has been made in the construction of the exact solutions for nonlinear partial differential equations, which have been a basic concern for both mathematicians and physicists [1–3].We do not attempt to characterize the general form of nonlinear dispersive wave equations [4, 5]. When an original nonlinear equation is directly calculated, the solution will preserve the actual physical characters of solutions [6]. The studies in finding exact solutions to nonlinear differential equation (NPDE), when they exist, are very important for the understanding of most nonlinear physical phenomena. There are many studies which obtain explicit solutions for nonlinear differential equations. Many explicit exact methods have been introduced in literature [7–21]. Some of them are generalized Miura transformation, Darboux transformation, Cole-Hopf transformation, Hirota’s dependent variable transformation, the inverse scattering transform and the Bäcklund transformation, tanh method, sine-cosine method, Painleve method, homogeneous balance method (HB), similarity reduction method, improved tanh method and so on.

In this article, the first section presents the scope of the study as an introduction. In the second section contains analyze of a new method and balance term definition. In the third section, we will obtain wave solutions of RLW Burgers and Hirota Satsuma coupled equations by using a new method. In the last section, we implement the conclusion.

2. Method and Its Applications

Let us simply describe the method [22]. Consider a given partial differential equation in two variables 𝐻𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑥𝑥,…=0. (1) The fact that the solutions of many nonlinear equations can be expressed as a finite series of solutions of the auxiliary equation motivates us to seek for the solutions of (1) in the form 𝑢(𝑥,𝑡)=𝜆^𝑚_𝑖=0𝑎_𝑖𝐹(𝜉)^𝑖+𝑎_−𝑖𝐹(𝜉)^−𝑖, (2) where,𝜉=𝑘(𝑥−𝑐𝑡),  𝑘 and 𝑐 are the wave number and the wave speed respectively, 𝑚 is a positive integer that can be determined by balancing the linear term of highest order with the nonlinear term in (1), 𝜆 is balancing coefficient that will be defined in a new “Balance term” definition and 𝑎₀,  𝑎₁,  𝑎₂,… are parameters to be determined. Substituting (2) into (1) yields a set of algebraic equations for 𝑎₀,  𝑎₁,  𝑎₂,… because all coefficients of 𝐹 have to vanish. From these relations 𝑎₀,  𝑎₁,  𝑎₂,… can be determined. The main idea of our method is to take full advantage of the new auxiliary equation. The desired auxiliary equation presents as following 𝐹^′=𝐴𝐹+𝐵𝐹+𝐶𝐹³, (3) where 𝑑𝐹/𝑑𝜉=𝐹^′ and 𝐴, 𝐵, 𝐶 are constants.

Case 1. If 𝐴=−1/4, 𝐵=1/2, 𝐶=−1/2 then (3) has the solution 𝐹=1/1+tan(𝜉)+sec(𝜉).

Case 2. If 𝐴=1/4, 𝐵=−1/2, 𝐶=0 then (3) has the solutions 𝐹=1/1+cscℎ(𝜉)+coth(𝜉) or 𝐹=1/1+𝑖secℎ(𝜉)+tanh(𝜉).

Case 3. If 𝐴=1/2, 𝐵=−1,𝐶=0 then (3) has the solutions 𝐹=1/1+cotℎ(𝜉) or 1𝐹=(𝜉).1+tanh (4)

Remark 1. Depending on the 𝐴,𝐵 and 𝐶 coefficients in the (3), it could be reached only three cases.

In the following we present a new approach to the “Balance term” definition.

Definition 2. When (1) is transformed with𝑢(𝑥,𝑡)=𝑢(𝜉), 𝜉=𝑘(𝑥−𝑐𝑡), where 𝑘 and 𝑐 are real constants, we get a nonlinear ordinary differential equation for 𝑢(𝜉) as following 𝑄^′𝑢,𝑘𝑐𝑢^′,𝑘𝑢^′,𝑘²𝑢^′′,…=0. (5) Let 𝑢^(𝑝) is the highest order derivative linear term and 𝑢^𝑞𝑢^(𝑟) is the highest nonlinear term in (5) and 𝐹^′=𝑘₀+𝑘₁𝐹+𝑘₂𝐹²+⋯+𝑘_𝑛𝐹^𝑛 is the auxiliary equation that is used to solve the nonlinear partial differential equation then the “Balance term” 𝑚 can be decided by the balancing the nonlinear term 𝑢^𝑞𝑢^(𝑟) and the linear term 𝑢^(𝑝) with acceptances of 𝑢≅𝜆𝐹^𝑖 and 𝐹^′≅𝐹^𝑛 where 𝑛 is integer (𝑛≠1)and 𝜆 is the balance coefficient that can be determined later.

Example 1. For the KdV equation with the transform 𝑢(𝑥,𝑡)=𝑢(𝜉), 𝜉=𝑥−𝑐𝑡 we have the ordinary differential equation as following −𝑐𝑢^′+6𝑢𝑢^′+𝑢^′′′=0. (6) By the balancing linear term 𝑢^′′′ with nonlinear term 𝑢𝑢^′𝑢^′=𝜆𝐹^𝑚′=𝜆𝑚𝐹^𝑚−1𝐹^′=𝜆𝑚𝐹^𝑚−1𝐹^𝑛=𝜆𝑚𝐹^{𝑚+𝑛−1},𝑢^′′=𝜆𝑚𝐹^{𝑚+𝑛−1′}=𝜆𝑚(𝑚+𝑛−1)𝐹^{𝑚+𝑛−2}𝐹^′=𝜆𝑚(𝑚+𝑛−1)𝐹^{𝑚+𝑛−2}𝐹^𝑛=𝜆𝑚(𝑚+𝑛−1)𝐹^{𝑚+2𝑛−2},𝑢^′′′=𝜆𝑚(𝑚+𝑛−1)𝐹^{𝑚+2𝑛−2′}=𝜆𝑚(𝑚+𝑛−1)(𝑚+2𝑛−2)𝐹^{𝑚+2𝑛−3}𝐹^′=𝜆𝑚(𝑚+𝑛−1)(𝑚+2𝑛−2)𝐹^{𝑚+2𝑛−3}𝐹^𝑛=𝜆𝑚(𝑚+𝑛−1)(𝑚+2𝑛−2)𝐹^{𝑚+3𝑛−3},𝑢𝑢^′=𝜆𝐹^𝑚𝜆𝑚𝐹^{𝑚+𝑛−1}=𝜆²𝑚𝐹^{2𝑚+𝑛−1} (7) we have the equations above and the equating 𝑢𝑢^′ to 𝑢^′′′𝜆²𝑚𝐹^{2𝑚+𝑛−1}=𝜆𝑚(𝑚+𝑛−1)(𝑚+2𝑛−2)𝐹^{𝑚+3𝑛−3},𝜆=(𝑚+𝑛−1)(𝑚+2𝑛−2),𝑚=2(𝑛−1). (8) If it is noticed that our new balance term 𝑚(𝑚=2(𝑛−1)) is connected to 𝑛. Namely our new balance term definition is connected to chosen auxiliary equation.

3. Application of the Method

Example 2. Let’s consider RLW Burgers equation 𝑢_𝑡+𝑢_𝑥+12𝑢𝑢_𝑥−𝑢_𝑥𝑥−𝑢_𝑥𝑥𝑡=0, (9) with the transform 𝑢(𝑥,𝑡)=𝑢(𝜉), 𝜉=𝑥−𝑐𝑡 we have the following equation −𝑐𝑢^′+𝑢^′+12𝑢𝑢^′−𝑢^′′+𝑐𝑢^′′′=0. (10) From the Definition 2 we have the balance term of RLW Burgers equation by using the auxiliary equation (3) for “Case 1”, is equal to 4. Therefore, we may choose the following ansatz: 𝑢(𝑥,𝑡)=48𝑎₋₄𝐹⁻⁴+𝑎₋₃𝐹⁻³+𝑎₋₂𝐹⁻²+𝑎₋₁𝐹⁻¹+𝑎₀+𝑎₁𝐹+𝑎₂𝐹²+𝑎₃𝐹³+𝑎₄𝐹⁴. (11) Substituting (11) into (10) along with (3) and using Mathematica yields a system of equations w.r.t. 𝐹^𝑖. Setting the coefficients of 𝐹^𝑖 in the obtained system of equations to zero, we can deduce the following set of algebraic polynomials with the respect unknowns 𝑎₀, 𝑎₁, 𝑎₂,… for the Case 1: 3𝑐𝑎₋₄+12𝑎²₋₄=0,105𝑐𝑎₋₃64+21𝑎₋₄𝑎₋₃−=0,3𝑎₋₄2−27𝑐𝑎₋₄2−24𝑎²₋₄+9𝑎²₋₃+3𝑐𝑎₋₂4+18𝑎₋₄𝑎₋₂−=0,15𝑎₋₃−16225𝑐𝑎₋₃32−42𝑎₋₄𝑎₋₃+15𝑎₋₃𝑎₋₂+15𝑐𝑎₋₁64+15𝑎₋₄𝑎₋₁=0,6𝑎₋₄+28𝑐𝑎₋₄+24𝑎²₋₄−18𝑎²₋₃−𝑎₋₂2−3𝑐𝑎₋₂−36𝑎₋₄𝑎₋₂+6𝑎²₋₂+12𝑎₋₃𝑎₋₁+12𝑎₋₄𝑎₀=0,15𝑎₋₃4+429𝑐𝑎₋₃32+42𝑎₋₄𝑎₋₃−30𝑎₋₃𝑎₋₂−3𝑎₋₁−1627𝑐𝑎₋₁32−30𝑎₋₄𝑎₋₁+9𝑎₋₂𝑎₋₁+9𝑎₋₃𝑎₀−3𝑐𝑎₁64+9𝑎₋₄𝑎₁=0,8𝑎₋₄+20𝑐𝑎₋₄−3𝑎₋₂−4𝑐𝑎₋₂+12𝑎²₋₂+24𝑎₋₃𝑎₋₁−6𝑎²₋₁+24𝑎₋₄𝑎₀−12𝑎₋₂𝑎₀−12𝑎₋₃𝑎₁−12𝑎₋₄𝑎₂=0,−10𝑎₋₄−32𝑐𝑎₋₄+18𝑎²₋₃+2𝑎₋₂+5𝑐𝑎₋₂+36𝑎₋₄𝑎₋₂−12𝑎²₋₂−24𝑎₋₃𝑎₋₁+3𝑎²₋₁−24𝑎₋₄𝑎₀+6𝑎₋₂𝑎₀+6𝑎₋₃𝑎₁+6𝑎₋₄𝑎₂=0,−6𝑎₋₃−27𝑐𝑎₋₃2+30𝑎₋₃𝑎₋₂+3𝑎₋₁4+35𝑐𝑎₋₁32+30𝑎₋₄𝑎₋₁−18𝑎₋₂𝑎₋₁−18𝑎₋₃𝑎₀+3𝑎₋₁𝑎₀+𝑎₁+163𝑐𝑎₁32−18𝑎₋₄𝑎₁+3𝑎₋₂𝑎₁+3𝑎₋₃𝑎₂+3𝑐𝑎₃64+3𝑎₋₄𝑎₃=0,9𝑎₋₃2+105𝑐𝑎₋₃16−𝑎₋₁−𝑐𝑎₋₁2+18𝑎₋₂𝑎₋₁+18𝑎₋₃𝑎₀−6𝑎₋₁𝑎₀−𝑎₁4+𝑐𝑎₁32+18𝑎₋₄𝑎₁−6𝑎₋₂𝑎₁−3𝑎₀𝑎₁−6𝑎₋₃𝑎₂−3𝑎₋₁𝑎₂−3𝑎₃+169𝑐𝑎₃32−6𝑎₋₄𝑎₃−3𝑎₋₂𝑎₃−3𝑎₋₃𝑎₄=0,−2𝑎₋₄−6𝑐𝑎₋₄+2𝑎₋₂+𝑐𝑎₋₂+6𝑎²₋₁+12𝑎₋₂𝑎₀+12𝑎₋₃𝑎₁−3𝑎²₁−𝑐𝑎₂2+12𝑎₋₄𝑎₂−6𝑎₀𝑎₂−6𝑎₋₁𝑎₃−𝑎₄2+3𝑐𝑎₄2−6𝑎₋₂𝑎₄−=0,3𝑎₋₃4−9𝑐𝑎₋₃8+𝑎₋₁2−𝑐𝑎₋₁16+6𝑎₋₁𝑎₀+𝑐𝑎₁2+6𝑎₋₂𝑎₁+6𝑎₀𝑎₁+6𝑎₋₃𝑎₂+6𝑎₋₁𝑎₂−9𝑎₁𝑎₂+3𝑎₃4−105𝑐𝑎₃32+6𝑎₋₄𝑎₃+6𝑎₋₂𝑎₃−9𝑎₀𝑎₃+6𝑎₋₃𝑎₄−9𝑎₋₁𝑎₄=0,6𝑎²₁−𝑎₂+4𝑐𝑎₂+12𝑎₀𝑎₂−6𝑎²₂+12𝑎₋₁𝑎₃−12𝑎₁𝑎₃+2𝑎₄−10𝑐𝑎₄+12𝑎₋₂𝑎₄−12𝑎₀𝑎₄−3=0,8𝑐𝑎₋₃+𝑎₋₁4−3𝑐𝑎₋₁8+𝑎₁2−35𝑐𝑎₁16−6𝑎₀𝑎₁−6𝑎₋₁𝑎₂+18𝑎₁𝑎₂−3𝑎₃+27𝑐𝑎₃2−6𝑎₋₂𝑎₃+18𝑎₀𝑎₃−15𝑎₂𝑎₃−6𝑎₋₃𝑎₄+18𝑎₋₁𝑎₄−15𝑎₁𝑎₄=0,−6𝑎²₁+2𝑎₂−10𝑐𝑎₂−12𝑎₀𝑎₂+12𝑎²₂−12𝑎₋₁𝑎₃+24𝑎₁𝑎₃−9𝑎²₃−6𝑎₄+32𝑐𝑎₄−12𝑎₋₂𝑎₄+24𝑎₀𝑎₄−18𝑎₂𝑎₄−=0,105𝑐𝑎₃8−42𝑎₃𝑎₄=0,3𝑐𝑎₋₁8−3𝑎₁4+27𝑐𝑎₁8−18𝑎₁𝑎₂+9𝑎₃2−429𝑐𝑎₃16−18𝑎₀𝑎₃+30𝑎₂𝑎₃−18𝑎₋₁𝑎₄+30𝑎₁𝑎₄−21𝑎₃𝑎₄−=0,15𝑐𝑎₁8−15𝑎₃4+225𝑐𝑎₃8−30𝑎₂𝑎₃−30𝑎₁𝑎₄+42𝑎₃𝑎₄=0,−24𝑐𝑎₄−24𝑎²₄=0,−2𝑎₂+12𝑐𝑎₂−12𝑎²₂−24𝑎₁𝑎₃+18𝑎²₃+8𝑎₄−56𝑐𝑎₄−24𝑎₀𝑎₄+36𝑎₂𝑎₄−12𝑎²₄=0,−6𝑐𝑎₂−18𝑎²₃−6𝑎₄+54𝑐𝑎₄−36𝑎₂𝑎₄+24𝑎²₄=0. (12) From the system of (12) we have
(i) 𝑎₀=−11−𝑖60,𝑎12₁=𝑎₂=𝑎₃=𝑎₄=𝑎₋₁=𝑎₋₃𝑎=0,₋₂=1+𝑖10,𝑎10₋₄𝑖=−𝑖20,𝑐=5,𝑢(𝑥,𝑡)=−11+5𝑖+601+𝑖×1101+sec[𝑥−(𝑖/5)𝑡]+tan[𝑥−(𝑖/5)𝑡]⁻²−𝑖1201+sec[𝑥−(𝑖/5)𝑡]+tan[𝑥−(𝑖/5)𝑡]⁻⁴. (13)
(ii) 𝑎₀=1+5𝑖60,𝑐=−𝑎₄,𝑎₁=𝑎₋₂=𝑎₃=𝑎₋₄=𝑎₋₁=𝑎₋₃𝑎=0,₂=−1+𝑖5,𝑎₄𝑖=±5,𝑢(𝑥,𝑡)=1+5𝑖−601+𝑖5×𝑖1+sec𝑥+5𝑡𝑖+tan𝑥+5𝑡²+𝑖5𝑖1+sec𝑥+5𝑡𝑖+tan𝑥+5𝑡⁴. (14) From the Definition 2 we have the balance term of RLW Burgers equation by using the auxiliary equation (3) for “Case 2”, is equal to −4 then we have the following system of equations −3𝑐𝑎₋₄−12𝑎²₋₄−=0,3𝑎₋₄2+27𝑐𝑎₋₄2+24𝑎²₋₄−9𝑎²₋₃−3𝑐𝑎₋₂4−18𝑎₋₄𝑎₋₂−=0,15𝑎₋₃+16225𝑐𝑎₋₃32+42𝑎₋₄𝑎₋₃−15𝑎₋₃𝑎₋₂−15𝑐𝑎₋₁64−15𝑎₋₄𝑎₋₁=0,4𝑎₋₄−18𝑐𝑎₋₄+18𝑎²₋₃−𝑎₋₂2+3𝑐𝑎₋₂+36𝑎₋₄𝑎₋₂−6𝑎²₋₂−12𝑎₋₃𝑎₋₁−12𝑎₋₄𝑎₀=0,9𝑎₋₃4−135𝑐𝑎₋₃16+30𝑎₋₃𝑎₋₂−3𝑎₋₁+1627𝑐𝑎₋₁32+30𝑎₋₄𝑎₋₁−9𝑎₋₂𝑎₋₁−9𝑎₋₃𝑎₀=0,−2𝑎₋₄+6𝑐𝑎₋₄+𝑎₋₂−3𝑐𝑎₋₂+12𝑎²₋₂+24𝑎₋₃𝑎₋₁−3𝑎²₋₁+24𝑎₋₄𝑎₀−6𝑎₋₂𝑎₀−=0,3𝑎₋₃4+15𝑐𝑎₋₃8+𝑎₋₁4−9𝑐𝑎₋₁16+18𝑎₋₂𝑎₋₁+18𝑎₋₃𝑎₀−3𝑎₋₁𝑎₀−=0,10564𝑐𝑎₋₃−21𝑎₋₄𝑎₋₃=0,6𝑎²₋₁+12𝑎₋₂𝑎₀𝑎=0,₋₁4−3𝑐𝑎₋₁8+6𝑎₋₁𝑎₀=0. (15) From the system of (15) we have
(i) 𝑎₀=𝑎₋₁=0,𝑎₋₂1=−5,𝑎₋₃=0,𝑎₋₄=1120,𝑐=−5,𝑢()1𝑥,𝑡=−511+𝑖sech[𝑥+(1/5)𝑡]+tanh[𝑥+(1/5)𝑡]⁻²+11201+𝑖sech[𝑥+(1/5)𝑡]+tanh[𝑥+(1/5)𝑡]⁻⁴, (16) or 𝑢()1𝑥,𝑡=−511+csch[𝑥+(1/5)𝑡]+coth[𝑥+(1/5)𝑡]⁻²+11201+csch[𝑥+(1/5)𝑡]+coth[𝑥+(1/5)𝑡]⁻⁴. (17)

Example 3. Let’s consider Hirota Satsuma coupled equation 𝑢_𝑡−3𝑢𝑢_𝑥+6𝑣𝑣_𝑥−12𝑢_𝑥𝑥𝑥𝑣=0,_𝑡+3𝑢𝑣_𝑥+𝑣_𝑥𝑥𝑥=0, (18) with the transform 𝑢(𝑥,𝑡)=𝑢(𝜉), 𝑣(𝑥,𝑡)=𝑣(𝜉), 𝜉=𝑥−𝑐𝑡 we have −𝑐𝑢^′−3𝑢𝑢^′+6𝑣𝑣^′−12𝑢^′′′=0,−𝑐𝑣^′+3𝑢𝑣^′+𝑣^′′′=0. (19) From the Definition 2 we have the balance term of Hirota Satsuma coupled equation by using the auxiliary equation (3) for “Case 1”, is equal to 4 for 𝑢 and 𝑣. Therefore, we may choose the following ansatz: 𝑢(𝑥,𝑡)=48𝑎₀+𝑎₁𝐹+𝑎₂𝐹²+𝑎₃𝐹³+𝑎₄𝐹⁴,𝑣(𝑥,𝑡)=48𝑏₀+𝑏₁𝐹+𝑏₂𝐹²+𝑏₃𝐹³+𝑏₄𝐹⁴. (20) Substituting (20) into (19) along with (3) and using Mathematica yields a system of equations w.r.t. 𝐹^𝑖. Setting the coefficients of 𝐹^𝑖 in the obtained system of equations to zero, we can deduce the following set of algebraic polynomials with the respect unknowns 𝑎₀, 𝑎₁, 𝑎₂,… for the Case 1: 3𝑎₁−128=0,3𝑎₁−643𝑎₃128=0,7𝑎₁+64𝑐𝑎₁4+3𝑎₀𝑎₁4−9𝑎₃−643𝑏₀𝑏₁2=0,3𝑎²₁4+𝑎₂2+𝑐𝑎₂2+3𝑎₀𝑎₂2−3𝑎₄4−3𝑏²₁2−3𝑏₀𝑏₂−𝑎=0,₁2−𝑐𝑎₁2−3𝑎₀𝑎₁2+9𝑎₁𝑎₂4+129𝑎₃+643𝑐𝑎₃4+9𝑎₀𝑎₃4+3𝑏₀𝑏₁−9𝑏₁𝑏₂2−9𝑏₀𝑏₃2−=0,3𝑎²₁2−5𝑎₂2−𝑐𝑎₂−3𝑎₀𝑎₂+3𝑎²₂2+3𝑎₁𝑎₃+11𝑎₄2+𝑐𝑎₄+3𝑎₀𝑎₄+3𝑏²₁+6𝑏₀𝑏₂−3𝑏²₂−6𝑏₁𝑏₃−6𝑏₀𝑏₄=0,43𝑎₁+32𝑐𝑎₁2+3𝑎₀𝑎₁2−9𝑎₁𝑎₂2−15𝑎₃2−3𝑐𝑎₃2−9𝑎₀𝑎₃2+15𝑎₂𝑎₃4+15𝑎₁𝑎₄4−3𝑏₀𝑏₁+9𝑏₁𝑏₂+9𝑏₀𝑏₃−15𝑏₂𝑏₃2−15𝑏₁𝑏₄2=0,3𝑎²₁2+11𝑎₂2+𝑐𝑎₂+3𝑎₀𝑎₂−3𝑎²₂−6𝑎₁𝑎₃+9𝑎²₃4−17𝑎₄−2𝑐𝑎₄−6𝑎₀𝑎₄+9𝑎₂𝑎₄2−3𝑏²₁−6𝑏₀𝑏₂+6𝑏²₂+12𝑏₁𝑏₃−9𝑏²₃2+12𝑏₀𝑏₄−9𝑏₂𝑏₄=0,105𝑎₃+1621𝑎₃𝑎₄2−21𝑏₃𝑏₄−=0,27𝑎₁+169𝑎₁𝑎₂2+453𝑎₃+323𝑐𝑎₃2+9𝑎₀𝑎₃2−15𝑎₂𝑎₃2−15𝑎₁𝑎₄2+21𝑎₃𝑎₄4−9𝑏₁𝑏₂−9𝑏₀𝑏₃+15𝑏₂𝑏₃+15𝑏₁𝑏₄−21𝑏₃𝑏₄2=0,15𝑎₁−16225𝑎₃+1615𝑎₂𝑎₃2+15𝑎₁𝑎₄2−21𝑎₃𝑎₄2−15𝑏₂𝑏₃−15𝑏₁𝑏₄+21𝑏₃𝑏₄=0,12𝑎₄+6𝑎²₄−12𝑏²₄=0,−6𝑎₂+3𝑎²₂+6𝑎₁𝑎₃−9𝑎²₃2+29𝑎₄+2𝑐𝑎₄+6𝑎₀𝑎₄−9𝑎₂𝑎₄+3𝑎²₄−6𝑏²₂−12𝑏₁𝑏₃+9𝑏²₃−12𝑏₀𝑏₄+18𝑏₂𝑏₄−6𝑏²₄=0−3𝑏₁64=0,3𝑏₁+323𝑏₃−64=0,7𝑏₁+32𝑐𝑏₁4−3𝑎₀𝑏₁4+9𝑏₃−332=0,4𝑎₁𝑏₁−𝑏₂+𝑐𝑏₂2−3𝑎₀𝑏₂2+3𝑏₄2𝑏=0,₁−𝑐𝑏₁2+3𝑎₀𝑏₁2−3𝑎₂𝑏₁4−3𝑎₁𝑏₂2−129𝑏₃+323𝑐𝑏₃4−9𝑎₀𝑏₃4=0,3𝑎₁𝑏₁2−3𝑎₃𝑏₁4+5𝑏₂−𝑐𝑏₂+3𝑎₀𝑏₂−3𝑎₂𝑏₂2−9𝑎₁𝑏₃4−11𝑏₄+𝑐𝑏₄−3𝑎₀𝑏₄−=0,43𝑏₁+16𝑐𝑏₁2−3𝑎₀𝑏₁2+3𝑎₂𝑏₁2−3𝑎₄𝑏₁4+3𝑎₁𝑏₂−3𝑎₃𝑏₂2+15𝑏₃−3𝑐𝑏₃2+9𝑎₀𝑏₃2−9𝑎₂𝑏₃4−3𝑎₁𝑏₄−3=0,2𝑎₁𝑏₁+3𝑎₃𝑏₁2−11𝑏₂+𝑐𝑏₂−3𝑎₀𝑏₂+3𝑎₂𝑏₂−3𝑎₄𝑏₂2+9𝑎₁𝑏₃2−9𝑎₃𝑏₃4+34𝑏₄−2𝑐𝑏₄+6𝑎₀𝑏₄−3𝑎₂𝑏₄−=0,105𝑏₃8−9𝑎₄𝑏₃2−6𝑎₃𝑏₄=0,27𝑏₁8−3𝑎₂𝑏₁2+3𝑎₄𝑏₁2−3𝑎₁𝑏₂+3𝑎₃𝑏₂−453𝑏₃+163𝑐𝑏₃2−9𝑎₀𝑏₃2+9𝑎₂𝑏₃2−9𝑎₄𝑏₃4+6𝑎₁𝑏₄−3𝑎₃𝑏₄−=0,15𝑏₁8−3𝑎₄𝑏₁2−3𝑎₃𝑏₂+225𝑏₃8−9𝑎₂𝑏₃2+9𝑎₄𝑏₃2−6𝑎₁𝑏₄+6𝑎₃𝑏₄=0,−24𝑏₄−6𝑎₄𝑏₄−3=0,2𝑎₃𝑏₁+12𝑏₂−3𝑎₂𝑏₂+3𝑎₄𝑏₂−9𝑎₁𝑏₃2+9𝑎₃𝑏₃2−58𝑏₄+2𝑐𝑏₄−6𝑎₀𝑏₄+6𝑎₂𝑏₄−3𝑎₄𝑏₄=0,−6𝑏₂−3𝑎₄𝑏₂−9𝑎₃𝑏₃2+54𝑏₄−6𝑎₂𝑏₄+6𝑎₄𝑏₄=0,3𝑎₂+9𝑎²₃2−27𝑎₄+9𝑎₂𝑎₄−6𝑎²₄−9𝑏²₃−18𝑏₂𝑏₄+12𝑏²₄=0. (21) From the system of (21) we have
(i) 1𝑐=45−6𝑏₀,𝑎₀=14−5−2𝑏₀,𝑎₁=𝑎₃=0,𝑏₁=𝑏₃𝑎=0,₂=4,𝑏₂𝑎=−2,₄=−4,𝑏₄𝑢()=1=2,𝑥,𝑡4−5−2𝑏₀1+41+sec𝑥−45−6𝑏₀𝑡1+tan𝑥−45−6𝑏₀𝑡^−1/2−121−41+sec𝑥−45−6𝑏₀𝑡1+tan𝑥−45−6𝑏₀𝑡^−1/2−14,𝑣(𝑥,𝑡)=𝑏₀1−21+sec𝑥−45−6𝑏₀𝑡1+tan𝑥−45−6𝑏₀𝑡^−1/2−121+21+sec𝑥−45−6𝑏₀𝑡1+tan𝑥−45−6𝑏₀𝑡^−1/2−14. (22)
(ii) 1𝑐=4−1−3𝑏²₂,𝑎₀=14−3−𝑏²₂,𝑎₁=𝑎₃=0,𝑏₁=𝑏₃=𝑏₄𝑎=0,₂=2,𝑏₂≠0,𝑎₄𝑢()=1=−2,𝑥,𝑡4−3−𝑏²₂1+21+sec𝑥−4−1−3𝑏²₂𝑡1+tan𝑥−4−1−3𝑏²₂𝑡^−1/2−121−21+sec𝑥−4−1−3𝑏²₂𝑡1+tan𝑥−4−1−3𝑏²₂𝑡^−1/2−14,𝑣()𝑏𝑥,𝑡=−²2−𝑏₂11+sec𝑥−4−1−3𝑏²₂𝑡1+tan𝑥−4−1−3𝑏²₂𝑡^−1/2−12. (3)(23) From the Definition 2 we have the balance term of Hirota Satsuma coupled equation by using the auxiliary equation (3) for “Case 2”, is equal to −4 then we have the following system of equations 3𝑎₋₄2+3𝑎²₋₄−6𝑏²₋₄=0,105𝑎₋₃+128214𝑎₋₄𝑎₋₃−212𝑏₋₄𝑏₋₃−𝑎=0,₋₁−16𝑐𝑎₋₁2−32𝑎₋₁𝑎₀+3𝑏₋₁−=0,27𝑎₋₄4−6𝑎²₋₄+9𝑎²₋₃4+3𝑎₋₂8+92𝑎₋₄𝑎₋₂+12𝑏²₋₄−9𝑏²₋₃2−9𝑏₋₄𝑏₋₂−=0,225𝑎₋₃−64212𝑎₋₄𝑎₋₃+154𝑎₋₃𝑎₋₂+15𝑎₋₁+128154𝑎₋₄𝑎₋₁+21𝑏₋₄𝑏₋₃−152𝑏₋₃𝑏₋₂−152𝑏₋₄𝑏₋₁=0,19𝑎₋₄2+𝑐𝑎₋₄−9𝑎²₋₃2−3𝑎₋₂2−9𝑎₋₄𝑎₋₂+3𝑎²₋₂2+3𝑎₋₃𝑎₋₁+3𝑎₋₄𝑎₀−6𝑏₋₄+9𝑏²₋₃+18𝑏₋₄𝑏₋₂−3𝑏²₋₂−6𝑏₋₃𝑏₋₁=0,147𝑎₋₃+323𝑐𝑎₋₃4−152𝑎₋₃𝑎₋₂−27𝑎₋₁−64152𝑎₋₄𝑎₋₁+94𝑎₋₂𝑎₋₁+94𝑎₋₃𝑎₀−9𝑏₋₃2+15𝑏₋₃𝑏₋₂+15𝑏₋₄𝑏₋₁−92𝑏₋₂𝑏₋₁−=0,27𝑎₋₃−163𝑐𝑎₋₃2+13𝑎₋₁+32𝑐𝑎₋₁4−92𝑎₋₂𝑎₋₁−92𝑎₋₃𝑎₀+34𝑎₋₁𝑎₀+9𝑏₋₃−3𝑏₋₁2+9𝑏₋₂𝑏₋₁=0,−4𝑎₋₄−2𝑐𝑎₋₄+7𝑎₋₂4+𝑐𝑎₋₂2−3𝑎²₋₂−6𝑎₋₃𝑎₋₁+3𝑎²₋₁4−6𝑎₋₄𝑎₀+32𝑎₋₂𝑎₀+12𝑏₋₄−3𝑏₋₂+6𝑏²₋₂+12𝑏₋₃𝑏₋₁−3𝑏²₋₁2−𝑎=0,₋₂2−𝑐𝑎₋₂−3𝑎²₋₁2−3𝑎₋₂𝑎₀+6𝑏₋₂+3𝑏²₋₁=0,−3𝑏₋₄−3𝑎₋₄𝑏₋₄=0,−3𝑎₋₃𝑏₋₄−105𝑏₋₃−9644𝑎₋₄𝑏₋₃=0,27𝑏₋₄2+6𝑎₋₄𝑏₋₄−3𝑎₋₂𝑏₋₄−94𝑎₋₃𝑏₋₃−3𝑏₋₂4−32𝑎₋₄𝑏₋₂=0,6𝑎₋₃𝑏₋₄−3𝑎₋₁𝑏₋₄+225𝑏₋₃+9322𝑎₋₄𝑏₋₃−94𝑎₋₂𝑏₋₃−32𝑎₋₃𝑏₋₂−15𝑏₋₁−3644𝑎₋₄𝑏₋₁=0,−19𝑏₋₄+𝑐𝑏₋₄+6𝑎₋₂𝑏₋₄−3𝑎₀𝑏₋₄+92𝑎₋₃𝑏₋₃−94𝑎₋₁𝑏₋₃+3𝑏₋₂+3𝑎₋₄𝑏₋₂−32𝑎₋₂𝑏₋₂−34𝑎₋₃𝑏₋₁=0,6𝑎₋₁𝑏₋₄−147𝑏₋₃+163𝑐𝑏₋₃4+92𝑎₋₂𝑏₋₃−94𝑎₀𝑏₋₃+3𝑎₋₃𝑏₋₂−32𝑎₋₁𝑏₋₂+27𝑏₋₁+3322𝑎₋₄𝑏₋₁−34𝑎₋₂𝑏₋₁=0,8𝑏₋₄−2𝑐𝑏₋₄+6𝑎₀𝑏₋₄+92𝑎₋₁𝑏₋₃−7𝑏₋₂2+𝑐𝑏₋₂2+3𝑎₋₂𝑏₋₂−32𝑎₀𝑏₋₂+32𝑎₋₃𝑏₋₁−34𝑎₋₁𝑏₋₁𝑏=0,₋₂−𝑐𝑏₋₂+3𝑎₀𝑏₋₂+32𝑎₋₁𝑏₋₁=0,27𝑏₋₃8−3𝑐𝑏₋₃2+92𝑎₀𝑏₋₃+3𝑎₋₁𝑏₋₂−13𝑏₋₁+16𝑐𝑏₋₁4+32𝑎₋₂𝑏₋₁−34𝑎₀𝑏₋₁𝑏=0,₋₁8−𝑐𝑏₋₁2+32𝑎₀𝑏₋₁=0. (24) From the system of (3) we have
(i) 𝑎₋₁=𝑎₋₃=𝑏₋₁=𝑏₋₃𝑏=0,₋₂=±1,𝑏₋₄𝑏=−⁻²2,𝑎₋₂𝑎=2,₋₄1=−1,𝑐=41+6𝑏₋₂,𝑎₀=14−1+2𝑏₋₂,𝑢()=1𝑥,𝑡41+21+𝑖sech[𝑥−(7/4)𝑡]+tanh[𝑥−(7/4)𝑡]⁻²−11+𝑖sech[𝑥−(7/4)𝑡]+tanh[𝑥−(7/4)𝑡]⁻⁴,𝑣(𝑥,𝑡)=𝑏₀+11+𝑖sech[𝑥−(7/4)𝑡]+tanh[𝑥−(7/4)𝑡]⁻²−1211+𝑖sech[𝑥−(7/4)𝑡]+tanh[𝑥−(7/4)𝑡]⁻⁴, (25) or 𝑢()=1𝑥,𝑡41+21+csch[𝑥−(7/4)𝑡]+coth[𝑥−(7/4)𝑡]⁻²−11+csch[𝑥−(7/4)𝑡]+coth[𝑥−(7/4)𝑡]⁻⁴,𝑣(𝑥,𝑡)=𝑏₀+11+csch[𝑥−(7/4)𝑡]+coth[𝑥−(7/4)𝑡]⁻²−1211+csch[𝑥−(7/4)𝑡]+coth[𝑥−(7/4)𝑡]⁻⁴. (26) Figures 1, 2, 3 and 4 gives to us 3D graphics for RLW Burgers and RLW Burgers and Hirota Satsuma coupled equations.

(a)

(b)

Figure 1 

Graph of the solution 𝑢(𝑥,𝑡) from left to the right for (13) and (14).

(a)

(b)

Figure 2 

Graph of the solution 𝑢(𝑥,𝑡) from left to the right for (15) and (16).

(a)

(b)

(c)

(d)

Figure 3 

Graph of the solutions 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡) corresponding to the value 𝑏0=1 from left to the right for (22), (3), and (23).

(a)

(b)

(c)

(d)

Figure 4 

Graph of the solutions 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡) corresponding to the value 𝑏0=1 from left to the right for (25) and (26).

4. Conclusion

We have presented a new method and balance term definition and used it to solve the RLW Burgers and Hirota Satsuma coupled equations. In fact, this method is readily applicable to a large variety of nonlinear PDEs. First, all the nonlinear PDEs which can be solved by the other methods can be solved by our method. Second, we used only the special solutions of (3). If we use other solutions of (3), we can obtain more travelling wave solutions. Third it is a computerizable method, which allow us to perform complicated and tedious algebraic calculation on computer and so our balance term definition is effectively useful for any to chosen auxiliary equation.