Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics Using the Modified Simple Equation Method

Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics Using the Modified Simple Equation Method

The modified simple equation method is employed to construct the exact solutions involving parameters of nonlinear evolution equations via the (1+1)-dimensional modified KdV equation, and the (1+1)-dimensional reaction-diffusion equation. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

^{Nonlinear phenomena play crucial roles in ap-plied mathematics and physics.} Calculating exact and numerical solutions, in particular the traveling wave solutions of nonlinear equations in mathematical physics, plays an important role in soliton theory.^[1] Recently, it has become more interesting to obtain ex-act solutions of nonlinear evolution equations by us-ing symbolical computer programs such as Matlab, Maple, and Mathematica that facilitate complicated and tedious algebrical computations. These equa-tions are mathematical models of complex physical phenomena that arise in engineering, chemistry, bi-ology, mechanics, and applied physics. Several power-ful methods for obtaining explicit traveling wave soli-tary solutions of nonlinear equations have been pro-posed, such as Hirota’s bilinear method,^[2] Painleve expansions,^[3] the inverse scattering transform,^[4] the homogeneous balance method,^[5] the F-expansion method,^[6] the Jacobi-elliptic function method,^[7,8] the tanh-function method,^[9,10] the Backlund transform method,^[11] the (𝐺^′/𝐺)-expansion method,^[12,13] the exp-function method,^[14,15] and the modified simple equation method.^[16,17] In this Letter, we apply the modified simple equation method to seek the exact so-lutions of some nonlinear evolution equations via the (1+1)-dimensional modified KdV equation and the (1+1)-dimensional reaction-diffusion equation. The proposed method can be described as follows: sup-pose that we have a nonlinear evolution equation in the form^[16,17]

𝐹(𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑥𝑥,...) = 0, (1)

where 𝐹 is a polynomial in 𝑢 and its partial derivatives in which the highest order derivatives and nonlinear terms are involved. In the following, we give the main steps of this method.
Step 1. Using the wave transformation

𝑢(𝑥,𝑡) = 𝑢(𝜉), 𝜉 = 𝑥 + 𝑡, (2)

to reduce Eq.(1) to the following ODE:

𝑃(𝑢,𝑢^′,𝑢^′′,...) = 0, (3)

where 𝑃 is a polynomial in 𝑢 and its total derivatives with respect to 𝜉.

∑︁
Step 2. We suppose that Eq.(3) has the formal solution _{𝑁 [︂} ′ _]︂𝑘
𝑢(𝜉) = 𝑘=0 𝐴_{𝑘 𝜓(𝜉)} , (4)
where 𝐴_𝑘 are constants to be determined, such that 𝐴_𝑁 = 0, and 𝜓(𝜉) is an unknown function to be de-termined later.
Step 3. We determine the positive integer 𝑁 in Eq.(4) by considering the homogeneous balance be-tween the highest order derivatives and the nonlinear terms in Eq.(3).
Step 4. We substitute Eq.(4) into Eq.(3) and cal-culate all the necessary derivatives 𝑢^′,𝑢^′′,... of the unknown function 𝑢(𝜉) and we account the function 𝜓(𝜉). As a result of this substitution, we obtain a polynomial of ^𝜓′^(𝜉) and its derivatives. In this poly-
nomial, we gather all the terms of the same power of ^𝜓′^(𝜉) and its derivatives, and we equate with zero
all the coeficients of this polynomial. This operation yields a system of equations which can be solved to find 𝐴_𝑘 and 𝜓(𝜉). Consequently, we can obtain the exact solutions to Eq.(1). Let us now describe the exp-function method^[15] as follows: We assume that the solution to Eq.(3) can be expressed in the form

𝑑

𝑢(𝜉) =
^∑︀_𝑛=−𝑐 𝑎_𝑛 exp(𝑛𝜉) _𝑚=−𝑝 𝑏_𝑚 exp(𝑚𝜉)

where 𝑐,𝑑,𝑝,𝑞 are positive integers which could be freely chosen, 𝑎_𝑛 and 𝑏_𝑛 are unknown constants to be determined. To determine the values of 𝑐 and 𝑝, we balance the linear term of the highest order with

^*Corresponding author. Email: [e-mail]; [e-mail] ©2012 Chinese Physical Society and IOP Publishing Ltd

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the highest-order nonlinear term in Eq.(3). Similarly, to determine the values of 𝑑 and 𝑞, we balance the linear term of the lowest order in Eq.(3) with the low-est order nonlinear term. To determine 𝑎_𝑛 and 𝑏_𝑛, we substitute Eq.(5) into Eq.(3) and collect all the terms with the same order of exp(𝑖𝜉)(𝑖 = 0,1,2,...) and set them to zero, yielding a set of algebraic equations which can be solved using computer programs such as Maple or Mathematica to get 𝑎_𝑛 and 𝑏_𝑚. Conse-quently, we can obtain the exact solutions to Eq.(1).
Further, the well-known (𝐺^′/𝐺)-expansion method^[12] can be described as follows: we assume that the solution to Eq.(3) can be expressed in the form

𝑁
𝑢(𝜉) = ^∑︁𝛼_𝑘 (︂^𝐺′^(𝜉))︂𝑘, 𝛼_𝑁 = 0, (6) 𝑘=0
where 𝐺 = 𝐺(𝜉) satisfies the second-order linear ODE

𝐺′′ (𝜉) + 𝜆𝐺^′(𝜉) + 𝜇𝐺(𝜉) = 0, (7)

where 𝛼_𝑘 (𝑘 = 0,...,𝑁), 𝜆 and 𝜇 are constants to be determined, the primes represent the derivatives with respect to 𝜉.
To determine 𝑢(𝜉) explicitly, we give the following steps.
Step 1. Determine the positive integer 𝑁 in Eq.(6) by balancing the highest-order derivatives with the nonlinear terms in Eq.(3).
Step 2. Substitute Eq.(6) along with Eq.(7) into Eq.(3), collect all the terms with the same order of (𝐺^′/𝐺) together and set them to zero, yielding a set of algebraic equations which can be solved by using computer programs such as Maple or Mathematica to get the values 𝛼_𝑘,𝜆,𝜇. Consequently, we can obtain the exact solutions to Eq.(1) since we know the solu-tions to Eq.(7). Furthermore, we can summarize the projective Riccati equation method^[18] as follows: we assume that the solution to Eq.(3) can be expressed in the form


𝑁
_{𝑢(𝜉) =} ∑︁_{𝜎𝑖−1(𝜉){𝐴}𝑖_{𝜎(𝜉) + 𝐵}𝑖_{𝜏(𝜉)} + 𝐴}0_{, (8)} 𝑖=1
where 𝐴₀, 𝐴_𝑖 and 𝐵_𝑖, (𝑖 = 1,2,...,𝑁) are constant to be determined, while 𝜎(𝜉) and 𝜏(𝜉) satisfy

𝜎^′(𝜉) = 𝑒𝜎(𝜉)𝜏(𝜉), 𝜏^′(𝜉) = 𝑒𝜏²(𝜉) − 𝜇𝜎(𝜉) + 𝑟, (9)

where 𝑒, 𝑟 and 𝜇 are constants, such that 𝑒 = ±1, 𝑟 = 0, and the prime represents the derivative with respect to 𝜉.
To determine 𝑢(𝜉) explicitly, we give the main steps.
Step 1. Determine the positive integer 𝑁 in Eq.(8) by balancing the highest-order derivatives with the nonlinear terms in Eq.(3).
Step 2. Substitute Eq.(8) along with Eq.(9) into Eq.(3) and collect all terms with the same order of 𝜎^𝑖(𝜉) 𝜏^𝑖(𝜉), (𝑖,𝑗 = 0,1,...) and set them to zero, yield-ing a set of algebraic equations, which can be solved with the aid of the symbolic Maple software. Conse-quently, we can obtain the exact solutions to Eq.(1). Let us now apply the modified simple equation method to find the exact solutions and then the solitary wave solutions of two nonlinear evolution equations.
The first equation is the nonlinear modified KdV equation
𝑢_𝑡 − 𝛽𝑢²𝑢_𝑥 + 𝑢_𝑥𝑥𝑥 = 0, (10)
where 𝛽 ia a positive constant. The solutions to Eq.(10) have been investigated by using differ-ent methods, e.g. the exp-function method,^[15] and the (𝐺^′/𝐺)-expansion method.^[12] Let us now solve Eq.(10) using the modified simple equation method. To this end, we use the wave transformation (2) to reduce Eq.(10) to the following ODE:

𝑢 − ₃𝛽𝑢³ + 𝑢^′′ = 0, (11)
with zero constant of integration. Balancing 𝑢^′′ with 𝑢³ yields 𝑁 = 1. Consequently, Eq.(10) has the for-mal solution _[︂ ′ _]︂
𝑢(𝜉) = 𝐴₀ + 𝐴_{1 𝜓(𝜉)} , (12)

where 𝐴₀ and 𝐴₁ are constants to be determined such that 𝐴₁ = 0. It is easy to see

′
′′ ′²
𝑢 = 𝐴₁( _𝜓 − _𝜓2 ), (13)


′′
′′′ ′ ′′ ′³
𝑢 = 𝐴₁( _𝜓 − 3 _𝜓2 + 2 _𝜓3 ), (14)
and so on. Substituting Eqs.(12) and (14) into Eq.(11) and equating all the coeficients of 𝜓⁰, 𝜓⁻¹, 𝜓⁻², 𝜓⁻³ to zero, we obtain


𝐴₀ ^(︀3 − 𝛽𝐴₀^)︀ = 0, (15) 𝐴₁𝜓^′ 1 − 𝛽𝐴² + 𝐴₁𝜓^′′′ = 0, (16) 𝐴₁𝜓^′ (𝛽𝐴₀𝐴₁𝜓^′ + 3𝜓^′′) = 0, (17) 𝐴₁𝜓^′3 6 − 𝛽𝐴² = 0. (18)
Equations (15) and (18) yield 𝐴₀ = 0, 𝐴₀ = ±^√︀3/𝛽, 𝐴₁ = ± 6/𝛽.
Let us now discuss the following cases:
Case 1. If 𝐴₀ = 0, then we can obtain the trivial solution, which is rejected.


Case 2. If 𝐴₀ = ± 3/𝛽, then we can deduce that 𝜓^′′′/𝜓^′′ = −^√2. (19)

Integrating Eq.(19) yields

𝜓^′′ = 𝑐₁ exp(−^√2𝜉), (20)

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and consequently, we obtain

√
^√ (︁ )︁

√
𝜓^′ = −𝑐_{1 2} exp − 2𝜉 , (21) (︁ )︁
𝜓 = 𝑐₂ + ₂𝑐₁ exp − 2𝜉 , (22) where 𝑐₁ and 𝑐₂ are constants of integration.
Now, the exact solution to Eq.(10) has the form
√︂ √︂

[︀

]︀
𝑢(𝑥,𝑡) = ± _𝛽 ∓ 2𝑐_{1 𝛽}
^{︁ exp − 2(𝑥 + 𝑡) ^}︁
₂𝑐₂ + 𝑐₁ exp − 2(𝑥 + 𝑡) ₍₂₃₎
Integrating Eq.(32) yields
√︂
𝜓^′′ = 𝑐₁ exp[− _{𝛼 + 1} 𝜉]. (33)

Consequently, we have

[︃ ]︃
𝜓^′ = −𝑐₁ ^𝛼_2𝛽¹ exp − _{𝛼 + 1} 𝜉 , (34)



𝜓 = 𝑐₂ + 𝑐₁ (︂^{𝛼 + 1})︂exp[︃−√︂^𝛼2_{𝛽 1} 𝜉]︃, (35)

If we set 𝑐₁ = ±1, 𝑐₂ = 1/2 in Eq.(23) we will have the following solitary wave solutions:

√
[︃ ]︃
𝑢₁(𝑥,𝑡) = ± _𝛽 tanh ₂²(𝑥 + 𝑡) , (24)

√

√︂
[︃ ]︃
𝑢₂(𝑥,𝑡) = ± ³ coth ₂²(𝑥 + 𝑡) . (25)

The second equation is the nonlinear reaction-diffusion equation

𝑢_𝑡𝑡 + 𝛼𝑢_𝑥𝑥 + 𝛽𝑢 − 𝛾𝑢³ = 0, (26)

where 𝑐₁ and 𝑐₂ are constants of integration. Now, the exact solution of Eq.(26) has the form

√︃ √︃
𝑢(𝑥,𝑡) = ± ^𝛽 ∓ 2𝑐₁ ^𝛽
[︁ √︁ ]︁
exp − (𝑥 + 𝑡)
· [︁ √︁ ]︁ . _𝛼+1𝑐₂ + 𝑐₁ exp − _𝛼+1 (𝑥 + 𝑡) ₍₃₆₎


2𝛽
If we set 𝑐₁ = ±1 and 𝑐₂ = ^𝛼+1 in Eq.(36) we will have the following solitary wave solutions:

where 𝛼, 𝛽 and 𝛾 are positive constants. The solu-tion to Eq.(26) has been investigated by using differ-ent methods namely, the projective Riccati equation method,^[18] and the (𝐺^′/𝐺)-expansion method.^[13] Let us now solve Eq.(26) using the modified simple equa-tion method. To this end, we use the wave transfor-mation (2) to reduce Eq.(26) to the following ODE:


√︃ _{[︃ √︂ ]︃}
𝑢₁(𝑥,𝑡) = ± _𝛾 tanh _{2 𝛼 + 1} (𝑥 + 𝑡) , (37)

√︃ _{[︃ √︂ ]︃}
𝑢₂(𝑥,𝑡) = ± _𝛾 coth _{2 𝛼 + 1} (𝑥 + 𝑡) . (38)

(𝛼 + 1)𝑢^′′ + 𝛽𝑢 − 𝛾𝑢³ = 0. (27)


]︀
Balancing 𝑢^′′ with 𝑢³ yields 𝑁 = 1. Consequently, Eq.(27) has the same formal solution (12). Substitut-ing Eqs.(12) and (14) into Eq.(27) and equating all the coeficients of 𝜓⁰,𝜓⁻¹,𝜓⁻²,𝜓⁻³ to zero, we obtain 𝐴₀ ^(︀𝛽 − 𝛾𝐴₀^)︀ = 0, (28)
𝐴₁ (𝛼 + 1)𝜓^′′′ + 𝛽 − 3𝛾𝐴² 𝜓^′ = 0,
(29)


3𝐴₁𝜓^′ [(𝛼 + 1)𝜓^′′ + 𝛾𝐴₀𝐴₁𝜓^′] = 0, (30) 𝜓^′3 2(𝛼 + 1) − 𝛾𝐴₁ = 0. (31)
Equations (28) and (31) yield 𝐴₀ = 0, 𝐴₀ = ±^√︀𝛽/𝛾, 𝐴₁ = ± 2(𝛼 + 1)/𝛾.
Now, let us now discuss the following cases:
Case 1. If 𝐴₀ = 0, then we will obtain the trivial solution which is rejected.
Case 2. If 𝐴₀ = ± 𝛽/𝛾, then we can deduce that
√︂
𝜓^′′′/𝜓^′′ = − _{𝛼 + 1}. (32)

In summary, the modified simple equation method has been proposed to find exact solutions and soli-tary wave solutions of the (1+1)-dimensional nonlin-ear modified KdV equation and the (1+1)-dimensional nonlinear reaction-diffusion equation. Comparing the presently proposed method with other meth-ods, namely, the exp-function method, the (𝐺^′/𝐺)-expansion method and the projective Riccati equa-tion method, we can conclude: (i) the exact solutions to Eqs.(10) and (26) can be investigated using these methods with the aid of the computer programs, such as Matlab, Maple, and Mathematica to facilitate the complicated algebraic computations. (ii) The exact solutions to these equations have been obtained here without using the computer programs since the com-putations are simple. From (i) and (ii) we can con-clude that the modified simple equation method is much more simple than the other methods. Also, we can see that the proposed method is direct, effective, and can be applied to many other nonlinear evolution equations.

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