Solving Nonlinear Partial Differential Equations by the sn-ns Method

Volume 2012, Article ID 340824,25pages doi:10.1155/2012/340824

Research Article

Solving Nonlinear Partial Differential Equations by the sn-ns Method

Alvaro H. Salas

FIZMAKO Research Group and Department of Mathematics, University of Caldas/ National University of Colombia, Campus la Nubia, Manizales, Colombia

Correspondence should be addressed to Alvaro H. Salas,asalash2002@yahoo.com Received 2 January 2012; Accepted 30 January 2012

Academic Editor: Muhammad Aslam Noor

Copyrightq2012 Alvaro H. Salas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We present the application of the sn-ns method to solve nonlinear partial differential equations.

We show that the well-known tanh-coth method is a particular case of the sn-ns method.

1. Introduction

The search of explicit solutions to nonlinear partial differential equationsNLPDEsby using computational methods is one of the principal objectives in nonlinear science problems. Some powerful methods have been extensively used in the past decade to handle nonlinear PDEs.

Some of them are the tanh-method1, the tanh-coth method2, the exp-function method 3, the projective Riccati equation method 4, and the Jacobi elliptic functions method.

Practically, there is no unified method that could be used to handle all types of nonlinear problems.

The main purpose of this work consists in solving nonlinear polynomial PDE starting from the idea of the projective Riccati equations method. We derive exact solutions to the following equations: Duffing equation, cubic nonlinear Schrodinger equation, Klein- Gordon-Zakharov equations, quadratic Duffing equation, KdV equation, Gardner equation, Boussinesq equation, symmetric regular long wave equation, generalized shallow water wave equation, Klein-Gordon equation with quadratic nonlinearity, Fitzhugh-Nagumo- Huxley equation, and double sine-Gordon equation.

2. The Main Idea

In the search of the traveling wave solutions to nonlinear partial differential equation of the form

Pu, ux, ut, uxx, uxt, utt, . . . 0, 2.1 the first step consists in considering the wave transformation

ux, t v φξ

, ξx λt ξ0, ξ0 arbitrary constant, 2.2 for a suitable functionφφξ, whereλis a constant. Usually,φξ ξthe identity function.

Using2.2,2.1converts to an ordinary differential equationODEwith respect to shortly, w.r.t.the functionvξ

Q

v, v, v, . . .

0, 2.3

withQbeing a polynomial with respect to variablesv, v, v,. . ..

To find solutions to2.3, we suppose thatvξcan be expressed as vξ H

fξ, gξ

, 2.4

whereHf, gis a rational function in the new variablesf fξ, g gξ, which satisfy the system

fξ rfξgξ, g²ξ S

fξ ,

2.5

withr /0 being some constant to be determined andSfa rational function in the variable ffξ. We show that the system2.5may be solved exactly in certain cases. In fact, taking

fξ ϕ^Nξ, 2.6

whereϕξ/0 andN /0, system2.5reduces to ϕξ r

Nϕξgξ, g²ξ S

ϕ^Nξ .

2.7

From2.7we get

ϕξ₂ r²

N²ϕ²ξS ϕ^N

. 2.8

Equation2.8is of elliptic type. ChoosingSfand Nadequately, we may obtain distinct methods to solve nonlinear PDEs. More exactly, suppose we have solved2.8. Then, in view of2.5and2.6functionsfandgmay be computed by formulae

fξ ϕ^Nξ, gξ fξ rfξ N

r ϕξ

ϕξ. 2.9

To solve2.3we try one of the following ansatz:

vξ a₀ n j1

f^j−1

a_jf b_jg

, 2.10

vξ a₀ n

j1

a_jf^j,

vξ a₀ n

j1

a_jf^j b_jf^−j .

2.11

We substitute any of these ansatz into 2.3 and we obtain a polynomial equation either in the variables f fξ and g gξ or f fξ. We equate the coefficients of fⁱg^ji, j 0,1,2,3, . . . to zero, and we obtain a system of polynomial equations in the variablesaj, bi, λ, . . . .Solving this system with the aid of a symbolic computational package such as Mathematica 8 or Maple 15, we obtain the desired solutions. Sometimes, we replace ξwithkξand then the corresponding system is regarded w.r.t. the variablesf fkξand ggkξ, wherekconst.

We also may solve coupled systems of nonlinear equations. Indeed, suppose that we have a coupled system of two equations in the form

Pu,u, u _x,u_x, u_t,u_t, u_xx,u_xx, u_xt,u_xt, u_tt,u_tt, . . . 0, Pu, u, u x,ux, ut,ut, uxx,uxx, uxt,uxt, utt,utt, . . . 0.

2.12

We first apply the wave transformation ux, t v

φξ

, ux, t v φξ

, ξx λt ξ0, ξ0arbitrary constant,

2.13

for a suitable pair of functionsφ φξandφφξ in order to obtain a coupled system of two ODEs in the form

Q

v,v, v ,v, v,v, v,v, . . . 0, Q

v,v, v ,v, v,v, v,v, . . . 0,

2.14

whereQandQ are polynomials w.r.t. the variablesv,v, v ,v, v,v, v,v, . . ..

We seek solutions to system2.14in the forms

vξ a0

n j1

f^j−1

ajf bjg ,

vξ a0

n j1

f^j−1

ajf bjg ,

2.15

vξ a0

n j1

ajf^j,

vξ a0

n j1

aifⁱ,

2.16

vξ a0

n j1

ajf^j bjf^−j ,

vξ a₀

n i1

a_ifⁱ b_if⁻ⁱ .

2.17

The integersnandnare determined by the balancing method.

We substitute any of ansatz 2.15, 2.16, or 2.17 into 2.14, and we obtain a system of two polynomial equations either w.r.t. the variables f fξ and g gξ or w.r.t. the variable f fξ. We equate the coefficients offⁱg^j resp., the coefficients of fⁱ i, j0,1,2,3, . . .to zero, and we obtain a system of polynomial equations w.r.t the variables ai, bj,ai,bj, λ, . . . .Solving this system with the aid of a symbolic computational package such as Mathematica or Maple, we obtain the desired solutions. Sometimes, with the aim to add an additional parameterk, we replaceξwithkξand then the corresponding system is regarded w.r.t. the variablesffkξandggkξ, wherekconst.

The same technique is applied for solving systems of three or more equations.

3. The sn-ns Method and Its Derivation

LetN−1,r1 andSf af⁻² b cf², where

Δ b²−4ac >0. 3.1

This choice gives us2.8in the form

ϕ₂

aϕ⁴ bϕ² c. 3.2

We may express the general solution of this equation in terms of the Jacobi elliptic functions ns or nd as follows:

ϕξ ±

−b √ Δ

2a nskξ C|m, Carbitrary constant, 3.3 where

k

−b √ Δ

2 , m

b²−2ac b√ Δ

2ac . 3.4

Solution3.3is valid fora >0,b <0, and 0< c≤b²/4a.

On the other hand, functionξ → √

−1ns√

−1kξ C|mis real valued for any real k,m,ξ, andC. We may verify that function

φξ ±√

−1

−b √ Δ 2a ns√

−1kξ ξ₀|m

, 3.5

wherekandmare given by3.4, is a solution to equation

ϕ2aϕ⁴−bϕ² c, a >0, b <0, 0< c≤ b²

4a. 3.6

Thus, we always may find a solution to3.2whena >0 and 0< c≤b²/4afor anyb /0.

Now, let us assume thata <0. It may be verified that a solution to3.2is

ϕξ ±

−b−√ Δ

2a ndkξ C|m, Carbitrary constant, 3.7 where

k

b √ Δ

2 , m

−b²−4ac b√ Δ

2ac , forb /0, c >0. 3.8 Observe that for anyA >0 functionξ →nd√

−Aξ C|mis real valued for any real values ofm,ξ, andC. We conclude that3.2has exact solutions for anya /0 andb /0.

Projective equations are

fξ fξgξ, g²ξ a

f²ξ b cf²ξ. 3.9

TakingC0 in3.3we see that solutions to system3.9are fξ ϕ⁻¹ξ

√a

k snkξ|m, gξ fξ

fξ ksnkξ|mcskξ|mdskξ|m.

3.10

This motivates us to seek solutions to2.3in the form

vξ a0

n j1

ajsn^jkξ bjns^jkξ

. 3.11

Usually,n1,2 and thenvξhas the forms

vξ a0 a1snkξ|m b1nskξ|m, 3.12

vξ a₀ a₁snkξ|m b₁nskξ|m a₂sn²kξ|m b₂ns²kξ|m. 3.13

In the case whenm 1, this gives the tanh-coth method since snkξ | 1 tanhkξand nskξ|1 cothkξ.

Another possible ansatz suggested from2.10is

vξ a₀ ⁿ

j1sn^j−1kξ

a_jsnkξ|m b_jsnkξ|mcskξ|mdskξ|m

. 3.14

Ifn1,2, this ansatz reads

vξ a0 a1snkξ|m b1snkξ|m cskξ|m dskξ|m, 3.15 vξ a₀ a₁snkξ|m b₁snkξ|mcskξ|mdskξ|m

a₂sn²kξ|m b₂sn²kξ|mcskξ|mdskξ|m. 3.16 We may consider similar ansatz by replacing sn by dn and ns by nd, respectively. We will call this the dn-nd method. Thus, two possible ansatz for this method are

vξ a₀ a₁dnkξ|m b₁ndkξ|m,

vξ a₀ a₁dnkξ|m b₁ndkξ|m a₂dn²kξ|m b₂nd²kξ|m. 3.17 An other useful ansatz to handle other equations of the form2.3is

vξ a0 a1cnkξ|m b1nckξ|m. 3.18

We will call this the cn-nc method.

We also may try the following ansatz:

vξ a₀ a₁cnkξ|m

1 b1cnkξ|m. 3.19

For example, this ansatz may be successfully applied to the cubic-quintic Duffing equation, which is defined by

vξ pvξ qv³ξ rv⁵ξ 0. 3.20

4. Examples

In this section we solve various important models related to nonlinear science by the methods described in previous sections.

4.1. Duffing Equation

vξ pvξ qv³ξ 0, 4.1

wherep andqare nonzero constants. This equation is very important since some relevant physical models described by a nonlinear PDEs may be studied once this equation is solved.

Two of them are related to cubic nonlinear the Schrodinger equation and the Klein-Gordon- Zakharov equations.

To find solutions we multiply4.1byvξand we integrate it w.r.t.ξ. The resulting equation is

dv dξ

2

q

2v⁴ξ−pv²ξ−2C, 4.2

whereCis the constant of integration. This equation has form3.2and we already know that there exists an exact solution to it for anypandq. Instead, we may apply directly the sn-ns methodresp., the dn-nd methodto it. Balancing givesn1. We seek solutions in the form

vξ a0 a1snkξ|m b1nskξ|m. 4.3

Inserting4.3into4.1, we obtain a polynomial equation w.r.t. the variableζsnkξ|m.

Equating to zero the coefficients ofζ^jj0,1,2, . . .yields the following algebraic system:

3a₀a²₁q0, a₁

a²₁q 2k²m² 0, a₁b₁

3a₁b₁q 3a²₀q−k²m²−k² p 0, a₀

6a₁b₁q a²₀q p 0, b₁

b₁²q 2k² 0.

4.4

Solving system4.4gives solutions as follows:

ia₀ 0,a₁

−2pm/

m² 6m 1q,b₁

−2p/

m² 6m 1q,k

√p/√

m² 6m 1,

vξ ±

−2p m² 6m 1q

msn

√p

√m² 6m 1ξ|m

ns

√p

√m² 6m 1ξ|m

, 4.5

ia₀ 0,a₁

−2pm/

m²−6m 1q,b₁ −

−2p/

m²−6m 1q,k

√p/√

m²−6m 1,

vξ ±

−2p m²−6m 1q

msn

√p

m²−6m 1ξ|m

−ns

√p

√m²−6m 1ξ|m

, 4.6

iia₀0,a₁m

−2p/

qm² 1,b₁0, k√p/√ m² 1,

vξ ±m

− 2p qm² 1sn

p

m² 1ξ|m

, 4.7

iiia₀0,a₁0,b₁

−2p/

qm² 1,k√p/√ m² 1,

vξ ±

−2p qm² 1ns

√p

√m² 1ξ|m

. 4.8

Lettingm → 1, we obtain trigonometric and hyperbolic solutions:

vξ

− p 2qtanh

√p 2√

2ξ

− p 2qcoth

√p 2√

2ξ

,

vξ −

−p 2qtan

√p 2 ξ

−

−p 2qcot

√p 2 ξ

−2p qcsc

pξ

,

vξ ±

−p qcoth

p 2ξ

,

4.9

vξ ±

−p qtanh

p 2ξ

. 4.10

4.2. Cubic Nonlinear Schrodinger Equation

iut uxx μ|u|²u0, 4.11

where u ux, t is a complex-valued function of two real variables x and t and μ is a nonzero real parameter andi√

−1. The physical model of the cubic nonlinear Schrodinger equationshortly, NLS equation 4.11and its generalized variants occur in various areas of physics such as nonlinear optics, water waves, plasma physics, quantum mechanics, superconductivity, and the Bose-Einstein condensate theory. It also has applications in optics since it models many nonlinearity effects in a fiber, including but not limited to self-phase modulation, four-wave mixing, second harmonic generation, stimulated Raman scattering, and so forth. For water waves, the NLS equation 4.11 describes the evolution of the envelope of modulated nonlinear wave groups. All these physical phenomena can be better understood with the help of exact solutions for a givenμ. Whenμ > 0, the equation is said to be attractive. Ifμ <0 we say that it is repulsive. Recently, Ma and Chen5obtained some solutions to4.11.

We seek solutions to4.11in the form

ux, t vξexp i

αx γt

, ξx−2αt ξ0, 4.12

where α and γ are some real constants to be determined. Inserting4.12 into 4.11 and simplifying, we obtain

vξ− γ α²

vξ μv³ξ 0. 4.13

This last equation has the form4.1with

p− γ α²

, qμ. 4.14

Thus, making use of solutions4.5–4.10 for the choices given by4.14 we obtain exact solutions to the Schrodinger equation4.11in the form4.12.

4.3. Quadratic Duffing Equation

Let us consider the following second-order and second-degree nonlinear ODE:

vξ pv²ξ qvξ r, 4.15

wherep,q, andrare constants andp /0. Solutions to this equation may be used to study some important physical models whose associated PDEs may be solved after making the traveling wave transformation2.2. As we will show in next subsections some examples of nonlinear partial differential equations where this equation arises are the following.

iKdV equation:ut 6uux uxxx0.

iiGardner equationalso called combined KdV-mKdV equation:ut αuux βu²ux

γuxxx0,βγ /0,

iiiBoussinesq equation:utt αuuxx αu²_x βuxxxx0,

ivsymmetric regular long wave equation:utt uxx uuxt uxut uxxtt0,

vgeneralized shallow water wave equation:u_xxtt αu_xu_xt βu_tu_xx−u_xt−u_xx 0, viKlein-Gordon equation with quadratic nonlinearity:u_tt−α²u_xx βu−γu²0.

Balancing givesn2. We seek solutions to4.15in the form3.13, that is,

vξ a0 a1snkξ|m b1nskξ|m a2sn²kξ|m b2ns²kξ|m. 4.16

Inserting this ansatz into4.15gives the following algebraic system:

a1

k²m²−a2p 0,

a₂

6k²m²−a₂p 0,

4a2k²m² 4a2k² a²₁p 2a0a2p a2q0, 2a₂b₁p a₁k²m² a₁k² 2a₀a₁p a₁q0,

b1

k²−b2p 0,

b2

6k²−b2p 0,

2a0b2p 4b2k²m² 4b2k² b₁²p b2q0, 2a₀b₁p 2a₁b₂p b₁k²m² b₁k² b₁q0, 2a₁b₁p 2a₂b₂p−2a₂k² a²₀p a₀q−2b₂k²m² r 0.

4.17

Solving this system, we obtain the following solutions:

ik 1/2⁴

q²−4pr/m⁴−m² 1, a₀ q/2p − 1/2pm²

1

q²−4pr/m⁴−m² 1,a10,a2 3m²/2p

q²−4pr/m⁴−m² 1,b1 0,b₂0,

vξ a₀ 3m² 2p

q²−4pr m⁴−m² 1sn²

⎛

⎝1 2

4

q²−4pr m⁴−m² 1ξ|m

⎞

⎠, 4.18

iik 1/2⁴

q²−4pr/m⁴−m² 1, a0 −q/2p − 1/2pm²

1

q²−4pr/m⁴−m² 1, a₁ 0, a₂ 0, b₁ 0, b₂ 3/2p

q²−4pr/m⁴−m² 1,

vξ a0

3 2p

q²−4pr m⁴−m² 1ns²

⎛

⎝1 2

4

q²−4pr m⁴−m² 1ξ|m

⎞

⎠, 4.19

iiik 1/2⁴

q²−4pr/m⁴ 14m² 1, a0 −q/2p − 1/2pm²

1

q²−4pr/m⁴−m² 1, a₁ 0, a₂ 3m²/2p

q²−4pr/m⁴ 14m² 1, b₁0,b2 3/2p

q²−4pr/m⁴ 14m² 1,

vξ a0 3 2p

q²−4pr m⁴ 14m² 1

⎛

⎝3m²sn²

⎛

⎝1 2

4

q²−4pr

m⁴ 14m² 1ξ|m

⎞

⎠

ns²

⎛

⎝1 2

4

q²−4pr

m⁴ 14m² 1ξ|m

⎞

⎠

⎞

⎠,

4.20

ivk √

−1/2⁴

q²−4pr/m⁴−m² 1, a₀ 1/2pm²

1

q²−4pr/m⁴−m² 1 − q, a1 0, a2 0, b1 0, b2

−3/2p

q²−4pr/m⁴−m² 1,

vξ a0− 3 2p

q²−4pr m⁴−m² 1ns²

⎛

⎝√

−1 2

4

q²−4pr m⁴−m² 1ξ|m

⎞

⎠, 4.21

vk √

−1/2⁴

q²−4pr/m⁴−m² 1, a₀ q/2p − 1/2pm²

1

q²−4pr/m⁴−m² 1,a1 0,a2−3m²/2p

q²−4pr/m⁴−m² 1,b1 0,b₂0,

vξ a₀−3m² 2p

q²−4pr m⁴−m² 1sn²

⎛

⎝√

−1 2

4

q²−4pr m⁴−m² 1ξ|m

⎞

⎠, 4.22

vik √

−1/2⁴

q²−4pr/m⁴ 14m² 1,a₀ q/2p − 1/2pm²

1

q²−4pr/m⁴ 14m² 1,a10,a2−3m²/2p

q²−4pr/m⁴ 14m² 1, b₁0,b₂−3/2p

q²−4pr/m⁴ 14m² 1,

vξ a₀− 3 2p

q²−4pr m⁴ 14m² 1

⎛

⎝3m²sn²

⎛

⎝√

−1 2

4

q²−4pr

m⁴ 14m² 1ξ|m

⎞

⎠

ns²

⎛

⎝√

−1 2

4

q²−4pr

m⁴ 14m² 1ξ|m

⎞

⎠

⎞

⎠,

4.23

Lettingm → 1, we obtain trigonometric and hyperbolic solutions:

vξ a0 3 2p

q²−4prtanh² 1

2

4

q²−4prξ

.

vξ a0 3 2p

q²−4prcoth² 1

2

4

q²−4prξ

. vξ a0

3 4p

q²−4pr

3tanh² 1

4

4

q²−4prξ

coth² 1

4

4

q²−4prξ

.

4.24

Now, let us seek solutions in the ansatz form3.16, that is,

vξ a₀ a₁snkξ|m b₁nskξ|mcnkξ|mdnkξ|m

a2sn²kξ|m b2cnkξ|mdnkξ|m. 4.25

Inserting this ansatz into 4.15 and solving the resulting algebraic system yields the following solutions to4.15:

ik √

−1⁴

q²−4pr/m⁴ 14m² 1, a0 −q/2p 1/2pm²

1

q²−4pr/m⁴ 14m² 1,a10,a2−3m²/p

q²−4pr/m⁴ 14m² 1, b₁0,b₂±3m/p

q²−4pr/m⁴ 14m² 1,

vξ − q 2p

1 2p

m² 1

q²−4pr m⁴ 14m² 1

−3m² p

q²−4pr m⁴ 14m² 1sn²

⎛

⎝√

−1⁴

q²−4pr

m⁴ 14m² 1ξ|m

⎞

⎠

±3m p

q²−4pr m⁴ 14m² 1cn

⎛

⎝√

−1⁴

q²−4pr

m⁴ 14m² 1ξ|m

⎞

⎠

×dn

⎛

⎝√

−1⁴

q²−4pr

m⁴ 14m² 1ξ|m

⎞

⎠,

4.26

iik ⁴

q²−4pr/m⁴ 14m² 1, a0 −q/2p − 1/2pm²

1

q²−4pr/m⁴ 14m² 1,a₁ 0,a₂ 3m²/p

q²−4pr/m⁴ 14m² 1, b₁0,b₂±3m²/p

q²−4pr/m⁴ 14m² 1,

vξ − q 2p− 1

2p

m² 1

q²−4pr m⁴ 14m² 1

− 3m² p

q²−4pr m⁴ 14m² 1sn²

⎛

⎝⁴

q²−4pr

m⁴ 14m² 1ξ|m

⎞

⎠

± 3m p

q²−4pr m⁴ 14m² 1cn

⎛

⎝⁴

q²−4pr

m⁴ 14m² 1ξ|m

⎞

⎠dn

⎛

⎝⁴

q²−4pr

m⁴ 14m² 1ξ|m

⎞

⎠. 4.27

We may obtain other solutions by making use of the dn-nd method.

4.4. KdV Equation

ut 6uux uxxx0. 4.28

Ifuvξ,ξx λt ξ0, this equation takes the form

λvξ 6vξvξ vξ 0. 4.29

Integrating this equation w.r.t.ξyields

vξ −3v²ξ λvξ C, 4.30

whereCis the constant of integration. Equation4.30has the form4.15with

p−3, qλ, rC. 4.31

Exact solutions to KdV equation may be derived from4.18–4.25and4.31.

The KdV equation may also be solved by the Weierstrass elliptic functions method.

4.5. Gardner Equation

ut αuux βu²ux γu_xxx0, βγ /0. 4.32

This equation reduces to mKdV equation if we apply the similarity transformations

tγt, xx α²

4βt, u β

γ u α

2β

4.33

to the Gardner equation4.32. We obtain the mKdV equation

u_t εu²_xu γu_xxx0, 4.34

whereεsignβγ ±1.

This means that ifu u t,x is a solution to the mKdV equation, then the function defined by

uut, x γ

β u

γt, x α² 4βt

− α

2β 4.35

is a solution to Gardner equation4.32. For solutions to MkdV equation, see6.

4.6. Boussinesq Equation

utt αuuxx αu²_x βu_xxxx0, αβ /0. 4.36

After the traveling wave transformationuvξ,ξx λt ξ₀and integrating twice w.r.tξ, we obtain the following ODE:

vξ − α

2βv²ξ− λ²

βvξ−Dξ C

β , 4.37

whereCandDare the constants of integration. SettingD 0, we obtain an equation of the form4.15with

p− α

2β, q−λ²

β, r−C

β. 4.38

As we can see, exact solutions to the Boussinesq equation are calculated from4.18–4.25 and4.38.

4.7. Symmetric Regular Long Wave Equation

u_tt u_xx uu_xt u_xu_t u_xxtt0. 4.39 Letuvξ,ξ x λt ξ₀. Applying this transformation and integrating twice w.r.t.ξ, we obtain the ODE

vξ − 1

2λv²ξ−λ² 1

λ² vξ−Dξ C

λ² , 4.40

whereCandDare the constants of integration. SettingD 0, we obtain an equation of the form4.15with

p− 1

2λ, q−λ² 1

λ² , r−C

λ². 4.41

Thus, exact solutions to symmetric regular long wave equation are easily found from4.18–

4.25taking into account4.41.

4.8. Generalized Shallow Water Wave Equation

uxxtt αuxuxt βutuxx−uxt−uxx 0. 4.42 Let

uux, t Vξ, Vξ

vξdζ, ξx λt. 4.43

Substituting4.43into4.42and integrating once w.r.t.ξ, we obtain

vξ −α β

2λ v²ξ λ 1

λ² vξ C

λ², 4.44

whereCis the constant of integration. Equation4.15has the form4.15with

p−α β

2λ , q λ 1

λ² , r C

λ². 4.45

It is evident that we may find exact solutions to generalized shallow water wave equation 4.42from4.18–4.25for the choices given by4.45.

4.9. Klein-Gordon Equation with Quadratic Nonlinearity

u_tt−α²u_xx βu fu 0. 4.46

In the case whenfu −γu²we obtain the so-called Klein-Gordon equation with quadratic nonlinearity:

u_tt−α²u_xx βu−γu²0. 4.47 Letuvξ,ξx λt ξ₀. After this traveling wave transformation,4.47reduces to

vξ − γ

α²−λ²v²ξ β

α²−λ²vξ, α²/λ², 4.48 which is an equation of the form4.15with

p− γ

α²−λ², q β

α²−λ², r 0. 4.49

Again, exact solutions to4.47are obtained from4.18–4.25taking into account4.49.

4.10. Fitzhugh-Nagumo-Huxley Equation

u_t−u_xx u1−uα−u 0, αconst. 4.50

This equation is an important model in the study of neuron axon7. Let

uux, t vξ, ξx λt ξ0. 4.51

The corresponding ODE is

vξ−λvξ−vξvξ−1vξ−α 0. 4.52 Application of the sn-ns method gives only trivial solutions since we getλ 0. Instead, we may use other methods. If we apply the tanh-coth method or the exp method, we obtain nontrivial solutions. Indeed, balancing givesn1. Following the tanh-coth method, we try the ansatz

vξ a0 a1tanhkξ b1cothkξ. 4.53

Inserting 4.53 into 4.50 and solving the corresponding algebraic system gives the following solutions to4.50:

ux, t 1 2

1 tanh 1

2√ 2

x−2α−1

√2 t ξ₀

, ux, t 1

2

1 coth 1

2√ 2

x−2α−1

√2 t ξ0

, ux, t 1

4

2 tanh 1

4√ 2

x−2α−1

√2 t ξ0

coth

1 4√

2

x−2α−1

√2 t ξ0

, ux, t α

2

1 tanh α

2√ 2

x−2−α

√2 t ξ₀

, ux, t α

2

1 coth α

2√ 2

x−2−α

√2 t ξ0

, ux, t α

4

2 tanh α

4√ 2

x− 2α−1

√2 t ξ0

coth

α 4√ 2

x− 2α−1

√2 t ξ0

.

4.54

4.11. Double Sine-Gordon Equation

Our last example deals with the double sine-Gordon equation. This equation in a normalized form reads

u_tt−u_xx sinu−1

2sin2u 0. 4.55

This equation is an important model in the study of the DNA molecule8.

The application of the tanh-coth method gives only the trivial solutionu 0. If we apply the sn-ns method, we obtain nontrivial solutions. Indeed, let

u2 arctanv, vvξ, ξx λt ξ₀. 4.56

Inserting ansatz4.56into4.55gives the ODE

2v³ξ−2 λ²−1

vξ

vξ₂ λ²−1

vξ λ²−1

v²ξvξ 0. 4.57

Let

vξ a₀ a₁snkξ|m b₁nskξ|m. 4.58

Substituting 4.58 into 4.57 and solving the corresponding algebraic system gives the following solutions to4.55:

ia00, a1

√2/2√

m² 1√

−1, b10, λ

k²m²−1²−2m² 1/km²−1,

vξ

√2 2

m² 1√

−1sn

⎛

⎜⎝k

⎛

⎜⎝x

k²m²−1²−2m² 1 km²−1 t ξ₀

⎞

⎟⎠|m

⎞

⎟⎠, 4.59

iia₀ 0,a₁0,b₁ √

2/2m√

m² 1√

−1,λ

k²m²−1²−2m² 1/km²−1,

vξ

√2 2m

m² 1√

−1ns

⎛

⎜⎝k

⎛

⎜⎝x

k²m²−1²−2m² 1 km²−1 t ξ0

⎞

⎟⎠|m

⎞

⎟⎠, 4.60

iiia0 0,a1

mm²−6m 1/2√

2m−1,b1 −

mm²−6m 1/2√

2mm− 1,λ−

k²m 1⁴−2m²−6m 1/km 1²,

vξ

mm²−6m 1 2√

2m−1 snkξ|m−

mm²−6m 1 2√

2mm−1 nskξ|m,

ξx−

k²m 1⁴−2m²−6m 1 km 1² t ξ₀.

4.61

5. Comparison with Other Methods to Solve Nonlinear PDEs

There are some other powerful and systematical approaches for solving nonlinear partial differential equations, such as the expansion along the integrable ODE 9, 10, the transformed rational function method11, and the multiple expfunction method12. Even about linear DEs, there is some recent study on solution representations13and the linear superposition principle has been applied to bilinear equations14.

5.1. Multiple Exp Function Method

u_t u_xxxxx 30uu_xxx 30u_xu_xx 180u²u_x0. 5.1

Equation 5.1 is also called the Sawada-Kotera equation 15. In a recent work 16, the authors obtained multisoliton solutions to5.1by using Hirota’s bilinear approach.

Introducing the potentialw, defined by

uwx, 5.2

5.1may be written in the form

w_tx w_xxxxxx 30w_xw_xxxx 30w_xxw_xxx 180w_x²w_xx0. 5.3

Integrating 5.3 once w.r.t x and taking the constant of integration equal to zero, the following partial differential equation is obtained:

w_t 60w³_x 30w_xw_xxx w_xxxxx0. 5.4

We will call5.4the potential Caudrey-Dodd-Gibbon equation associated with5.1.

In view of the multiple exp method one-soliton solutions to5.4have the form

wx, t a0 a1exp η 1 b1exp

η , ηηx, t kx−ωt, 5.5

wherea0,a1, andb1are some constants to be determined. Inserting ansatz5.5into5.4and simplifying, we obtain the following polynomial equation in the variableζexpη:

k⁵−ω 2

−15a0b₁k⁴ 15a₁k⁴−13b₁k⁵−2b₁ω ζ 6

20a₀b²₁k⁴−20a₁b₁k⁴ 10a²₀b²₁k³−20a₀a₁b₁k³ 10a²₁k³ 11b₁²k⁵−b²₁ω 2b²₁

−15a0b₁k⁴ 15a₁k⁴−13b₁k⁵−2b₁ω

ζ³ b⁴₁

k⁵−ω ζ⁴0.

ζ² 5.6

Equating the coefficients of different powers ofζto zero gives an algebraic system. Solving it with either Mathematica 8 or Maple 15, we obtain

wk⁵, a₁b₁a0 k. 5.7

Thus, the following is a one-soliton solutionor one wave solution in Ma’s terminologyto 5.4:

wx, t a₀ b₁a0 kexp η 1 b1exp

η , ηkx−k⁵t. 5.8

In view of5.2one-soliton solution to Caudrey-Dodd-Gibbon equation5.1is

ux, t b1k²exp

kx−k⁵t

1 b1expkx−k⁵t₂. 5.9

Observe that solution5.9is the same solution obtained in 16by using Hirota’s bilinear method. We conclude that Hirota’s method and multiple exp method give the same result

for one-soliton solutions. Let us examine two-soliton solutions. In view of Ma’s method the two-soliton solutions to5.4are of the form

wx, t k1η1 k2η2 Rk1 k2η1η2

Δ η1 η2 Rη1η2 , Δ∈ {0,1}, η1expk1x−ω1t, η2expk2x−ω2t.

5.10

Let Δ 1. Inserting ansatz 5.10 into 5.4 and simplifying, we obtain the following polynomial equation w.r.t the variablesζ₁expη1andζ₂expη2:

c²₁c₂k₂R

k⁵₂−ω₂

ζ²₁ζ₂ c₁c²₂k₁R

k⁵₁−ω₁ ζ₁ζ²₂ c₁c₂k1 k₂

k₁⁵ 5k₂k₁⁴ 10k²₂k₁³ 10k³₂k₁² 5k₂⁴k₁ k⁵₂−ω₁−ω₂ R c1c₂k1−k₂

k⁵₁−5k₂k⁴₁ 10k²₂k³₁−10k³₂k²₁ 5k₂⁴k₁−k⁵₂−ω₁ ω₂ ζ₁ζ₂ c1k1

k⁵₁−ω1

ζ1 c2k2

k₂⁵−ω2

ζ2 0.

5.11

Equating the coefficients of ζ₁²ζ₂, ζ₁ζ²₂, ζ₁ζ₂, ζ₁,andζ₂ to zero gives a system of algebraic equations. Solving it, we obtain the following nontrivial solution:

w1k⁵₁, w2k₂⁵, R k1−k₂²

k₁²−k₁k₂ k²₂ k1 k₂²

k₁² k₁k₂ k²₂. 5.12 We see that solution5.12coincides with solutions obtained in16. The same is valid if we setΔ 0.We conclude that Ma’s method does not give any new solutions compared with Hirota’s method.

5.2. The Transformed Rational Function Method

Given a nonlinear ODE in the unknownu ux, t,u ux, y, toru ux, y, z, t. we search for traveling wave solutions determined by

u^rξ v η

, v

η p

η q

η p_mη^m p_m−1η^m−1 · · · p₀

q_nηⁿ q_n−1ηⁿ⁻¹ · · · q₀ , 5.13 wherem andnare two natural numbers andp_i0 ≤ i ≤ m, qj 0 ≤ j ≤ nare normally constants but could be functions of the independent variables andηis a solution to equation

Aη Bη Cη² Dη E0, 5.14

with A, B, C, D, and E being some constants. Observe that equations ηη²,

ηα η²

Riccati equation

, 5.15

vξ pv²ξ qvξ r

quadratic Duffing equation4.15

5.16

are particular cases of5.14. This method was applied in11to solve the3 1- dimensional Jimbo-Miwa equation

u_xxxy 3u_yu_xx 3u_xu_xy 2u_yt−3u_xz0, 5.17 which converts into the nonlinear ODE

a³bu⁴ 6a²buu−2bω 3acu 0 5.18 after the traveling wave transformation

u

x, y, z, t

uξ, ξax by cz−ωt. 5.19 If we integrate5.14once w.r.tx, then the following equation follows:

vξ −3

av²ξ 2bω 3ac

a³b vξ, whereuξ

vξdξ. 5.20

Observe that5.20is a quadratic Duffing equation5.16withp−3/a, q 2bω 3ac/a³b, andr0.

Solutions to this equation are given by4.18–4.27. These solutions were not reported in11. On the other hand, it is clear from4.24that the sn-ns method covers the solutions obtained in11.

6. Conclusions

We successfully obtained exact solutions for some important physical models by techniques based on projective equations. Mainly, we have used the sn-ns method. In our opinion, this is the most appropriate of all methods we have studied since it provides elliptic function solutions as well as trigonometric and hyperbolic solutions. However, in the cases when the sn-ns method does not workthis occurs for the Fitzhugh-Nagumo-Huxley equation, we may try other methods, such as the tanh-coth method. On the other hand, there are some equations for which the tanh-coth method gives only trivial solutionsthis is the case of the double sine-Gordon equation.

We think that some of the results we obtained are new in the open literature. Other results concerning exact solutions of nonlinear PDEs may be found in6,15,17–52.

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