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eISSN: 2576-4543

Physics & Astronomy International Journal

Research Article Volume 1 Issue 1

The backlund transformation of the generalized Riccati equation and its applications to the nonlinear KPP equation

Elsayed ME Zayed,1 Khaled AE Alurrfi,2 Abdul Ghani Al Nowehy3

1Department of Mathematics, Faculty of Sciences, Zagazig University, Egypt
2Department of Mathematics, Faculty of Arts & Science, Mergib University, Libya
3Department of Mathematics, Faculty of Education and Science, Taiz University, Yemen

Correspondence: Elsayed ME Zayed, Department of Mathematics, Faculty of Sciences, Zagazig University, Zagazig, Egypt

Received: June 17, 2017 | Published: August 30, 2017

Citation: Zayed EME, Alurrfi KAE, Al-Nowehy AG. The backlund transformation of the generalized Riccati equation and its applications to the nonlinear KPP equation. Phys Astron Int J. 2017;1(1):39-47. DOI: 10.15406/paij.2017.01.00007

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Abstract

The Bäcklund transformation of the generalized Riccati equation is applied in this article to construct many new exact traveling wave solutions for the nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation. Solutions, trigonometric and rational solutions of this equation are obtained. This transformation is straightforward and concise. It gives much more general results than the well-known results obtaining by other methods. With the aid of Maple, some graphical representations for some results are presented by choosing suitable values of parameters.

Keywords: exact traveling wave solutions, bäcklund transformation of generalized Riccati equation, kolmogorov-petrovskii-piskunov equation, soliton solutions, trigonometric solutions, rational solutions

Mathematics subject classification

35K99, 35P05, 35P99, 35C05

Introduction

The investigation of exact traveling wave solutions to nonlinear PDEs plays an important role in the study of nonlinear physical phenomena. Nonlinear wave phenomena appears in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics and geochemistry. Nonlinear wave phenomena of dispersion, dissipation, diffusion, reaction and convection are very important in nonlinear wave equations. In recent decades, many effective methods have been established to obtain exact solutions of nonlinear PDEs, such as the inverse scattering transform,1 the Hirota method,2 the truncated Painlevé expansion method,3 the Bäcklund transform method,1,4,5 the exp-function method,68 the simplest equation method,9,10 the Weierstrass elliptic function method,11 the Jacobi elliptic function method,1214 the tanh-function method,15,16 the(G/G) expansion method,1722 the modified simple equation method,2326 the Kudryashov method,2729 the multiple exp-function algorithm method,30,31 the transformed rational function method,32 the Frobenius decomposition technique,33 the local fractional variation iteration method,34 the local fractional series expansion method,35 the(GG,1G) expansion method,3640 the generalized Riccati equation mapping method4145 and so on.

The objective of this article is to use the Bäcklund transformation of the generalized Riccati equation to construct new exact traveling wave solutions of the following nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation.22,26,46

utuxx+μu2+γu2+δu3=0  (1.1)

Where μ, γ, δ are real constants. Equation (1.1) includes the Fisher equation, Huxley equation, Burgers-Huxley equation, Chaffee-Infanfe equation and Fitzhugh-Nagumo equation as special cases. Recently, Feng et al.22 have discussed Equation (1.1) using the (G/G) -expansion method and found its exact solutions, while Zayed et al.26,46 have applied two methods via the modified simple equation method and the Riccati equation method combined with the (G/G) -expansion method respectively, to Equation (1.1) and determined the exact traveling wave solutions of it.

This paper is organized as follows: In Section 2, the description of the Bäcklund transformation of the generalized Riccati equation is given. In Section 3, we use the given method described in Section 2, to find many new exact traveling wave solutions of the nonlinear KPP equation. In Section 4, physical explanations of some results are presented. In Section 5, some conclusions are obtained.

Description of the bäcklund transformation of the generalized riccati equation

Suppose that we have the following nonlinear PDE:

F(u,ut,ux,utt,uxx,...)=0,  (2.1)

Where F  is a polynomial in u(x,t)  and its partial derivatives, in which the highest order derivatives and the nonlinear terms are involved. In the following, we give the main steps of this method [5]:

Step 1: Using the wave transformation

u(x,t)=u(ξ),                ξ=kx+ωt,  (2.2)

where k  and ω  are constants, to reduce Equation (2.1) to the following ODE:

P(u,u,u,...)=0,  (2.3)

where P  is a polynomial in u(ξ)  and its total derivatives while =d/dξ.

Step 2: Assume that Equation (2.3) has the formal solution

u(ξ)=Ni=0aiψ(ξ)i,  (2.4)

where ai  are constants to be determined, such that aN0  , while ψ(ξ)  comes from the following Bäcklund transformation

ψ(ξ)=rB+Aφ(ξ)A+Bφ(ξ),  (2.5)

where r,A,B  are constants with B0,  while φ(ξ)  satisfies the generalized Riccati equation:

φ(ξ)=r+pφ(ξ)+qφ(ξ)2,  (2.6)

where p,q  are constants, such that q0 .

It is well-known41-45 that Equation (2.6) has many families of solutions as follows:

Family 1: When p24qr>0  and pq0  or qr0 , we have

φ1(ξ)=12q(p+p24qrtanh(p24qr2ξ)),

 φ2(ξ)=12q(p+p24qrcoth(p24qr2ξ)),

 φ3(ξ)=12q(p+p24qr(coth(p24qrξ)±csch(p24qrξ))),

 φ4(ξ)=14q(2p+p24qr(tanh(p24qr4ξ)+coth(p24qr4ξ))),

 φ5(ξ)=12q(p+±(R2+M2)(p24qr)Ap24qrcosh(p24qrξ)Rsinh(p24qrξ)+M),

 φ6(ξ)=12q(p±(M2R2)(p24qr)+Ap24qrsinh(p24qrξ)Rcosh(p24qrξ)+M),

Where R  and M  are nonzero real constants satisfyingM2R2>0 .

φ7(ξ)=2rcosh(p24qr2ξ)p24qrsinh(p24qr2ξ)pcosh(p24qr2ξ),  

φ8(ξ)=2rsinh(p24qr2ξ)psinh(p24qr2ξ)p24qrcosh(p24qr2ξ),  φ9(ξ)=2rsinh(p24qrξ)psinh(p24qrξ)+p24qrcosh(p24qrξ)±p24qr,  φ10(ξ)=4rsinh(14p24qrξ)cosh(14p24qrξ)2psinh(14p24qrξ)cosh(14p24qrξ)+2p24qrcosh2(14p24qrξ)p24qr.

Family 2: When p24qr<0  and pq0  orqr0 , we have

φ11(ξ)=12q(p+4qrp2tan(4qrp22ξ)),

φ12(ξ)=12q(p+4qrp2cot(4qrp22ξ)),  

φ13(ξ)=12q(p+4qrp2(tan(4qrp2ξ)±sec(4qrp2ξ))),  φ14(ξ)=12q(p+4qrp2(cot(4qrp2ξ)±csc(4qrp2ξ))),  φ15(ξ)=14q(2p+4qrp2(tan(4qrp24ξ)cot(4qrp24ξ))),  φ16(ξ)=12q(p+±(R2M2)(4qrp2)A4qrp2cos(4qrp2ξ)Rsin(4qrp2ξ)+M),  

φ17(ξ)=12q(p±(R2M2)(4qrp2)+A4qrp2sin(4qrp2ξ)Rcos(4qrp2ξ)+M),

where R  and M  are two nonzero real constants satisfying R2M2>0 .

φ18(ξ)=2rcos(4qrp22ξ)4qrp2sin(4qrp22ξ)+pcos(4qrp22ξ),  

φ19(ξ)=2rsin(4qrp22ξ)psin(4qrp22ξ)+4qrp2cos(4qrp22ξ),  φ20(ξ)=2rcos(4qrp2ξ)4qrp2sin(4qrp2ξ)+pcos(4qrp2ξ)±4qrp2.  φ21(ξ)=2rsin(4qrp2ξ)psin(4qrp2ξ)+4qrp2cos(4qrp2ξ)±4qrp2,  φ22(ξ)=4rsin(144qrp2ξ)cos(144qrp2ξ)2psin(144qrp2ξ)cos(144qrp2ξ)+24qrp2cos2(144qrp2ξ)4qrp2.

Family 3: When r=0  andpq0 , we have

φ23(ξ)=pdq(d+cosh(pξ)sinh(pξ)),

φ24(ξ)=p(cosh(pξ)+sinh(pξ))q(d+cosh(pξ)+sinh(pξ)),

where d  is an arbitrary constant.

Family 4: When q0  and r=p=0  , we have

φ25(ξ)=1qξ+c1,

where c1  is an arbitrary constant.

Step 3: We determine the positive integer N  in (2.4) by using the homogeneous balance between the highest-order derivatives and the nonlinear terms in Equation (2.3). More precisely we define the degree of u(ξ)  as D[u(ξ)]=N  which gives rise to the degree of other expressions as follows:

D[dludξl]=N+l,D[um(dludξl)s]=Nm+s(l+N).  (2.7)

Therefore, we can get the value of N  in (2.4).

Step 4: We substitute (2.4) along with Equations (2.5) and (2.6) into Equation (2.3), collect all the terms with the same powers of φi(ξ) and set them to zero, we obtain a system of algebraic equations, which can be solved by Maple to get the values of ai  ,k andω . Consequently, we obtain the exact traveling wave solutions of Equation (2.1).

An application

In this section, we will apply the method described in Section 2 to find the exact traveling wave solutions of the nonlinear KPP equation (1.1). To this end, we use the wave transformation (2.2) to reduce Equation (1.1) to the following ODE:

ωu(ξ)k2u(ξ)+μu(ξ)+γu2(ξ)+δu3(ξ)=0.  (3.1)

By balancing u  with u3  in Equation (3.1), we getN=1 . Consequently, we have the formal solution

u(ξ)=a0+a1ψ(ξ),  (3.2)

where a0,a1  are constants to be determined, such that a1 0,  while ψ(ξ)  is given by (2.5).

Now, substituting (3.2) along with Equations (2.5) and (2.6) into (3.1), collecting the coefficients of φi(ξ)  and setting them to zero, we get the following system of algebraic equations:

φ3:δA3a312A3k2q2a1+pA2Bk2qa1+ωA2Bqa1+3δA2Ba0a21+γA2Ba212rAB2k2q2a1+3δAB2a20a1+2γAB2a0a1+μAB2a1+prB3k2qa1+rωB3qa1+δB3a30+γB3a20+μB3a0=0,

+3δAB2a20a1+2γAB2a0a1+μAB2a1+prB3k2qa1+rωB3qa1+δB3a30+γB3a20+μB3a0=0,

φ2:3δA3a0a213qA3k2pa1+γA3a21+qωA3a1+A2Bk2p2a1+2qA2Bk2ra1+ωA2Bpa1

3δA2Bra31+6δA2Ba20a1+4γA2Ba0a1+2μA2Ba13qAB2k2pra16δAB2ra0a21

2γAB2ra21+qωAB2ra1+3δAB2a30+3γAB2a20+3μAB2a0+B3k2p2ra1

+2qB3k2r2a1+ωB3pra13δB3ra20a12γB3ra0a1μB3ra1=0,

φ:ωA3pa12qA3k2ra1A3k2p2a1+3δA3a20a1+2γA3a0a1+μA3a1+3A2Bk2pra1

6δA2Bra0a212γA2Bra21+ωA2Bra1+3δA2Ba30+3γA2Ba20+3μA2Ba0AB2k2p2ra1

2qAB2k2r2a1+ωAB2pra1+3δAB2r2a316δAB2ra20a14γAB2ra0a12μAB2ra1

+3B3k2pr2a1+3δB3r2a0a21+γB3r2a21+ωB3r2a1=0,

φ0:ωA3ra1pA3k2ra1+δA3a30+γA3a20+μA3a0+2A2Bk2r2a13δA2Bra20a12γA2Bra0a1 μA2Bra1pAB2k2r2a1+3δAB2r2a0a21+γAB2r2a21+ωAB2r2a1+2B3k2r3a1δB3r3a31=0.

On solving the above algebraic equations with the aid of Maple or Mathematical, we have the following results:

Result 1:

a0=0a1=2k2pγr=ω24k4, p=p, q=1, δ=γ2k22ω2, μ=12k2p2,A=ωB2k2, B=B, k=k, ω=ω.  (3.3)

Form this result, we have p24qr=p2+ω2k4>0.  

Consequently, we have the following exact solutions:

u1(ξ)=pωγ[(ω+k2p)+k2p2+ω2k4tanh(12p2+ω2k4ξ)(ω+k2p)+k2p2+ω2k4tanh(12p2+ω2k4ξ)],

u2(ξ)=pωγ[(ω+k2p)+k2p2+ω2k4coth(12p2+ω2k4ξ)(ω+k2p)+k2p2+ω2k4coth(12p2+ω2k4ξ)],

u3(ξ)=pωγ[(ω+k2p)+k2p2+ω2k4(coth(p2+ω2k4ξ)±csch(p2+ω2k4ξ))(ω+k2p)+k2p2+ω2k4(coth(p2+ω2k4ξ)±csch(p2+ω2k4ξ))],

 u4(ξ)=pωγ[2(ω+k2p)+k2p2+ω2k4(tanh(14p2+ω2k4ξ)+coth(14p2+ω2k4ξ))2(ω+k2p)+k2p2+ω2k4(tanh(14p2+ω2k4ξ)+coth(14p2+ω2k4ξ))],

 u5(ξ)=pωγ[(ω+k2p)k2(±(R2+M2)(p2k4+ω2)k4Rp2+ω2k4cosh(p2+ω2k4ξ)Rsinh(p2+ω2k4ξ)+M)(ω+k2p)k2(±(R2+M2)(p2k4+ω2)k4Rp2+ω2k4cosh(p2+ω2k4ξ)Rsinh(p2+ω2k4ξ)+M)],

 u6(ξ)=pωγ[(ω+k2p)+k2(±(R2+M2)(p2k4+ω2)k4+Rp2+ω2k4sinh(p2+ω2k4ξ)Rcosh(p2+ω2k4ξ)+M)(ω+k2p)+k2(±(R2+M2)(p2k4+ω2)k4+Rp2+ω2k4sinh(p2+ω2k4ξ)Rcosh(p2+ω2k4ξ)+M)],  u7(ξ)=pωγ[p2k4+ω2sinh(12p2+ω2k4ξ)+(ωpk2)cosh(12p2+ω2k4ξ)p2k4+ω2sinh(12p2+ω2k4ξ)+(ω+pk2)cosh(12p2+ω2k4ξ)],

 u8(ξ)=pωγ[(ωpk2)sinh(12p2+ω2k4ξ)+p2k4+ω2cosh(12p2+ω2k4ξ)(ω+pk2)sinh(12p2+ω2k4ξ)p2k4+ω2cosh(12p2+ω2k4ξ)],

 u9(ξ)=pωγ[(ωpk2)sinh(p2+ω2k4ξ)+p2k4+ω2(cosh(p2+ω2k4ξ)±1)(ω+pk2)sinh(p2+ω2k4ξ)p2k4+ω2(cosh(p2+ω2k4ξ)±1)],

 u10(ξ)=pωγ[2(ωpk2)sinh(14p2+ω2k4ξ)cosh(14p2+ω2k4ξ)+p2k4+ω2(2cosh2(14p2+ω2k4ξ)1)2(ω+pk2)sinh(14p2+ω2k4ξ)cosh(14p2+ω2k4ξ)p2k4+ω2(2cosh2(14p2+ω2k4ξ)1)],

where ξ=kx+ωt.  

Result 2:

a0=0a1=(ω+3k2p)(AqBp)γAr=0, p=p, q=q, δ=2γ2k2(ω+3k2p)2, A=A, B=B, μ=p(ω+k2p),k=k, ω=ω.  (3.4)

Since r=0  andpq0 , then we have the following exact solutions:

u11(ξ)=(ω+3k2p)(AqBp)γ[pd(AqBp)d+Aq(cosh(pξ)sinh(pξ))],

u12(ξ)=(ω+3k2p)(AqBp)γ[p(cosh(pξ)+sinh(pξ))Aqd+(AqBp)(cosh(pξ)+sinh(pξ))],

,p>where ξ=kx+ωt.  

Result 3:

a0=0a1=±k(AqBp)2δAr=0, p=p, q=q, A=A, B=B, μ=kp(2γ2δ2kp),k=k, ω=k(2γ2δ3kp),    δ>0.  (3.5)

,p>Since r=0  and pq0  , then we have the following exact solutions:

u13(ξ)=±k(AqBp)2δ[pd(AqBp)d+Aq(cosh(pξ)sinh(pξ))],

u14(ξ)=±k(AqBp)2δ[p(cosh(pξ)+sinh(pξ))Aqd+(AqBp)(cosh(pξ)+sinh(pξ))],

where ξ=kxk(2γ2δ3kp)t.

Result 4:

a0=Aγ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)a1=Bγ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)r=r, p=p, q=q, A=A, B=B, μ=qγ2(ABp+qA2+rB2)δ(pB+2Aq)2,k=2Bγ28δpB+2Aq, ω=Bγ22δ(pB+2Aq),    δ>0.  (3.6)

In this case, we deduce that Equation (1.1) has many types of the exact traveling wave solutions as follows:

Type 1: When p24qr>0  and pq0  or qr0 , we have

u15(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(2rqB+pA+Ap24rqtanh(12p24rqξ))2AqpBBp24rqtanh(12p24rqξ)],

 u16(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(2rqB+pA+Ap24rqcoth(12p24rqξ))2AqpBBp24rqcoth(12p24rqξ)],

 u17(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(2rqB+pA+Ap24rq(coth(p24rqξ)±csch(p24rqξ)))2AqpBBp24rq(coth(p24rqξ)±csch(p24rqξ))],

 u18(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(4rqB+2pA+Ap24rq(coth(14p24rqξ)+tanh(14p24rqξ)))4Aq2pBBp24rq(coth(14p24rqξ)+tanh(14p24rqξ))],

 u19(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(2rqB+pAAp24rq(Rcosh(p24rqξ)±R2+M2)Rsinh(p24rqξ)+M)2AqpB+Bp24rq(Rcosh(p24rqξ)±R2+M2)Rsinh(p24rqξ)+M],

 u20(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(2rqB+pA+Ap24rq(Rsinh(p24rqξ)±R2+M2)Rcosh(p24rqξ)+M)2AqpBBp24rq(Rsinh(p24rqξ)±R2+M2)Rcosh(p24rqξ)+M],

 u21(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(r(pB+2A)cosh(12p24rqξ)Brp24rqsinh(12p24rqξ))(pA2rB)cosh(12p24rqξ)Ap24rqsinh(12p24rqξ)],

 u22(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(r(pB+2A)sinh(12p24rqξ)Brp24rqcosh(12p24rqξ))(pA2rB)sinh(12p24rqξ)Ap24rqcosh(12p24rqξ)],

 u23(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(r(pB+2A)sinh(p24rqξ)Brp24rq(cosh(p24rqξ)±1))(pA2rB)sinh(p24rqξ)Ap24rq(cosh(p24rqξ)±1)],

u24(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)×[A+B(2r(pB+2A)sinh(14p24rqξ)cosh(14p24rqξ)Brp24rq(2cosh2(14p24rqξ)1))2(pA2rB)sinh(14p24rqξ)cosh(14p24rqξ)Ap24rq(2cosh2(14p24rqξ)1)].

Type 2: When p24qr<0  and pq0  or qr0 , we have

u25(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(2rqB+pAA4rqp2tan(124rqp2ξ))2AqpB+B4rqp2tan(124rqp2ξ)],  u26(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(2rqB+pA+A4rqp2cot(124rqp2ξ))2AqpBB4rqp2cot(124rqp2ξ)],  u27(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(2rqB+pAA4rqp2(tan(4rqp2ξ)±sec(4rqp2ξ)))2AqpB+B4rqp2(tan(4rqp2ξ)±sec(4rqp2ξ))],  u28(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(2rqB+pA+A4rqp2(cot(4rqp2ξ)±csc(4rqp2ξ)))2AqpBB4rqp2(cot(4rqp2ξ)±csc(4rqp2ξ))],  u29(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(2(2rqB+pA)+A4rqp2(cot(144rqp2ξ)tan(144rqp2ξ)))2(2AqpB)B4rqp2(cot(144rqp2ξ)tan(144rqp2ξ))],  u30(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)×[A+B((2rqB+pA)(M+Rsin(4rqp2ξ))A4rqp2(Rcos(4rqp2ξ)±R2M2))(2AqpB)(M+Rsin(4rqp2ξ))+B4rqp2(Rcos(4rqp2ξ)±R2M2)],

u31(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)×[A+B((2rqB+pA)(M+Rsin(4rqp2ξ))+A4rqp2(Rcos(4rqp2ξ)±R2M2))(2AqpB)(M+Rsin(4rqp2ξ))B4rqp2(Rcos(4rqp2ξ)±R2M2)],

u32(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(r(2A+pB)cos(124rqp2ξ)+Br4rqp2sin(124rqp2ξ))(Ap2rB)cos(124rqp2ξ)+A4rqp2sin(124rqp2ξ)],  u33(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(r(2A+pB)sin(124rqp2ξ)Br4rqp2cos(124rqp2ξ))(Ap2rB)sin(124rqp2ξ)A4rqp2cos(124rqp2ξ)],  u34(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(r(2A+pB)cos(4rqp2ξ)+Br4rqp2(sin(4rqp2ξ)±1))(Ap2rB)cos(4rqp2ξ)+A4rqp2(sin(4rqp2ξ)±1)],  u35(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)[A+B(r(2A+pB)sin(4rqp2ξ)Br4rqp2(cos(4rqp2ξ)±1))(Ap2rB)sin(4rqp2ξ)A4rqp2(cos(4rqp2ξ)±1)],  u36(ξ)=γ(ABp+qA2+rB2)δ(pB+2Aq)(A2+rB2)×[A+B(2r(2A+pB)sin(144rqp2ξ)cos(144rqp2ξ)Br4rqp2(2cos2(144rqp2ξ)1))2(Ap2rB)sin(144rqp2ξ)(144rqp2ξ)A4rqp2(2cos2(144rqp2ξ)1)].

Type 3: When r=0  and pq0 , we have

u37(ξ)=γ(Bp+qA)δ(pB+2Aq)[1+Bpd(AqBp)d+Aq(cosh(pξ)sinh(pξ))],

u38(ξ)=γ(Bp+qA)δ(pB+2Aq)[1+Bp(cosh(pξ)+sinh(pξ))Aqd+(AqBp)(cosh(pξ)+sinh(pξ))],

where ξ=(2BpB+2Aqγ28δ)x+(Bγ22δ(pB+2Aq))t .

Type 4: When r=p=0  and q0 , we have

u39(ξ)=γ2δ[1+BAqξ+Ac1B],

where ξ=BAqγ28δx+Bγ24δAqt  .

Physical explanations of our obtained solutions

The obtained exact traveling wave solutions for the nonlinear KPP equation (1.1) are hyperbolic, trigonometric and rational. In this section, we have presented some graphs of the exact solutions u1(x,t) , u7(x,t) , u11(x,t) , u14(x,t) , u16(x,t) , u19(x,t) , u25(x,t)  and u31(x,t)  constructed by taking suitable values of involved unknown parameters to visualize the mechanism of the original equation (1.1). These solutions are kink, singular kink-shaped soliton solution, hyperbolic solutions and trigonometric solutions. For more convenience the graphical representations of these solutions are shown in the following figures 1 to 8:

Figure 1 Plot of the solution u1(x,t)  when k=2,p=ω=1,γ=1.

Figure 2 Plot of the solution u7(x,t)  when k=1,p=3,ω=2,γ=1.  

Figure 3 Plot of the solution u11(x,t)  when k=1,p=3,q=4,ω=1, γ=3,d=1, B=2,A=1.

Figure 4 Plot of the solution u14(x,t)  when k=1,p=1,δ=2,q=3, γ=1,d=3,B=1,A=2.

Figure 5 Plot of the solution u16(x,t)  when p=3,δ=4,q=1, γ=3,r=1,B=2,A=1.

Figure 6 Plot of the solution u19(x,t)  when p=5, δ=1,q=1,γ=3,r=1,B=2,A=2,R=2,M=2.

Figure 7 Plot of the solution u25(x,t)  when p=1,δ=1,q=5, γ=1,r=4,B=3,A=2.

Figure 8 Plot of the solution u31(x,t)  when p=1, δ=1,q=2, γ=1,r=4, B=2,A=2,R=3,M=2.

Conclusion

In this article, we have employed the Bäcklund transformation of the generalized Riccati equation to obtain many new exact traveling wave solutions of the nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation (1.1). On comparing our results in this paper with the well-known results obtained in22,26,46 we deduce that our results in this article are new and are not published elsewhere. The Bäcklund transformation of the generalized Riccati equation obtained in this article is more effective and gives more exact solutions than the generalized Riccati equation mapping method obtained in.4145 Further, all solutions obtained in this article have been checked with the Maple by putting them back into the original equations. Finally, the proposed method in this article can be applied to many other nonlinear PDEs in mathematical physics, which will be done in forthcoming papers.

Acknowledgments

None.

Conflicts of interest

Authors declare that there are no conflicts of interests.

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