Research Article Volume 1 Issue 1
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
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
35K99, 35P05, 35P99, 35C05
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,6–8 the simplest equation method,9,10 the Weierstrass elliptic function method,11 the Jacobi elliptic function method,12–14 the tanh-function method,15,16 the(G′/G) expansion method,17–22 the modified simple equation method,23–26 the Kudryashov method,27–29 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(G′G,1G) expansion method,36–40 the generalized Riccati equation mapping method41–45 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
ut−uxx+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 aN≠0 , 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 B≠0, 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 q≠0 .
It is well-known41-45 that Equation (2.6) has many families of solutions as follows:
Family 1: When p2−4qr>0 and pq≠0 or qr≠0 , we have
φ1(ξ)=−12q(p+√p2−4qrtanh(√p2−4qr2ξ)),
φ2(ξ)=−12q(p+√p2−4qrcoth(√p2−4qr2ξ)),
φ3(ξ)=−12q(p+√p2−4qr(coth(√p2−4qrξ)±csch(√p2−4qrξ))),
φ4(ξ)=−14q(2p+√p2−4qr(tanh(√p2−4qr4ξ)+coth(√p2−4qr4ξ))),
φ5(ξ)=12q(−p+±√(R2+M2) (p2−4qr)−A√p2−4qrcosh(√p2−4qrξ)Rsinh(√p2−4qrξ)+M),
φ6(ξ)=12q(−p−±√(M2−R2) (p2−4qr)+A√p2−4qrsinh(√p2−4qrξ)Rcosh(√p2−4qrξ)+M),
Where R and M are nonzero real constants satisfyingM2−R2>0 .
φ7(ξ)=2rcosh(√p2−4qr2ξ)√p2−4qrsinh(√p2−4qr2ξ)−pcosh(√p2−4qr2ξ),
φ8(ξ)=−2rsinh(√p2−4qr2ξ)psinh(√p2−4qr2ξ)−√p2−4qrcosh(√p2−4qr2ξ), φ9(ξ)=2rsinh(√p2−4qrξ)−psinh(√p2−4qrξ)+√p2−4qrcosh(√p2−4qrξ)±√p2−4qr, φ10(ξ)=4rsinh(14√p2−4qrξ) cosh(14√p2−4qrξ)−2psinh(14√p2−4qrξ) cosh(14√p2−4qrξ)+2√p2−4qrcosh2(14√p2−4qrξ)−√p2−4qr.
Family 2: When p2−4qr<0 and pq≠0 orqr≠0 , we have
φ11(ξ)=12q(−p+√4qr−p2tan(√4qr−p22ξ)),
φ12(ξ)=−12q(p+√4qr−p2cot(√4qr−p22ξ)),
φ13(ξ)=12q(−p+√4qr−p2(tan(√4qr−p2ξ)±sec(√4qr−p2ξ))), φ14(ξ)=−12q(p+√4qr−p2(cot(√4qr−p2ξ)±csc(√4qr−p2ξ))), φ15(ξ)=14q(−2p+√4qr−p2(tan(√4qr−p24ξ)−cot(√4qr−p24ξ))), φ16(ξ)=12q(−p+±√(R2−M2) (4qr−p2)−A√4qr−p2cos(√4qr−p2ξ)Rsin(√4qr−p2ξ)+M),
φ17(ξ)=12q(−p−±√(R2−M2) (4qr−p2)+A√4qr−p2sin(√4qr−p2ξ)Rcos(√4qr−p2ξ)+M),
where R and M are two nonzero real constants satisfying R2−M2>0 .
φ18(ξ)=−2rcos(√4qr−p22ξ)√4qr−p2sin(√4qr−p22ξ)+pcos(√4qr−p22ξ),
φ19(ξ)=2rsin(√4qr−p22ξ)−psin(√4qr−p22ξ)+√4qr−p2cos(√4qr−p22ξ), φ20(ξ)=−2rcos(√4qr−p2ξ)√4qr−p2sin(√4qr−p2ξ)+pcos(√4qr−p2ξ)±√4qr−p2. φ21(ξ)=2rsin(√4qr−p2ξ)−psin(√4qr−p2ξ)+√4qr−p2cos(√4qr−p2ξ)±√4qr−p2, φ22(ξ)=4rsin(14√4qr−p2ξ) cos(14√4qr−p2ξ)−2psin(14√4qr−p2ξ) cos(14√4qr−p2ξ)+2√4qr−p2cos2(14√4qr−p2ξ)−√4qr−p2.
Family 3: When r=0 andpq≠0 , 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 q≠0 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).
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 : δA3a31−2A3k2q2a1+pA2Bk2qa1+ωA2Bqa1+3δA2Ba0a21+γA2Ba21−2rAB2k2q2a1+3δAB2a20a1+2γAB2a0a1+μAB2a1+prB3k2qa1+rωB3qa1+δB3a30+γB3a20+μB3a0=0,
+3δAB2a20a1+2γAB2a0a1+μAB2a1+prB3k2qa1+rωB3qa1+δB3a30+γB3a20+μB3a0=0,
φ2 : 3δA3a0a21−3qA3k2pa1+γA3a21+qωA3a1+A2Bk2p2a1+2qA2Bk2ra1+ωA2Bpa1
−3δA2Bra31+6δA2Ba20a1+4γA2Ba0a1+2μA2Ba1−3qAB2k2pra1−6δAB2ra0a21
−2γAB2ra21+qωAB2ra1+3δAB2a30+3γAB2a20+3μAB2a0+B3k2p2ra1
+2qB3k2r2a1+ωB3pra1−3δB3ra20a1−2γB3ra0a1−μB3ra1=0,
φ : ωA3pa1−2qA3k2ra1−A3k2p2a1+3δA3a20a1+2γA3a0a1+μA3a1+3A2Bk2pra1
−6δA2Bra0a21−2γA2Bra21+ωA2Bra1+3δA2Ba30+3γA2Ba20+3μA2Ba0−AB2k2p2ra1
−2qAB2k2r2a1+ωAB2pra1+3δAB2r2a31−6δAB2ra20a1−4γAB2ra0a1−2μAB2ra1
+3B3k2pr2a1+3δB3r2a0a21+γB3r2a21+ωB3r2a1=0,
φ0 : ωA3ra1−pA3k2ra1+δA3a30+γA3a20+μA3a0+2A2Bk2r2a1−3δA2Bra20a1−2γA2Bra0a1 −μA2Bra1−pAB2k2r2a1+3δAB2r2a0a21+γAB2r2a21+ωAB2r2a1+2B3k2r3a1−δB3r3a31=0.
On solving the above algebraic equations with the aid of Maple or Mathematical, we have the following results:
Result 1:
a0=0, a1=2k2pγ, r=ω24k4, p=p, q=−1, δ=γ2k22ω2, μ=−12k2p2, A=−ωB2k2, B=B, k=k, ω=ω. (3.3)
Form this result, we have p2−4qr=p2+ω2k4>0.
Consequently, we have the following exact solutions:
u1(ξ)=−pωγ[(ω+k2p)+k2√p2+ω2k4tanh(12√p2+ω2k4ξ)(−ω+k2p)+k2√p2+ω2k4tanh(12√p2+ω2k4ξ)],
u2(ξ)=−pωγ[(ω+k2p)+k2√p2+ω2k4coth(12√p2+ω2k4ξ)(−ω+k2p)+k2√p2+ω2k4coth(12√p2+ω2k4ξ)],
u3(ξ)=−pωγ[(ω+k2p)+k2√p2+ω2k4(coth(√p2+ω2k4ξ)±csch(√p2+ω2k4ξ))(−ω+k2p)+k2√p2+ω2k4(coth(√p2+ω2k4ξ)±csch(√p2+ω2k4ξ))],
u4(ξ)=−pωγ[2(ω+k2p)+k2√p2+ω2k4(tanh(14√p2+ω2k4ξ)+coth(14√p2+ω2k4ξ))2(−ω+k2p)+k2√p2+ω2k4(tanh(14√p2+ω2k4ξ)+coth(14√p2+ω2k4ξ))],
u5(ξ)=−pωγ[(ω+k2p)−k2(±√(R2+M2)(p2k4+ω2)k4−R√p2+ω2k4cosh(√p2+ω2k4ξ)Rsinh(√p2+ω2k4ξ)+M)(−ω+k2p)−k2(±√(R2+M2)(p2k4+ω2)k4−R√p2+ω2k4cosh(√p2+ω2k4ξ)Rsinh(√p2+ω2k4ξ)+M)],
u6(ξ)=−pωγ[(ω+k2p)+k2(±√(R2+M2)(p2k4+ω2)k4+R√p2+ω2k4sinh(√p2+ω2k4ξ)Rcosh(√p2+ω2k4ξ)+M)(−ω+k2p)+k2(±√(R2+M2)(p2k4+ω2)k4+R√p2+ω2k4sinh(√p2+ω2k4ξ)Rcosh(√p2+ω2k4ξ)+M)], u7(ξ)=−pωγ[√p2k4+ω2sinh(12√p2+ω2k4ξ)+(ω−pk2)cosh(12√p2+ω2k4ξ)−√p2k4+ω2sinh(12√p2+ω2k4ξ)+(ω+pk2)cosh(12√p2+ω2k4ξ)],
u8(ξ)=−pωγ[(ω−pk2)sinh(12√p2+ω2k4ξ)+√p2k4+ω2cosh(12√p2+ω2k4ξ)(ω+pk2)sinh(12√p2+ω2k4ξ)−√p2k4+ω2cosh(12√p2+ω2k4ξ)],
u9(ξ)=−pωγ[(ω−pk2)sinh(√p2+ω2k4ξ)+√p2k4+ω2(cosh(√p2+ω2k4ξ)±1)(ω+pk2)sinh(√p2+ω2k4ξ)−√p2k4+ω2(cosh(√p2+ω2k4ξ)±1)],
u10(ξ)=−pωγ[2(ω−pk2)sinh(14√p2+ω2k4ξ) cosh(14√p2+ω2k4ξ)+√p2k4+ω2(2cosh2(14√p2+ω2k4ξ)−1)2(ω+pk2)sinh(14√p2+ω2k4ξ) cosh(14√p2+ω2k4ξ)−√p2k4+ω2(2cosh2(14√p2+ω2k4ξ)−1)],
where ξ=kx+ωt.
Result 2:
a0=0, a1=(−ω+3k2p)(Aq−Bp)γA, r=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 andpq≠0 , then we have the following exact solutions:
u11(ξ)=(−ω+3k2p)(Aq−Bp)γ[−pd(Aq−Bp)d+Aq(cosh(pξ)−sinh(pξ))],
u12(ξ)=(−ω+3k2p)(Aq−Bp)γ[−p(cosh(pξ)+sinh(pξ))Aqd+(Aq−Bp)(cosh(pξ)+sinh(pξ))],
,p>where ξ=kx+ωt.Result 3:
a0=0, a1=±k(Aq−Bp)√2δA, r=0, p=p, q=q, A=A, B=B, μ=kp(√2γ2δ−2kp), k=k, ω=−k(√2γ2δ−3kp), δ>0. (3.5)
,p>Since r=0 and pq≠0 , then we have the following exact solutions:u13(ξ)=±k(Aq−Bp)√2δ[−pd(Aq−Bp)d+Aq(cosh(pξ)−sinh(pξ))],
u14(ξ)=±k(Aq−Bp)√2δ[−p(cosh(pξ)+sinh(pξ))Aqd+(Aq−Bp)(cosh(pξ)+sinh(pξ))],
where ξ=kx−k(√2γ2δ−3kp) 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=2B√γ28δ−pB+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 p2−4qr>0 and pq≠0 or qr≠0 , we have
u15(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(2rqB+pA+A√p2−4rqtanh(12√p2−4rqξ))2Aq−pB−B√p2−4rqtanh(12√p2−4rqξ)],
u16(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(2rqB+pA+A√p2−4rqcoth(12√p2−4rqξ))2Aq−pB−B√p2−4rqcoth(12√p2−4rqξ)],
u17(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(2rqB+pA+A√p2−4rq(coth(√p2−4rqξ)±csch(√p2−4rqξ)))2Aq−pB−B√p2−4rq(coth(√p2−4rqξ)±csch(√p2−4rqξ))],
u18(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(4rqB+2pA+A√p2−4rq(coth(14√p2−4rqξ)+tanh(14√p2−4rqξ)))4Aq−2pB−B√p2−4rq(coth(14√p2−4rqξ)+tanh(14√p2−4rqξ))],
u19(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(2rqB+pA−A√p2−4rq(−Rcosh(√p2−4rqξ)±√R2+M2)Rsinh(√p2−4rqξ)+M)2Aq−pB+B√p2−4rq(−Rcosh(√p2−4rqξ)±√R2+M2)Rsinh(√p2−4rqξ)+M],
u20(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(2rqB+pA+A√p2−4rq(Rsinh(√p2−4rqξ)±√R2+M2)Rcosh(√p2−4rqξ)+M)2Aq−pB−B√p2−4rq(Rsinh(√p2−4rqξ)±√R2+M2)Rcosh(√p2−4rqξ)+M],
u21(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(r(pB+2A)cosh(12√p2−4rqξ)−Br√p2−4rqsinh(12√p2−4rqξ))(pA−2rB)cosh(12√p2−4rqξ)−A√p2−4rqsinh(12√p2−4rqξ)],
u22(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(r(pB+2A)sinh(12√p2−4rqξ)−Br√p2−4rqcosh(12√p2−4rqξ))(pA−2rB)sinh(12√p2−4rqξ)−A√p2−4rqcosh(12√p2−4rqξ)],
u23(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(r(pB+2A)sinh(√p2−4rqξ)−Br√p2−4rq(cosh(√p2−4rqξ)±1))(pA−2rB)sinh(√p2−4rqξ)−A√p2−4rq(cosh(√p2−4rqξ)±1)],
u24(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)× [A+B(2r(pB+2A)sinh(14√p2−4rqξ) cosh(14√p2−4rqξ)−Br√p2−4rq(2cosh2(14√p2−4rqξ)−1))2(pA−2rB)sinh(14√p2−4rqξ) cosh(14√p2−4rqξ)−A√p2−4rq(2cosh2(14√p2−4rqξ)−1)].
Type 2: When p2−4qr<0 and pq≠0 or qr≠0 , we have
u25(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(2rqB+pA−A√4rq−p2tan(12√4rq−p2ξ))2Aq−pB+B√4rq−p2tan(12√4rq−p2ξ)], u26(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(2rqB+pA+A√4rq−p2cot(12√4rq−p2ξ))2Aq−pB−B√4rq−p2cot(12√4rq−p2ξ)], u27(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(2rqB+pA−A√4rq−p2(tan(√4rq−p2ξ)±sec(√4rq−p2ξ)))2Aq−pB+B√4rq−p2(tan(√4rq−p2ξ)±sec(√4rq−p2ξ))], u28(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(2rqB+pA+A√4rq−p2(cot(√4rq−p2ξ)±csc(√4rq−p2ξ)))2Aq−pB−B√4rq−p2(cot(√4rq−p2ξ)±csc(√4rq−p2ξ))], u29(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(2(2rqB+pA)+A√4rq−p2(cot(14√4rq−p2ξ)−tan(14√4rq−p2ξ)))2(2Aq−pB)−B√4rq−p2(cot(14√4rq−p2ξ)−tan(14√4rq−p2ξ))], u30(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)×[A+B((2rqB+pA)(M+Rsin(√4rq−p2ξ))−A√4rq−p2(−Rcos(√4rq−p2ξ)±√R2−M2))(2Aq−pB)(M+Rsin(√4rq−p2ξ))+B√4rq−p2(−Rcos(√4rq−p2ξ)±√R2−M2)],
u31(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)×[A+B((2rqB+pA)(M+Rsin(√4rq−p2ξ))+A√4rq−p2(Rcos(√4rq−p2ξ)±√R2−M2))(2Aq−pB)(M+Rsin(√4rq−p2ξ))−B√4rq−p2(Rcos(√4rq−p2ξ)±√R2−M2)],
u32(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(r(2A+pB)cos(12√4rq−p2ξ)+Br√4rq−p2sin(12√4rq−p2ξ))(Ap−2rB)cos(12√4rq−p2ξ)+A√4rq−p2sin(12√4rq−p2ξ)], u33(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(r(2A+pB)sin(12√4rq−p2ξ)−Br√4rq−p2cos(12√4rq−p2ξ))(Ap−2rB)sin(12√4rq−p2ξ)−A√4rq−p2cos(12√4rq−p2ξ)], u34(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(r(2A+pB)cos(√4rq−p2ξ)+Br√4rq−p2(sin(√4rq−p2ξ)±1))(Ap−2rB)cos(√4rq−p2ξ)+A√4rq−p2(sin(√4rq−p2ξ)±1)], u35(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)[A+B(r(2A+pB)sin(√4rq−p2ξ)−Br√4rq−p2(cos(√4rq−p2ξ)±1))(Ap−2rB)sin(√4rq−p2ξ)−A√4rq−p2(cos(√4rq−p2ξ)±1)], u36(ξ)=−γ(−ABp+qA2+rB2)δ(−pB+2Aq) (A2+rB2)×[A+B(2r(2A+pB)sin(14√4rq−p2ξ) cos(14√4rq−p2ξ)−Br√4rq−p2(2cos2(14√4rq−p2ξ)−1))2(Ap−2rB)sin(14√4rq−p2ξ) (14√4rq−p2ξ)−A√4rq−p2(2cos2(14√4rq−p2ξ)−1)].
Type 3: When r=0 and pq≠0 , we have
u37(ξ)=−γ(−Bp+qA)δ(−pB+2Aq)[1+Bpd(Aq−Bp)d+Aq(cosh(pξ)−sinh(pξ))],
u38(ξ)=−γ(−Bp+qA)δ(−pB+2Aq)[1+Bp(cosh(pξ)+sinh(pξ))Aqd+(Aq−Bp)(cosh(pξ)+sinh(pξ))],
where ξ=(2B−pB+2Aq√γ28δ) x+(Bγ22δ(−pB+2Aq)) t .
Type 4: When r=p=0 and q≠0 , we have
u39(ξ)=−γ2δ[1+BAqξ+Ac1−B],
where ξ=BAq√γ28δx+Bγ24δAqt .
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:
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.41–45 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.
None.
Authors declare that there are no conflicts of interests.
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