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10.1038/s41598-024-72610-w
Article
Explicit solutions of the generalized Kudryashov’s equation with truncated M-fractional derivative
Gu Musong msgu@cdu.edu.cn

Liu Fanming
Li Jiale
Peng Chen
Li Zhao
https://ror.org/034z67559 grid.411292.d 0000 0004 1798 8975 College of Computer Science, Chengdu University, Chengdu, 610106 People’s Republic of China
17 9 2024
17 9 2024
2024
14 2171427 3 2024
9 9 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
The main purpose of this article is to study the generalized Kudryashov’s equation with truncated M-fractional derivative, which is commonly used to describe the propagation of wide pulses in nonlinear optical fibers. By employing the complete discriminant system of fourth-order polynomials, various types of explicit solutions are systematically classified, which include periodic solutions, the trigonometric functions, the double-period solutions, and the elliptic function solutions. Additionally, a series of 2D, 3D, and contour plots are generated to visually depict the spatial distribution and evolution of various solutions. This not only advances the development of nonlinear equations in theory but also provides valuable guidance in practical applications.

Keywords

Generalized Kudryashov’s equation
Explicit solution
M-fractional derivative
Complete discriminant system
Subject terms

Mathematics and computing
Physics
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Nonlinear science is a discipline that studies nonlinear phenomena1–11. Many nonlinear sciences are modeled through nonlinear fractional partial differential equations (NLFPDE)12–14. However, constructing explicit solutions for NLFPDE have been a hot topic of research by many experts and scholars in recent years. Researchers worldwide are committed to continually proposing new methods and improving existing ones, gradually offering effective approaches for constructing exact solutions to NLFPDE. With the further development of symbolic computing software such as Matlab, Maple, and Mathematica15, the construction of analytical solutions for NLFPDE have been promoted, which is a powerful computational tools for the study of NLFPDE. In recent years, many effective methods have been proposed to construct explicit solutions for this type of equation. These methods mainly include: the modified extended tanh-function method16, the He-Laplace algorithm17, the trial solution method18, the sine-Gordon expansion method19, the improved (G′/G) method20, the (G′/G)-expansion method21, the complete discrimination system for polynomial method22, the Exp-function method23.

The generalized Kudryashov’s equation with truncated M-fractional derivative is described as follows241.1 iDM,tμ,βu+aDM,x2μ,βu+(w1|u|4n+w2|u|3n+w3|u|2n+w4|u|n+w5|u|n+w6|u|2n+w7|u|3n+w8|u|4n)u=i[λDM,xμ,β(u|u|2m)+θuDM,xμ,β|u|2m+δ|u|2mDM,xμ,βu],0<μ<1,β>0,

where u=u(t,x) is a complex-valued wave profile. DM,tμ,β represents the truncated M-fractional derivative. a stands for the coefficient of chromatic dispersion. wi (1≤i≤8) are the coefficient of self-phase modulation. λ and θ represents the coefficient of self-steepening for short pules and the higher-order dispersion, respectively. m and n (0<n<12) stand for the maximum intensity and the power law non-linearity. The Kudryashov’s equation of integer order represented in25. The main purpose of this article is to study the generalized Kudryashov’s equation with truncated M-fractional derivative. By using the complete discrimination system for polynomial method, a series of explicit solutions to Eq. (1.1) will be obtained.

Definition 1.1

26,27 Let be f(t):[0,∞)→(-∞,+∞). Then, the truncated M-fractional derivative is given by1.2 DM,tμ,βf=limτ→∞f(tEβ(τt1-τ))τ,0<μ<1,β>0,

where Eβ(·) stands for the truncated Mittag-Leffler function. Eβ(z)=∑j=0izjΓ(βj+1), β>0 and z∈C. The properties of truncated M-fractional derivatives can be referred to in reference28.

This paper is organized as follows: In "Mathematical analysis", Eq. (1.1) is transformed into a nonlinear ordinary differential equation. In "Explicit solutions of equation ", the explicit solutions of Eq. (1.1) are obtained. In "Method description", a series of 2D, 3D, and contour plots are plotted. Finally, a brief summary is provided in "Explicit solutions of equation".

Mathematical analysis

Firstly, we consider the wave transformation2.1 u(t,x)=U(ξ)eiϑ(t,x),ξ=c(xμ-vtμ)Γ(β+1)μ,ϑ(t,x)=(-kxμ+wtμ+θ0)Γ(β+1)μ,

where k, w and θ0 represent the frequency, the wave number and the phase of the solion, respectively.

Plugging Eq. (2.1) into Eq. (1.1), we have the real part2.2 ac2U′′-(w-ak)U+w1U1-4n+w2U1-3n+w3U1-2n+w4U1-n+w5U1+n+w6U1+2n+w7U1+3n+w8U1+4n-k(λ-δ)U1+2m=0,

and imaginary part2.3 2ak+v+(2λm+δ+λ+2θm)=0.

From Eq. (2.3), we have2.4 2λm+δ+λ+2θm=0andv=-2ak.

Let be U=Y12n and m=2n. Then, we can obtain2.5 ac2[(1-2n)(Y′)2+2nYY′′]-4n2(ak2+ω)Y2+4n2ω6Y3+4n2ω3Y+4n2ω1+4n2[ω8-k(δ+λ)]Y4=0.

According to the principle of rank homogeneous balance, we assume that the approximate solution of Eq. (2.5) as2.6 Y′′=a0+a1Y+a2Y2+a3Y3.

Explicit solutions of equation (1.1)

The complete discriminant system method of polynomials is a very important method for constructing traveling wave solutions of nonlinear partial differential equations, which was first proposed by Professor Liu Chengshi29. In recent years, many experts and scholars have applied this method to the solution of traveling wave solutions for nonlinear partial differential equations, fractional order partial differential equations, and stochastic nonlinear partial differential equations.

Method description

Firstly, we present the following general fractional order partial differential equation3.1 P(u,u2,⋯,DM,tμ,β,DM,xμ,β,DM,t2μ,β,DM,x2μ,β,⋯)=0.

Applying the traveling wave transformation (2.1) to Eq. (3.1) yields3.2 P(U,U2,⋯,U′,U′′,⋯)=0.

If Eq. (3.2) can be transformed into the following ordinary differential equation,3.3 (U′)2=G(U,d1,d2,⋯,dm).

where d1,⋯,dm are parameters. Then, we have3.4 ξ-ξ0=±∫duG(U,d1,d2,⋯,dm).

Explicit solutions of equation (1.1)

Integrating both sides of Eq. (2.6) simultaneously once yields3.5 (Y′)2=12a3Y4+23a2Y3+a1Y2+2a0Y+d,

where d is the integral constant.

Inserting Eqs. (2.6) and (3.5) into Eq. (2.5), we can obtain a polynomial equation3.6 r4Y4+r3Y3+r2Y2+r1Y+r0=0,

where r4=a3ac2(12+n)+4n2[ω8-k(δ+Δ)], r3=23a2ac2(1+n)+4n2ω6, r2=a1ac2-4n2(ak2+ω), r1=2a0ac2(1-n)+4n2ω3, r0=dac2(1-2n)+4n2ω1.

In order to determine aj (j=0,...,3) and d, we assume that rj=0 (j=0,...,4), and then we obtain a3=8n2[k(δ+Δ)-ω8]ac2(1+2n), a2=-6n2ω6ac2(1+n), a1=4n2(ak2+ω)ac2, a0=-2n2ω3ac2(1-n), d=-4n2ω1ac2(1-2n).

When a3>0, we perform the following transformation3.7 W=(12a3)14(Y+a23a3),ξ1=(12a3)14ξ.

Inserting Eq. (3.7) into Eq. (3.5), we can obtain3.8 [Wξ1′]2=F(W)=W4+pW2+qW+r,

where p=a1-a223a312a3, q=(4a2327a32-2a1a23a3+2a0)(12a3)-14, r=-a2454a33+a1a229a32-2a0a23a3+d.

When a3<0, we perform the following transformation3.9 W=(-12a3)14(Y+a23a3)ξ1=(-12a3)14ξ.

Inserting Eq. (3.9) into Eq. (3.5), we can obtain3.10 [Wξ1′]2=-F(W)=-(W4+pW2+qW+r),

where p=-a1-a223a3-12a3, q=(-4a2327a32+2a1a23a3-2a0)(-12a3)-14, r=a2454a33-a1a229a32-2a0a23a3-d.

On the basis of the complete discriminant system for the quartic polynomial f(W)=W4+pW2+qW+r. it is presented as D1=4, D2=-p, D3=-2p3+8pr-9q2, D4=-p3q2+4p4r+36pq2r-32p2r2-274q4+64r3, E2=9p2-32pr.

From Eqs. (3.8) and (3.10), we can obtain3.11 [Wξ1′]2=ϵF(W)=ϵ(W4+pW2+qW+r),

where ϵ=±1.

Therefore, we can rewrite its integral expression as3.12 ±(ξ1-ξ)=∫dWϵ(W4+pW2+qW+r).

Based on the complete discriminant system for the polynomial F(W) as given in Eq. (3.12), the classification of all solutions o Eq. (3.12) can be obtained.

Case 1. D2<0,D3=0,D4=0. Here, F(W) has a pair of complex conjugate roots with multiplicity two, i.e. F(W)=[(W-l)2+s2]2, where l and s are real numbers, here s>0.

When ϵ=1, we can obtain from Eq. (3.12)3.13 ξ1-ξ0=∫dW(W-l)2+s2=1sarctanW-ls,

where ξ0 is the integral constant.

Therefore, the solutions to Eq. (1.1) are given byu1(t,x)=[(12a3)-14(stan(s((12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0))+l)-a23a3]eiϑ(t,x).

Case 2. D2=0,D3=0,D4=0. Here, F(W) has a quadruple real root at zero, i.e. F(W)=W4. Therefore, it follows that from Eq. (3.12) when a3>03.14 ξ1-ξ0=∫dWW2=-W-1.

Therefore, we can derive the solutions to Eq. (1.1)u2(t,x)=[-(12a3)-14((12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0)-1-a23a3]eiϑ(t,x),

which represents rational solutions.

Case 3. D2>0,D3=0,D4=0,E2>0. Obviously, F(W) has two distinct real roots with multiplicity two, i.e. F(W)=(W-α)2(W-β)2, where α and β are real numbers, here α>β.

If ϵ=1, we can obtain from Eq. (3.12) when W>α or W <β.3.15 ±(ξ1-ξ0)=∫dW(W-α)(W-β)=1α-βln|W-αW-β|.

The explicit solutions of (1.1) can be obtained from Eq. (3.12).u3(t,x)=[(12a3)-14[β-α2(coth(α-β)((12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0)2-1)+β]-a23a3]eiϑ(t,x).

When α>W>β, the explicit solutions of (1.1) can be obtained from Eq. (3.12)u4(t,x)=[(12a3)-14(β-α2(tanh(α-β)((12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0)2-1)+β)-a23a3]eiϑ(t,x).

Case 4. D2>0,D3>0,D4=0. Clearly, F(W) has a double real root and two distinct single real roots, i.e. F(W)=(W-β)(W-γ)(W+β+γ2)2, where β and γ are real numbers, here β>γ,γ≠-3β,γ≠-β3.

Denote θ=(3β+γ)(β+3γ), θ1=12(12a3)-14(β+γ)+a23a3, θ2=12(-12a3)-14(β+γ)+a23a3.

When a3>0, W>β or W<γ, if γ≤0andγ<β<-γ3orβ>-3γ or β>γ>0, the explicit solution of (1.1) can be obtainedu5(t,x)=[(12a3)-14θ2(β-γ)cosh[θ2(12a3)-14(c(xμ-vtμ)Γ(β+1)μ-ξ0)]+4(β+γ)-θ1]eiϑ(t,x).

If -3β<γ<-β3<0, the solution of Eq. (1.1) isu6(t,x)=[(12a3)-14θ4(β+γ)±2(β-γ)sin[-θ2(12a3)-14(c(xμ-vtμ)Γ(β+1)μ-ξ0)]-θ1]eiϑ(t,x),

when a3<0, β>W>γ.

If 0<-γ3<β<-3γ, the solution of Eq. (1.1) isu7(t,x)=[-(-12a3)-14θ2(β-γ)cosh[-θ2(-12a3)-14(c(xμ-vtμ)Γ(β+1)μ-ξ0)]-4(β+γ)-θ2]eiϑ(t,x).

If γ≤0andγ<β<-γ3orβ>-3γ or β>γ>0, the solution of Eq. (1.1) isu8(t,x)=[-(-12a3)-14θ±2(β-γ)sin[θ2(-12a3)-14(c(xμ-vtμ)Γ(β+1)μ-ξ0)]-4(β+γ)-θ2]eiϑ(t,x).

Case 5.D2>0,D3>0,D4>0. Obviously, F(W) has four real roots, i.e. F(W)=(W-β1)(W-β2)(W-β3)(W-β4), where β1,β2,β3,β4 are real numbers. β1>β2>β3>β4 and β1+β2+β3+β4=0.

Denote m1=(β2-β3)(β1-β4)(β1-β3)(β2-β4),m2=(β1-β2)(β3-β4)(β1-β3)(β2-β4),θ=12(β1-β3)(β2-β4). When a3>0, if W<β4 or W>β1, we make the following transformation:W=β2(β1-β4)sin2φ-β1(β2-β4)(β1-β4)sin2φ-(β2-β4).

If β3<W<β2, we make the following transformation:3.16 W=β4(β2-β3)sin2φ-β3(β2-β4)(β2-β3)sin2φ-(β2-β4).

From Eq. (3.12), we obtain3.17 ξ1-ξ0=∫dW(W-β1)(W-β2)(W-β3)(W-β4)=1θ∫dφ1-m12sin2φ.

By using Eq. (3.17) and the definition of the Jacobian elliptic sine function, we can obtainsn[12(β1-β3)(β2-β4)(ξ1-ξ0),m1]=sn(θξ1-ξ0,m1)=sinφ.

If W<β4, the solution of Eq. (1.1)u9(t,x)=[(β3-β1)β4+β3(β1-β4)sn2(θ(12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0,m1)(12a3)14(β3-β1+(β1-β4)sn2(θ(12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0,m1))-a23a3]eiϑ(t,x).

If W>β1, the solution of Eq. (1.1)u10(t,x)=[(β4-β2)β1+β2(β1-β4)sn2(θ(12a3)14ξ-ξ0,m1)(12a3)14(β4-β2+(β1-β4)sn2(θ(12a3)14ξ-ξ0,m1))-a23a3]eiϑ(t,x).

If β3<W<β2 the solution isu11(t,x)=[(β4-β2)β3+β4(β2-β3)sn2(θ(12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0,m1)(12a3)14(β4-β2+(β2-β3)sn2(θ(12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0,m1))-a23a3]eiϑ(t,x).

When a3<0, if β1>W>β2, we perform the following transformation:3.18 W=β3(β1-β2)sin2φ-β2(β1-β3)(β1-β2)sin2φ-(β1-β3).

If β4<W<β3, we perform the following transformation:W=β1(β3-β4)sin2φ-β4(β3-β1)(β3-β4)sin2φ-(β3-β1).

Then, the solution of Eq. (3.12)3.19 ξ1-ξ0=∫dW(W-β1)(W-β2)(W-β3)(W-β4)=1θ∫dφ1-m22sin2φ.

By using Eq. (3.19) and the definition of the Jacobian elliptic sine function, we can obtain:sn[12(β1-β3)(β2-β4)(ξ1-ξ0),m2]=sn(θξ1-ξ0,m2)=sinφ.

If β2<W<β1, the solution of Eq. (1.1)u12(t,x)=[(β4-β2)β1-β4(β1-β2)sn2(θ(-12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0,m2)(-12a3)14(β4-β2+(β2-β1)sn2(θ(-12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0,m2))-a23a3]eiϑ(t,x).

If β4<W<β3, the solution of Eq. (1.1)u13(t,x)=[(β1-β3)β4+β1(β3-β4)sn2(θ(-12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0,m2)(-12a3)14(β1-β3+(β3-β4)sn2(θ(-12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0,m2))-a23a3]eiϑ(t,x).

Case 6.D4<0,D2D3≥0. Evidently, F(W) has two distinct real roots and a pair of complex conjugate roots, i.e. F(W)=(W-β1)((W-β2)[(W+β1+β22)+β32], where β1,β2,β3 are real numbers. β1>β2 and β3≠0.

Denotep1=(3β1+β2)24+β32,p2=(β1+3β2)24+β32,m1=12(p1+p2)2-(β1-β2)2p1p2,m2=12-(p1-p2)2+(β1-β2)2p1p2,

when a3>0.

If W>β1 or W<β2, the solution of Eq. (1.1)u14(t,x)=[-β2p1+β1p2+(β2p1+β1p2)cn(p1p2(12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0,m1)(12a3)14[-p1+p2+(p1+p2)cn(p1p2(12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0,m1)]-a23a3]eiϑ(t,x).

If β1<W<β2, the solution of Eq. (1.1)u15(t,x)=[β2p1+β1p2+(β2p1-β1p2)cn(p1p2(-12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0,m2)(-12a3)14[p1+p2+(p1-p2)cn(p1p2(-12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0,m2)]-a23a3]eiϑ(t,x).

Case 7.D2>0,D3=D4=E2=0. Apparently, F(W) has one triple real root and one single real root, i.e. p<0, q=±-827p3, r=2p3+9q28p, then F(W)=(W-β)3((W+3β), where β2=-p6.

When a3>0, if W>max(β,-3β) or W<min(β,-3β), the solution of Eq. (1.1)u16(t,x)=[(12a3)14β3+4β2((12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0)2-1+4β2((12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0)2-a23a3]eiϑ(t,x).

When a3<0, if min(β,-3β)<W<max(β,-3β), the solution of Eq. (1.1)u17(t,x)=[(-12a3)14β-3+4β2((-12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0)21+4β2((-12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0)2-a23a3]eiϑ(t,x).

Case 8. D4=0,D2D3<0. Obviously, F(W) has one double real root and a pair of complex conjugate roots, i.e. F(W) = (W-α)2[(W-l)2+s2].

When ϵ=1, Eq. (3.12) can be rewritten as3.20 ±(ξ1-ξ0)=∫dW(W-α)(W-l)2+s2=1(α-l)2+s2ln|γW+δ-(W-l)2+s2W-α|,

where γ=α-2l(α-l)2+s2,δ=(α-l)2+s2-α(α-2l)(α-l)2+s2.

Correspondingly, the solution of Eq. (1.1)u18(t,x)=[(12a3)-14[e±(α-l)2+s2((12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0)-γ]+(α-l)2+s2(2-γ)(e±(α-l)2+s2[(12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0]-γ)2-1-a23a3]eiϑ(t,x).

This represents a solitary wave solution.

Case 9.D4>0,D2D3≤0. Obviously, F(W) has two pairs of complex conjugate roots, i.e.F(W) = [(W-l1)2+s12][(W-l2)2+s22], where l1,l2,s1,s2 are real numbers, here s1≥s2>0.

If ϵ=1, we make the following transformation:W=atanφ+bctanφ+d,

where a=l1c+s1d,b=l1d-s1c,c=-s1-s2m1,d=l1-l2,m1=E+E2-1,E=(l1-l2)2+s12+s222s1s2.

From Eq. (3.12), we can obtain:3.21 ξ1-ξ0=∫dW[(W-l1)2+s12][(W-l2)2+s22]=c2+d2s2(c2+d2)+(m12c2+d2)∫dφ1-m2sin2φ,

where m2=m12-1m12.

Derived from Eq. (3.21) and the definitions of the Jacobian sine and cosine functions, we obtain:sn[s2(c2+d2)+(m12c2+d2)c2+d2(ξ1-ξ0),m]=sinφ.cn[s2(c2+d2)+(m12c2+d2)c2+d2(ξ1-ξ0),m]=cosφ.

So, the solution of Eq. (1.1)u19(t,x)=[asn[η((12a3)14ξ-ξ0),m]+bcn(η((12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0),m)(12a3)14(csn(η((12a3)14c(xμ-vtμ)Γ(β+1)μ-ξ0),m)+dcn(η((12a3)14ξ-ξ0),m))-a23a3]eiϑ(t,x).

where η=s2(c2+d2)+(m12c2+d2)c2+d2

Graphical illustrations

In this section, we provided specific values for some parameters and plotted three-dimensional, two-dimensional, and contour maps of the obtained solutions of Eq. (1.1) shown as Figs. 1, 2, 3. The solution shown in Fig. 1 is a trigonometric function solution. Figure 2 shows a bell shaped solitary wave. Figure 3 shows a Jacobian function solution, which is a periodic function solution.Fig. 1 The explicit solution |u1(t,x)| of Eq. (1.1) when a3=2,a2=3,a1=72,a0=54,d=-38,c=v=1,ξ0=0.

Fig. 2 The explicit solution |u3(t,x)| of Eq. (1.1) when a3=2,a2=3,a1=-12,a0=-34,d=-116,c=v=1,ξ0=0.

Fig. 3 The explicit solution |u9(t,x)| of Eq. (1.1) when a3=2,a2=3,a1=-72,a0=94,d=-8316,c=v=1,ξ0=0.

Conclusion

In this paper, we successfully employed a complete discriminant system of fourth-order polynomials to solve the traveling wave solutions of the generalized Kudryashov’s equation with truncated M-fractional derivative and classified them. What’s more, we presented 2D and 3D plots of the solutions under different parameter values by using Mathematical software. Compared to previous studies, our work further enriches the traveling wave solutions of the generalized Kudryashov’s equation with truncated M-fractional derivative. In the future, we plan to explore higher-order and more complex fractional partial differential equations to address problems in a broader and more complex range of engineering and physical fields.

Author contributions

All authors contributed equally to this paper.

Data availability 

The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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