
Sperm morphology analysis (SMA) is a significant factor in diagnosing male infertility. Therefore, healthy sperm detection is of great significance in this process. However, the traditional manual microscopic sperm detection methods have the disadvantages of a long detection cycle, low detection accuracy in large orders, and very complex fertility prediction. Therefore, it is meaningful to apply computer image analysis technology to the field of fertility prediction. Computer image analysis can give high precision and high efficiency in detecting sperm cells. In this article, first, we analyze the existing sperm detection techniques in chronological order, from traditional image processing and machine learning to deep learning methods in segmentation and classification. Then, we analyze and summarize these existing methods and introduce some potential methods, including visual transformers. Finally, the future development direction and challenges of sperm cell detection are discussed. We have summarized 44 related technical papers from 2012 to the present. This review will help researchers have a more comprehensive understanding of the development process, research status, and future trends in the field of fertility prediction and provide a reference for researchers in other fields.
Citation: Muhammad Suleman, Muhammad Ilyas, M. Ikram Ullah Lali, Hafiz Tayyab Rauf, Seifedine Kadry. A review of different deep learning techniques for sperm fertility prediction[J]. AIMS Mathematics, 2023, 8(7): 16360-16416. doi: 10.3934/math.2023838
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Sperm morphology analysis (SMA) is a significant factor in diagnosing male infertility. Therefore, healthy sperm detection is of great significance in this process. However, the traditional manual microscopic sperm detection methods have the disadvantages of a long detection cycle, low detection accuracy in large orders, and very complex fertility prediction. Therefore, it is meaningful to apply computer image analysis technology to the field of fertility prediction. Computer image analysis can give high precision and high efficiency in detecting sperm cells. In this article, first, we analyze the existing sperm detection techniques in chronological order, from traditional image processing and machine learning to deep learning methods in segmentation and classification. Then, we analyze and summarize these existing methods and introduce some potential methods, including visual transformers. Finally, the future development direction and challenges of sperm cell detection are discussed. We have summarized 44 related technical papers from 2012 to the present. This review will help researchers have a more comprehensive understanding of the development process, research status, and future trends in the field of fertility prediction and provide a reference for researchers in other fields.
Plasma physics simply refers to the study of a state of matter consisting of charged particles. Plasmas are usually created by heating a gas until the electrons become detached from their parent atom or molecule. In addition, plasma can be generated artificially when a neutral gas is heated or subjected to a strong electromagnetic field. The presence of free charged particles makes plasma electrically conductive with the dynamics of individual particles and macroscopic plasma motion governed by collective electromagnetic fields [1].
Nonlinear partial differential equations (NPDE) in the fields of mathematics and physics play numerous important roles in theoretical sciences. They are the most fundamental models essential in studying nonlinear phenomena. Such phenomena occur in plasma physics, oceanography, aerospace industry, meteorology, nonlinear mechanics, biology, population ecology, fluid mechanics to mention a few. We have seen in [2] that the authors studied a generalized advection-diffusion equation which is a NPDE in fluid mechanics, characterizing the motion of buoyancy propelled plume in a bent-on absorptive medium. Moreover, in [3], a generalized Korteweg-de Vries-Zakharov-Kuznetsov equation was studied. This equation delineates mixtures of warm adiabatic fluid, hot isothermal as well as cold immobile background species applicable in fluid dynamics. Furthermore, the authors in [4] considered a NPDE where they explored important inclined magneto-hydrodynamic flow of an upper-convected Maxwell liquid through a leaky stretched plate. In addition, heat transfer phenomenon was studied with heat generation and absorption effect. The reader can access more examples of NPDEs in [5,6,7,8,9,10,11,12,13,14,15,16].
In order to really understand these physical phenomena it is of immense importance to solve NPDEs which govern these aforementioned phenomena. However, there is no general systematic theory that can be applied to NPDEs so that their analytic solutions can be obtained. Nevertheless, in recent times scientists have developed effective techniques to obtain viable analytical solutions to NPDEs, such as inverse scattering transform [16], simple equation method [17], Bäcklund transformation [18], F-expansion technique [19], extended simplest equation method[20], Hirota technique [21], Lie symmetry analysis [22,23,24,25,26,27], bifurcation technique [28,29], the (G′/G)-expansion method [30], Darboux transformation [31], sine-Gordon equation expansion technique [32], Kudryashov's method [33], and so on.
The (2+1)-dimensional Bogoyavlensky-Konopelchenko (BK) equation given as
utx+6αuxuxx+3βuxuxy+3βuyuxx+αuxxxx+βuxxxy=0, | (1.1) |
where parameters α and β are constants, is a special case of the KdV equation in [34] which was introduced as a (2+1)-dimensional version of the KdV and it is described as an interaction of a long wave propagation along x-axis and a Riemann wave propagation along the y-axis [35]. In addition to that, few particular properties of the equation have been explored. The authors in [36] provided a Darboux transformation for the BK equation and the obtained transformation was used to construct a family of solutions of this equation. In [37], with 3β replaced by 4β and uy=vx in (1.1), the authors integrated the result once to get
ut+αuxxx+βvxxx+3αu2x+4βuxvx=0,uy−vx=0. | (1.2) |
Further, they utilized Lie group theoretic approach to obtain solutions of the system of Eq (1.2). They also engaged the concept of nonlinear self-adjointness of differential equations in conjunction with formal Lagrangian of (1.2) for constructing nonlocal conservation laws of the system. In addition, various applications of BK equation (1.1) were highlighted in [37]. Further investigations on certain particular cases of (1.1) were also carried out in [38,39].
In [40], the 2D generalized BK equation that reads
utx+k1uxxxx+k2uxxxy+2k1k3k2uxuxx+k3(uxuy)x+γ1uxx+γ2uxy+γ3uyy=0 | (1.3) |
was studied and lump-type and lump solutions were constructed by invoking the Hirota bilinear method. Liu et al. [41] applied the Lie group analysis together with (G′/G)-expansion and power series methods and obtained some analytic solutions of (1.3).
Yang et al. [42] recently examined a generalized combined fourth-order soliton equation expressed as
α(6uxuxx+uxxxx)+β[3(uxut)x+uxxxt]+γ[3(uxuy)x+uxxxy]+δ1uyt+δ2uxx+δ3uxt+δ4uxy+δ5uyy+δ6utt=0, | (1.4) |
with constant parameters α,β and γ which are not all zero, whereas all constant coefficients δi,1≤i≤6, are arbitrary. It was observed that Eq (1.4) comprises three fourth-order terms and second-order terms that consequently generalizes the standard Kadomtsev-Petviashvili equation. Soliton equations are known to have applications in plasma physics and other nonlinear sciences such as fluid mechanics, atomic physics, biophysics, nonlinear optics, classical and quantum fields theories.
Assuming α=0, β=1, γ=0 and δ1=δ2=1, δ3=δ4=δ5=δ6=0, the authors gain an integrable (1+2)-dimensional extension of the Hirota-Satsuma equation commonly referred to as the Hirota-Satsuma-Ito equation in two dimensions[43] given as
uty+uxx+3(uxut)x+uxxxt=0 | (1.5) |
that satisfies the Hirota three-soliton condition and also admits a Hirota bilinear structure under logarithmic transformation presented in the form
u=2(lnf)x,where(D3xDt+DyDt+D2x)f⋅f=0, | (1.6) |
whose lump solutions have been calculated in [44]. On taking parameters α=1, β=0, γ=0 along with δ1=δ4=δ6=0 whereas δ2=δ3=δ5=1, they eventually came up with a two dimensional equation[42]:
utx+6uxuxx+uxxxx+uxxxy+3(uxuy)x+uxx+uyy=0, | (1.7) |
which is called a two-dimensional generalized Bogoyavlensky-Konopelchenko (2D-gBK) equation. We notice that if one takes α=β=1 in Eq (1.1) with the introduction of two new terms uxx and uyy, the new generalized version (1.7) is achieved.
In consequence, we investigate explicit solutions of the new two-dimensional generalized Bogoyavlensky-Konopelchenko equation (1.7) of plasma physics in this study. In order to achieve that, we present the paper in the subsequent format. In Section 2, we employ Lie symmetry analysis to carry out the symmetry reductions of the equation. In addition, direct integration method will be employed in order to gain some analytic solutions of the equation by solving the resulting ordinary differential equations (ODEs) from the reduction process. We achieve more analytic solutions of (1.7) via the conventional (G′/G)-expansion method as well as Kudryashov's technique. In addition, by choosing suitable parametric values, we depict the dynamics of the solutions via 3-D, 2-D as well as contour plots. Section 3 presents the conservation laws for 2D-gBK equation (1.7) through the multiplier method and in Section 4, we give the concluding remarks.
In this section we in the first place compute the Lie point symmetries of Eq (1.7) and thereafter engage them to generate analytic solutions.
A one-parameter Lie group of symmetry transformations associated with the infinitesimal generators related to (\real{gbk}) can be presented as
ˉt=t+ϵξ1(t,x,y,u)+O(ϵ2),ˉx=x+ϵξ2(t,x,y,u)+O(ϵ2),ˉy=y+ϵξ3(t,x,y,u)+O(ϵ2),ˉu=u+ϵϕ(t,x,y,u)+O(ϵ2). | (2.1) |
We calculate symmetry group of 2D-gBK equation (1.7) using the vector field
R=ξ1(t,x,y,u)∂∂t+ξ2(t,x,y,u)∂∂x+ξ3(t,x,y,u)∂∂y+ϕ(t,x,y,u)∂∂u, | (2.2) |
where ξi,i=1,2,3 and ϕ are functions depending on t, x, y and u. We recall that (2.2) is a Lie point symmetry of Eq (1.7) if
R[4](utx+6uxuxx+uxxxx+uxxxy+3(uxuy)x+uxx+uyy)|Q=0=0, | (2.3) |
where Q=utx+6uxuxx+uxxxx+uxxxy+3(uxuy)x+uxx+uyy. Here, R[4] denotes the fourth prolongation of R defined by
R[4]=R+ηt∂ut+ηx∂ux+ηy∂uy+ηtx∂utx+ηxx∂uxx+ηyy∂uyy+ηxxxx∂uxxxx+ηxxxy∂uxxxy, | (2.4) |
where coefficient functions ηt, ηx, ηy, ηxt, ηxx, ηxy, ηyy, ηxxxx and ηxxxy can be calculated from [22,23,24].
Writing out the expanded form of the determining equation (2.3), splitting over various derivatives of u and with the help of Mathematica, we achieve the system of linear partial differential equations (PDEs):
ξ3x=0,ξ1x=0,ξ1y=0,ξ2u=0,ξ1u=0,ξ3u=0,ξ1tt+5ξ2yy=0,ξ1t+5ϕu=0,5ξ2x−ξ1t=0,5ξ2y−2ξ1t=0,5ξ3y−3ξ1t=0,3ϕxx−ξ3yy=0,ξ3t−3ϕx+2ξ2y=0,4ξ1t−5ξ2t+30ϕx+15ϕy=0,ϕtx+ϕxx+ϕxxxx+ϕxxxy+ϕyy=0. |
The solution of the above system of PDEs is
ξ1=A1+A2t,ξ2=F(t)+15A2(x+2y),ξ3=A4−45A2t+3A3t+35A2y,η=G(t)−15A2u+A3x−415A2y−2A3y+13yF′(t), |
where A1–A3 are arbitrary constants and F(t), G(t) are arbitrary functions of t. Consequently, we secure the Lie point symmetries of (1.7) given as
R1=∂∂t,R2=∂∂y,R3=3F(t)∂∂x+yF′(t)∂∂u,R4=3t∂∂y+(x−2y)∂∂u,R5=G(t)∂∂u,R6=15t∂∂t+(3x+6y)∂∂x+(9y−12t)∂∂y−(4y+3u)∂∂u. | (2.5) |
We contemplate the exponentiation of the vector fields (2.5) by computing the flow or one parameter group generated by (2.5) via the Lie equations [22,23]:
dˉtdϵ=ξ1(ˉt,ˉx,ˉy,ˉu),ˉt|ϵ=0=t,dˉxdϵ=ξ2(ˉt,ˉx,ˉy,ˉu),ˉx|ϵ=0=x,dˉydϵ=ξ3(ˉt,ˉx,ˉy,ˉu),ˉy|ϵ=0=y,dˉudϵ=ϕ(ˉt,ˉx,ˉy,ˉu),ˉu|ϵ=0=u. |
Therefore, by taking F(t)=G(t)=t in (2.5), one computes a one parameter transformation group of 2D-gBK (1.7). Thus, we present the result in the subsequent theorem.
Theorem 2.1. Let Tiϵ(t,x,y,u),i=1,2,3,…,6 be transformations group of one parameter generated by vectors R1,R2,R3…,R6 in (2.5), then, for each of the vectors, we have accordingly
T1ϵ:(˜t,˜x,˜y,˜u)⟶(t+ϵ1,x,y,u),T2ϵ:(˜t,˜x,˜y,˜u)⟶(t,x,y+ϵ2,u),T3ϵ:(˜t,˜x,˜y,˜u)⟶(t,3ϵ3t+x,y,ϵ3y+u),T4ϵ:(˜t,˜x,˜y,˜u)⟶(t,x,3ϵ4t+y,u+(x−2y)ϵ4−3ϵ24t),T5ϵ:(˜t,˜x,˜y,˜u)⟶(t,x,y,ϵ5t+u),T6ϵ:˜t,˜x,˜y,˜u)⟶(te15ϵ6,(2e9ϵ6−e3ϵ6−e15ϵ6)t+xe3ϵ6+(e9ϵ6−e3ϵ6)y,(2e9ϵ6−2e15ϵ6)t+ye9ϵ6,19[(4e18ϵ6−6e12ϵ6+2)t+(3−3e12ϵ6)y+9u]e−3ϵ6), |
where ϵ∈R is regarded as the group parameter.
Theorem 2.2. Hence, suppose u(t,x,y)=Θ(t,x,y) satisfies the 2D-gBK (1.7), in the same vein, the functions given in the structure
u1(t,x,y)=Θ(t−ϵ1,x,y,z),u2(t,x,y)=Θ(t,x,y−ϵ2,z,u),u3(t,x,y)=Θ(t,x−3ϵ3t,y)−ϵ3y,u4(t,x,y)=Θ(t,x,3ϵ4t+y)−(x−2y)ϵ4+3ϵ24t,u5(t,x,y)=Θ(t,x,y)−ϵ5t,u6(t,x,y)=Θ[te15ϵ6,(2e9ϵ6−e3ϵ6−e15ϵ6)t+xe3ϵ6+(e9ϵ6−e3ϵ6)y,(2e9ϵ6−2e15ϵ6)t+ye9ϵ6]−23te12ϵ6+49te18ϵ6+29t−13ye12ϵ6+13y |
will do, where ui(t,x,y)=Tϵi⋅Θ(t,x,y),i=1,2,3,…,6 with ϵ<<1 regarded as any positive real number.
In this subsection, we utilize symmetries (2.5) with a view to reduce Eq (1.7) to ordinary differential equations and thereafter obtain the analytic solutions of Eq (1.7) by solving the respective ODEs.
Case 1. Invariant solutions via R1–R3
Taking F(t)=1/3, we linearly combine translational symmetries R1–R3 as R=bR1+cR2+aR3 with nonzero constant parameters a, b and c. Subsequently utilizing the combination reduces 2D-gBK equation (1.7) to a PDE with two independent variables. Thus, solution to the characteristic equation associated with the symmetry R leaves us with invariants
r=ct−ay,s=cx−by,θ=u. | (2.6) |
Now treating θ above as the new dependent variable as well as r, s as new independent variables, (1.7) then transforms into the PDE:
c2θrs+6c3θsθss+c4θssss−3ac2θsθsr−6bc2θsθss−3ac2θrθss−ac3θsssr−bc3θssss+c2θss+a2θrr+2abθsr+b2θss=0. | (2.7) |
We now utilize the Lie point symmetries of (2.7) in a bid to transform it to an ODE. From (2.7), we achieve three translation symmetries:
Q1=∂∂r,Q2=∂∂s,Q3=∂∂θ. |
The linear combination Q=Q1+ωQ2 (ω≠0 being an arbitrary constant) leads to two invariants:
z=s−ωr,θ=Θ, | (2.8) |
that secures group-invariant solution Θ=Θ(z). Thus, on using these invariants, (2.7) is transformed into the fourth-order nonlinear ODE:
(c2−ωc2+a2ω2−2baω+b2)Θ″(z)−6(βbc2−βc2aω−c3)Θ′(z)Θ″(z)+(c3aω+c4−bc3)Θ⁗(z)=0, |
which we rewrite in a simple structure as
AΘ″(z)−BΘ′(z)Θ″(z)+CΘ⁗(z)=0, | (2.9) |
where A=c2−ωc2+a2ω2−2baω+b2, B=6(bc2−c2aω−c3), C=c3aω+c4−bc3 and z=cx+(aω−b)y−cωt.
In this section, we seek travelling wave solutions of the 2D-gBK equation (1.7).
A. Elliptic function solution of (1.7)
On integrating equation (2.9) once, we accomplish a third-order ODE:
AΘ′(z)−12BΘ′2(z)+CΘ‴(z)+C1=0, | (2.10) |
where C1 is a constant of integration. Multiplying Eq (2.10) by Θ″(z), integrating once and simplifying the resulting equation, we have the second-order nonlinear ODE:
12AΘ′(z)2−16BΘ′(z)3+12CΘ″(z)2+C1Θ′(z)+C2=0, |
where C2 is a constant of integration. The above equation can be rewritten as
Θ″(z)2=B3CΘ′(z)3−ACΘ′(z)2−2C1CΘ′(z)−2C2C. | (2.11) |
Letting U(z)=Θ′(z), Eq (2.11) becomes
U′(z)2=B3CU(z)3−ACU(z)2−2C1CU(z)−2C2C. | (2.12) |
Suppose that the cubic equation
U(z)3−3ABU(z)2−6C1BU(z)−6C2B=0 | (2.13) |
has real roots c1–c3 such that c1>c2>c3, then Eq (2.12) can be written as
U′(z)2=B3C(U(z)−c1)(U(z)−c2)(U(z)−c3), | (2.14) |
whose solution with regards to Jacobi elliptic function [45,46] is
U(z)=c2+(c1−c2)cn2{√B(c1−c2)12Cz,Δ2},Δ2=c1−c2c1−c3, | (2.15) |
with (cn) being the elliptic cosine function. Integration of (2.15) and reverting to the original variables secures a solution of 2D-gBK equation (1.7) as
u(t,x,y)=√12C(c1−c2)2B(c1−c3)Δ8{EllipticE[sn(B(c1−c3)12Cz,Δ2),Δ2]}+{c2−(c1−c2)1−Δ4Δ4}z+C3, | (2.16) |
with z=cx+(aω−b)y−cωt and C3 a constant of integration. We note that (2.16) is a general solution of (1.7), where EllipticE[p;q] is the incomplete elliptic integral [46,47] expressed as
EllipticE[p;q]=∫p0√1−q2r21−r2dr. |
We present wave profile of periodic solution (2.16) in Figure 1 with 3D, contour and 2D plots with parametric values a=−4, b=0.2, c=−0.1, ω=0.1, c1=100, c2=50.05, c3=−60, B=10, C=70, where t=1 and −10≤x,y≤10.
However, contemplating a special case of (2.9) with B=0, we integrate the equation twice and so we have
CΘ″(z)+AΘ(z)+K1z+K2=0, | (2.17) |
where K1 and K2 are integration constants. Solving the second-order linear ODE (2.17) and reverting to the basic variables, we achieve the trigonometric solution of 2D-gBK equation (1.7) as
u(t,x,y)=A1sin(√a2ω2−ω(2ab+c2)+b2+c2z√c3(aω−b+c))+A2cos(√a2ω2−ω(2ab+c2)+b2+c2z√c3(aω−b+c))−K1z+K2a2ω2−ω(2ab+c2)+b2+c2, | (2.18) |
with A1 and A2 as the integration constants as well as z=cx+(aω−b)y−cωt. We depict the wave dynamics of periodic solution (2.18) in Figure 2 via 3D, contour and 2D plots with dissimilar parametric values a=1, b=0.2, c=−0.1, ω=0.1, A1=20, A2=−2, K1=1, K2=10, where t=2 and −10≤x,y≤10.
B. Weierstrass elliptic solution of 2D-gBK equation (1.7)
We further explore Weierstrass elliptic function solution of (1.7), which is a technique often involved in getting general exact solutions to NPDEs [47,48]. In order to accomplish this, we use the transformation
U(z)=W(z)+AB | (2.19) |
and transform the nonlinear ordinary differential equation (NODE) (2.12) to
W2ξ=4W3−g2W−g3,ξ=√B12Cz, | (2.20) |
with the invariants g2 and g3 given by
g2=−12A2B2−24C1Bandg3=−8A3B3−24AC1B2−24C2B. |
Thus, we have the solution of NODE (2.12) as
U(z)=AB+℘(√112C(z−z0);g2;g3), | (2.21) |
where ℘ denotes the Weierstrass elliptic function [46]. In consequence, integration of (2.21) and reverting to the basic variables gives the solution of 2D-gBK equation (1.7) as
u(t,x,y)=AB(z−z0)−√12BCζ[√B12C(z−z0);g2,g3], | (2.22) |
with arbitrary constant z0, z=cx+(aω−b)y−cωt and ζ being the Weierstrass zeta function [46]. We give wave profile of Weierstrass function solution (2.22) in Figure 3 with 3D, contour and 2D plots using parameter values a=1, b=0.2, c=−0.1, ω=0.1, A=10, B=−2, z0=0, C=1, C1=1, C2=10, where t=2 and −10≤x,y≤10.
This part of the study furnishes the solution of 2D-gBK equation (1.7) through the use of Kudryashov's approach [33]. This technique is one of the most prominent way to obtain closed-form solutions of NPDEs. Having reduced Eq (1.7) to the NODE (2.9), we assume the solution of (2.9) as
Θ(z)=N∑n=0BnQn(z), | (2.23) |
with Q(z) satisfying the first-order NODE
Q′(z)=Q2(z)−Q(z). | (2.24) |
We observe that the solution of (2.24) is
Q(z)=11+exp(z). | (2.25) |
The balancing procedure for NODE (2.9) produces N=1. Hence, from (2.23), we have
Θ(z)=B0+B1Q(z). | (2.26) |
Now substituting (2.26) into (2.9) and using (2.24), we gain a long determining equation and splitting on powers of Q(z), we get algebraic equations for the coefficients B0 and B1 as
Q(z)5:2aB1c3ω+aB21c2ω−2bB1c3−bB21c2+2B1c4+B21c3=0,Q(z)4:2bB1c3−2aB1c3ω−aB21c2ω+bB21c2−2B1c4−B21c3=0,Q(z)3:a2B1ω2−2abB1ω+25aB1c3ω+12aB21c2ω+b2B1−25bB1c3−12bB21c2+25B1c4+12B21c3−B1c2ω+B1c2=0,Q(z)2:2abB1ω−a2B1ω2−5aB1c3ω−2aB21c2ω−b2B1+5bB1c3+2bB21c2−5B1c4−2B21c3+B1c2ω−B1c2=0,Q(z):a2B1ω2−2abB1ω+aB1c3ω+b2B1−bB1c3+B1c4−B1c2ω+B1c2=0. | (2.27) |
The solution of the above system gives
B0=0,B1=−2c,a=2bω−c3ω∓√c2ω2(c4−4c2+4ω−4)2ω2. | (2.28) |
Hence, the solution of 2D-gBK equation (1.7) associated with (2.28) is given as
u(t,x,y)=−2c1+exp{cx+(aω−b)y−cωt}. | (2.29) |
The wave profile of solution (2.29) is shown in Figure 4 with 3D, contour and 2D plots using parameter values a=1, b=−0.2, c=20, ω=0.05, B0=0 with t=7 and −6≤x,y≤6.
We reckon the (G′/G)-expansion technique [30] in the construction of analytic solutions of 2D-gBK equation (1.7) and so we contemplate a solution structured as
Θ(z)=M∑j=0Bj(Q′(z)Q(z))j, | (2.30) |
where Q(z) satisfies
Q″(z)+λQ′(z)+μQ(z)=0 | (2.31) |
with λ and μ taken as constants. Here, B0,…,BM are parameters to be determined. Utilization of balancing procedure for (2.9) produces M=1 and as a result, the solution of (1.7) assumes the form
Θ(z)=B0+B1(Q′(z)Q(z)). | (2.32) |
Substituting the value of Θ(z) from (2.32) into (2.9) and using (2.31) and following the steps earlier adopted, leads to an algebraic equation in B0 and B1, which splits over various powers of Q(z) to give the system of algebraic equations whose solution is secured as
B0=0,B1=2c,a=16bω−B31λ2ω±√Ω0+64B21ω3−64B21ω2+4B31μω16ω2, |
where Ω0=B61λ4ω2−8B61λ2μω2−16B41λ2ω2+16B61μ2ω2+64B41μω2. Thus, we have three types of solutions of the 2D-gBK equation (1.7) given as follows:
When λ2−4μ>0, we gain the hyperbolic function solution
u(t,x,y)=B0+B1(Δ1A1sinh(Δ1z)+A2cosh(Δ1z)A1cosh(Δ1z)+A2sinh(Δ1z)−λ2), | (2.33) |
with z=cx+(aω−b)y−cωt, Δ1=12√λ2−4μ together with A1, A2 being arbitrary constants. The wave profile of solution (2.33) is shown in Figure 5 with 3D, contour and 2D plots using parameter values a=3, b=0.5, c=10, ω=−0.1, B0=0, λ=−0.971, μ=10, A1=5, A2=1, where t=10 and −10≤x,y≤10.
When λ2−4μ<0, we achieve the trigonometric function solution
u(t,x,y)=B0+B1(Δ2A2cos(Δ2z)−A1sin(Δ2z)A1cos(Δ2z)+A2sin(Δ2z)−λ2), | (2.34) |
with z=cx+(aω−b)y−cωt, Δ2=12√4μ−λ2 together with A1 and A2 are arbitrary constants. The wave profile of solution (2.34) is shown in Figure 6 with 3D, contour and 2D plots using parameter values a=1, b=0.5, c=0.3, ω=0.3, B0=0, λ=−0.971, μ=2, A1=5, A2=1 with t=10 and −10≤x,y≤10.
When λ2−4μ=0, we gain the rational function solution
u(t,x,y)=B0+B1(A2A1+A2z−λ2), | (2.35) |
with z=cx+(aω−b)y−cωt and A1, A2 being arbitrary constants. We plot the graph of solution (2.35) in Figure 7 via 3D, contour and 2D plots using parametric values a=1, b=1.01, c=100, ω=0.1, B0=10, λ=10, A1=3, A2=10, where t=2.4 and −5≤x,y≤5.
Case 2. Group-invariant solutions via R4
Lagrange system associated with the symmetry R4=3t∂/∂y+(x−2y)∂/∂u is
dt0=dx0=dy3t=du(x−2y), | (2.36) |
which leads to the three invariants T=t,X=x,Q=u+(y2/3t)−(xy/3t). Using these three invariants, the 2D-gBK equation (1.7) is reduced to
18TQXQXX+3TQTX+3TQXX+3XQXX+3QX+3TQXXXX−2=0. | (2.37) |
Case 3. Group-invariant solutions via R1,R2andR5
We take G(t)=1 and by combining the generators R1,R2 as well as R5, we solve the characteristic equations corresponding to the combination and get the invariants X=x, Y=y−t with group-invariant u=Q(X,Y)+t. With these invariants, the 2D-gBK equation (1.7) transforms to the NPDE
QXX+QYY−QXY+3QXQXY+3QYQXX+6QXQXX+QXXXX+QXXXY=0, | (2.38) |
whose solution is given by
Q(X,Y)=2A2tanh[A2X+A2(12−12√16A42−24A22−3−2A22)Y+A1]+A3, | (2.39) |
with arbitrary constants A1–A3. Thus, we achieve the hyperbolic solution of (1.7) as
u(t,x,y)=t+2A2tanh[12A2(t−y)√16A42−24A22−3+12(4t−4y)A32+12(y+2x−t)A2+A1]+A3. | (2.40) |
The wave profile of solution (2.40) is shown in Figure 8 with 3D, contour and 2D plots using parameter values A1=70.1, A2=−30, A3=0, where t=0.5 and −10≤x,y≤10.
Besides, symmetries of (2.38) are found as
P1=∂∂X,P2=∂∂Y,P3=∂∂Q,P4=(13X+23Y)∂∂X+Y∂∂Y+(23X−2Y−13Q)∂∂Q. |
Now, the symmetry P1 furnishes the solution Q(X,Y)=f(z), z=Y. So, Eq (2.38) gives the ODE f″(z)=0. Hence, we have a solution of (1.7) as
u(t,x,y)=t+A0(y−t)+A1, | (2.41) |
with A0, A1 as constants. Further, the symmetry P2 yields Q(X,Y)=f(z), z=X and so Eq (2.38) reduces to
f″(z)+6f′(z)f″(z)+f⁗(z)=0. | (2.42) |
Integration of the above equation three times with respect to z gives
f′(z)2+2f(z)3+f(z)2+2A0f(z)+2A1=0, | (2.43) |
and taking constants A0=A1=0 and then integrating it results in the solution of (1.7) as
u(t,x,y)=t−12{1+tan(12A1−12x)2}. | (2.44) |
The wave profile of solution (2.44) is shown in Figure 9 with 3D, contour and 2D plots using parameter values A1=40, t=3.5 and −10≤x≤10.
On combining P1–P3 as P=c0P1+c1P2+c2P3, we accomplish
Q(X,Y)=c2c0X+f(z),wherez=c0Y−c1X. | (2.45) |
Using the newly acquired invariants (2.45), Eq (2.38) transforms to the NODE:
c0c21f″(z)+6c21c2f″(z)−3c0c1c2f″(z)+c20c1f″(z)+c30f″(z)+6c20c21f′(z)f″(z)−6c0c31f′(z)f″(z)+c0c41f⁗(z)−c20c31f⁗(z)=0. | (2.46) |
Engaging the Lie point symmetry P4, we obtain
Q(X,Y)=X−2Y+Y−1/3f(z)withz=Y−1/3(X−Y), | (2.47) |
and Eq (2.38) reduces to the NODE
6zf′(z)+z2f″(z)−18f′(z)2−9f(z)f″(z)+4f(z)−18zf′(z)f″(z)−12f‴(z)−3zf⁗(z)=0. | (2.48) |
Case 4. Group-invariant solutions via R6
Lie point symmetry R6 dissociates to the Lagrange system
dt15t=dx3x+6y=dy9y−12t=du−(4y+3u), |
which gives
u=t−1/5Q(T,X)−29t−13y,withT=(2t+y)t−3/5andX=(x−t−y)t−1/5. | (2.49) |
Substituting the expression of u in (1.7), we obtain the NPDE
5QTT−3TQTX−XQXX−2QX+15QXQTX+15QTQXX+5QTXXX=0, | (2.50) |
which has two symmetries:
P1=∂∂Q,P2=∂∂X+115T∂∂Q. |
The symmetry P2 gives Q(X,Y)=f(z)+(1/15)TX, z=T and hence (2.50) reduces to
75f″(z)−4z=0. |
Solving the above ODE and reverting to the basic variables gives the solution of (1.7) as
u(t,x,y)=15√t{(2t+y)(x−t−y)15t4/5+2(2t+y)3225t9/5+(2t+y)t3/5A1+A2}−2t9−y3, | (2.51) |
where A1 and A2 are integration constants. The wave profile of solution (2.51) is shown in Figure 10 with 3D, contour and 2D plots using parameter values A1=−0.3, A2=−50 with t=1.1 and −10≤x,y≤10.
Next, we invoke the symmetry P1+P2. This yields Q(X,Y)=f(z)+X+(1/15)TX,z=T. Consequently, we have the transformed version of (2.50) as
75f″(z)−4z−15=0. |
Solving the above ODE and reverting to basic variables gives the solution of (1.7) as
u(t,x,y)=15√t{(2t+y)(x−t−y)15t4/5+2(2t+y)3225t9/5+(2t+y)210t6/5+(2t+y)t3/5A1+A2}−2t9−y3. | (2.52) |
The wave profile of solution (2.52) is shown in Figure 11 with 3D, contour and 2D plots using parameter values A1=−3.6, A2=50 with t=1.1 and −10≤x,y≤10.
In this section, we construct the conservation laws for 2D-gBK equation (1.7) by making use of the multiplier approach [22,49,50], but first we give some basic background of the method that we are adopting.
Consider the n independent variables x=(x1,x2,…,xn) and m dependent variables u=(u1,u2,…,um). The derivatives of u with respect to x are defined as
uαi=Di(uα),uαij=DjDi(ui),…, | (3.1) |
where
Di=∂∂xi+uαi∂∂uα+uαij∂∂uαj+⋯,i=1,...,n, | (3.2) |
is the operator of total differentiation. The collection of all first derivatives uαi is denoted by u(1), i.e., u(1)={uαi},α=1,...,m,i=1,...,n. In the same vein u(2)={uαij},α=1,...,m,i,j=1,...,n and u(3)={uαijk} and likewise u(4) etc. Since uαij=uαji, u(2) contains only uαij for i≤j.
Now consider a kth-order system of PDEs:
Gα(x,u,u(1),…,u(k))=0,α=1,2,…,m. | (3.3) |
The {Euler-Lagrange operator}, for every α, is presented as
δδuα=∂∂uα+∑s≥1(−1)sDi1…Dis∂∂uαi1i2…is,α=1,…,m. | (3.4) |
An n-tuple T=(T1,T2,…,Tn), such that
DiTi=0 | (3.5) |
holds for all solutions of (3.3) is known as the conserved vector of system (3.3).
The multiplier Ωα(x,u,u(1),…) of system (3.3) has the property that
DiTi=ΩαGα | (3.6) |
holds identically [22]. The determining equations for multipliers are obtained by taking the variational derivative of (3.6), namely
δδuα(ΩαGα)=0. | (3.7) |
The moment multipliers are generated from (3.7), the conserved vectors can be derived systematically using (3.6) as the determining equation.
Conservation laws of 2D-gBK equation (1.7) are derived by utilizing second-order multiplier Ω(t,x,y,u,ut,ux,uy,uxx,uxy), in Eq (3.7), where G is given as
G≡utx+6uxuxx+uxxxx+uxxxy+3(uxuy)x+uxx+uyy, |
and the Euler operator δ/δu is expressed in this case as\newpage
δδu=∂∂u−Dt∂∂ut−Dx∂∂ux−Dy∂∂uy+DtDx∂∂utx+DxDy∂∂uxy+D2x∂∂uxx+D2y∂∂uyy+D4x∂∂uxxxx+D3xDy∂∂uxxxy. |
Expansion of Eq (3.7) and splitting on diverse derivatives of dependent variable u gives
Ωu=0,Ωx=0,Ωyy=0,Ωyux=0,Ωuxux=0,Ωtux=0,Ωut=0,Ωuxx=0,Ωuxy=0,Ωuy=0. | (3.8) |
Solution to the above system of equations gives Ω(t,x,y,u,ut,ux,uy,uxx,uxy) as
Ω(t,x,y,u,ut,ux,uy,uxx,uxy)=C1ux+f1(t)y+f2(t), | (3.9) |
with C1 being an arbitrary constant and f1(t), f2(t) being arbitrary functions of t. Using Eq (3.6), one obtains the following three conserved vectors of Eq (1.7) that correspond to the three multipliers ux,f1(t) and f2(t):
Case 1. For the first multiplier Q1=ux, the corresponding conserved vector (Tt1,Tx1,Ty1) is given by
Tt1=12u2x,Tx1=12u2x+2u3x−12u2xx−12uxxuxy+12uxuxxy+uxxxux+12uxxxuy+12uuxxxy+12uuyy+uuxuxy+2uyu2x,Ty1=12uyux−uuxuxx−12uuxy−12uuxxxx. |
Case 2. For the second multiplier Q2=f1(t), we obtain the corresponding conserved vector (Tt2,Tx2,Ty2) as
Tt2=uxf1(t)y,Tx2=3yf1(t)u2x+3yf1(t)uxuy−yf′1(t)u+yf1(t)ux+yf1(t)uxxx+yf1(t)uxxy,Ty2=uyf1(t)y−uf1(t). |
Case 3. Finally, for the third multiplier Q3=f2(t), the corresponding conserved vector (Tt3,Tx3,Ty3) is
Tt3=uxf2(t),Tx3=3u2xf2(t)+3uxuyf2(t)−uf′2(t)+uxf2(t)+uxxxf2(t)+uxxyf2(t),Ty3=uyf2(t). |
Remark 3.1. We notice that this method assists in the construction of conservation laws of (1.7) despite the fact that it possesses no variational principle [51]. Moreover, the presence of arbitrary functions in the multiplier indicates that 2D-gBK (1.7) has infinite number of conserved vectors.
In this paper, we carried out a study on two-dimensional generalized Bogoyavlensky-Konopelchenko equation (1.7). We obtained solutions for Eq (1.7) with the use of Lie symmetry reductions, direct integration, Kudryashov's and (G′/G)-expansion techniques. We obtained solutions of (1.7) in the form of algebraic, rational, periodic, hyperbolic as well as trigonometric functions. Furthermore, we derived conservation laws of (1.7) by engaging the multiplier method were we obtained three local conserved vectors. We note here that the presence of the arbitrary functions f1(t) and f2(t) in the multipliers, tells us that one can generate unlimited number of conservation laws for the underlying equation.
The authors state no conflicts of interest.
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