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Research article

Analysis of optical solitons solutions of two nonlinear models using analytical technique

  • Received: 11 May 2021 Accepted: 23 August 2021 Published: 15 September 2021
  • MSC : 35Q51, 35Q53

  • Looking for the exact solutions in the form of optical solitons of nonlinear partial differential equations has become very famous to analyze the core structures of physical phenomena. In this paper, we have constructed some various type of optical solitons solutions for the Kaup-Newell equation (KNE) and Biswas-Arshad equation (BAE) via the generalized Kudryashov method (GKM). The conquered solutions help to understand the dynamic behavior of different physical phenomena. These solutions are specific, novel, correct and may be beneficial for edifying precise nonlinear physical phenomena in nonlinear dynamical schemes. Graphical recreations for some of the acquired solutions are offered.

    Citation: Naeem Ullah, Muhammad Imran Asjad, Azhar Iqbal, Hamood Ur Rehman, Ahmad Hassan, Tuan Nguyen Gia. Analysis of optical solitons solutions of two nonlinear models using analytical technique[J]. AIMS Mathematics, 2021, 6(12): 13258-13271. doi: 10.3934/math.2021767

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  • Looking for the exact solutions in the form of optical solitons of nonlinear partial differential equations has become very famous to analyze the core structures of physical phenomena. In this paper, we have constructed some various type of optical solitons solutions for the Kaup-Newell equation (KNE) and Biswas-Arshad equation (BAE) via the generalized Kudryashov method (GKM). The conquered solutions help to understand the dynamic behavior of different physical phenomena. These solutions are specific, novel, correct and may be beneficial for edifying precise nonlinear physical phenomena in nonlinear dynamical schemes. Graphical recreations for some of the acquired solutions are offered.



    In 1982, Hamilton first introduced the Ricci soliton (RS) concept with [1], and Hamilton pointed out the RS serves as a self-similar outcome to the Ricci flow as long as it walks through a one single parameter family with modulo diffeomorphic mappings and grows on a space about Riemannian metrics in [2]. Since then, geometers and physicists turned their attention to discuss RS. For example, in [3], Rovenski found conditions for the existence of an Einstein manifold according to a similar Ricci tensor or generalized RS form in an extremely weak κ-contact manifold. In [4], Arfah presented a condition for RS on semi-Riemannian group manifold and illustrated the applications of group manifold that admit RS. There are some typical works on affine RS [5], algebraic RS [6], as well as generalized RS [7]. Nonetheless, a task about seeking out RS on manifolds is considerably challenging and often necessitates the imposition of limitations. These restrictions can typically be observed in several areas, such as the framework and dimensions of the manifold, the classification of metrics or the classification about vector fields used in the RS equation. An example of this is the utilization of homogeneous spaces, particularly Lie groups (LG) [8]. Following this, several mathematicians delved into the study of algebraic RS on LG, the area that had previously been explored by Lauret. During his research, he investigated the correlation between solvsolitons solitons and RS regarding Riemannian manifolds, ultimately proving that every Riemannian solvsoliton metric constitutes the RS in [9]. With these findings as a foundation, the author was able to derive both steady algebraic RS and diminishing algebraic RS in terms of Lorentzian geometries. It should be noted that Batat together with Onda subsequently investigated RS for three-dimensional Lorentzian Lie groups (LLG) in [10], examining all such Lie groups that qualify as algebraic RS. Furthermore, there also have been certain studies on the LG about Gauss Bonnet theorems in [11,12].

    Motivated by the above research, mathematicians undertook an investigation of algebraic RS that are associated with different affine connections. For example, in [13], Wang presented a novel product structure for three-dimensional LLG, along with a computation for canonical and Kobayashi-Nomizu connections as well as curvature tensor. He went ahead to define algebraic RSs that are related to the above statements. Furthermore, he categorized the algebraic RSs that are related to canonical as well as Kobayashi-Nomizu connections with this specific product structure. Wang also considered the distribution H=span{˜qY1,˜qY2} and its orthogonal complement H=span{˜qY3}, which are relevant to the three-dimensional LLG having a structure J:J˜qY1=˜qY1, J˜qY2=˜qY2 and J˜qY3=˜qY3. Moreover, other impressive results of RS are found in [14,15,16]. In [17], Calvaruso performed an in-depth analysis of three-dimensional generalized Rs with regards to Riemannian and Lorentzian frameworks. In order to study the properties associated with such solitons, they introduced a generalized RS in Eq. (1) [18] that can be regarded as the Schouten soliton, based on the Schouten tensor's definition mentioned in [19]. Drawing upon the works of [20], they also defined algebraic Schouten solitons(ASS). Moreover, the study in [21] introduced the concept of Yano connections (YC). Despite the substantial research on ASS, there is limited knowledge about their association with YC on LLG. Inspired by [22], and many studies provide extra incentives for solitons, see [23,24,25,26]. In this paper, we attempt to examine ASS associated with YC in the context of three-dimensional LG. The key to solving this problem is to find the existence conditions of ASS associated with YC. Based on this, by transforming equations of ASS into algebraic equations, the existence conditions of solitons are found. In particular, we calculate the curvature of YC and derive expressions for ASS to finish their categorization for three-dimensional LLG. Its main results demonstrate that ASS related to YC are present in G1, G2, G3, G5, G6 and G7, while they are not identifiable in G4.

    The paper is structured as follows. In Sec 2, fundamental concepts for three-dimensional LLG, specifically relating to YC as well ASS, will be introduced. Additionally, we present a succinct depiction of each three-dimensional connected LG, which is both unimodular and non-unimodular. In Sec 3, we obtain all formulas for YC as well their corresponding curvatures tensor in seven LLG. Using this Ricci operator and defining ASS associated to YC, we are able to fully classify three-dimensional LLG that admit the first kind ASS related to YC. In Sec 4, we use this soliton equation in an effort to finish a categorization about three-dimensional LLG that support ASS of the second kind related to YC. In Sec 5, we highlight certain important findings and talk about potential directions regarding research.

    In this section, fundamental concepts for three-dimensional LLG, specifically relating to YC as well ASS, will be introduced. Additionally, we present a succinct depiction of each three-dimensional connected LG, which is both unimodular and non-unimodular (for details see [27,28]).

    We designate {Gi}i=1,,7 as the collection for three-dimensional LLG, which is connected and simply connected, endowed with left-invariant Lorentzian metric gY. Furthermore, the respective Lie algebra(LA) for each group is denoted as {gYi}i=1,,7. The LCC will get represented by L. This is the definition of the YC:

    YUYVY=LUYVY12(LVYJ)JUY14[(LUYJ)JVY(LJUYJ)VY], (2.1)

    furthermore, {Gi}i=1,,7 having a structure J:J˜qY1=˜qY1, J˜qY2=˜qY2, J˜qY3=˜qY3, followed J2=id, then gY(J˜qYj,J˜qYj)=gY(˜qYj,˜qYj). This is the definitions of the curvature:

    RY(UY,VY)WY=YUYYVYWYYVYYUYWYY[UY,VY]WY. (2.2)

    This definition of the Ricci tensor for (Gi,gY), which is related to the YC, can be given as

    ρY(UY,VY)=gY(RY(UY,˜qY1)VY,˜qY1)gY(RY(UY,˜qY2)VY,˜qY2)+gY(RY(UY,˜qY3)VY,˜qY3), (2.3)

    the basis ˜qY1, ˜qY2 and ˜qY3 is pseudo-orthonormal, ˜qY3 is timelike vector fields. This definition of the Ricci operator RicY can be given as

    ρY(UY,VY)=gY(RicY(UY),VY). (2.4)

    One can define the Schouten tensor with the expression given by

    SY(˜qYi,˜qYj)=ρY(˜qYi,˜qYj)sY4gY(˜qYi,˜qYj), (2.5)

    where sY represents the scalar curvature. By extending the Schouten tensor's definition, we obtain

    SY(˜qYi,˜qYj)=ρY(˜qYi,˜qYj)sYλ0gY(˜qYi,˜qYj), (2.6)

    where λ0 is a real-valued constant. By referring to [29], we can obtain

    sY=ρY(˜qY1,˜qY1)+ρY(˜qY2,˜qY2)ρY(˜qY3,˜qY3), (2.7)

    for vector fields UY, VY, WY.

    Theorem 2.1. [27,28] Let (G,gY) be three-dimensional LG of connected unimodular that has a left-invariant Lorentzian metric. Thus the LA for G is one of the following if there exists a pseudo-orthonormal basis {˜qY1,˜qY2,˜qY3} with ˜qY3 timelike:

    (gY1):

    [˜qY1,˜qY2]=α˜qY1β˜qY3,[˜qY1,˜qY3]=α˜qY1β˜qY2,[˜qY2,˜qY3]=β˜qY1+α˜qY2+α˜qY3,α0.

    (gY2):

    [˜qY1,˜qY2]=γ˜qY2β˜qY3,[˜qY1,˜qY3]=β˜qY2γ˜qY3,[˜qY2,˜qY3]=α˜qY1,γ0.

    (gY3):

    [˜qY1,˜qY2]=γ˜qY3,[˜qY1,˜qY3]=β˜qY2,[˜qY2,˜qY3]=α˜qY1.

    (gY4):

    [˜qY1,˜qY2]=˜qY2+(2ηβ)˜qY2,η=±1,[˜qY1,˜qY3]=β˜qY2+˜qY3,[˜qY2,˜qY3]=α˜qY1.

    Theorem 2.2. [27,28] Let (G,gY) be three-dimensional LG of connected non-unimodular that has a left-invariant Lorentzian metric. Thus the LA for G is one of the following if there exists a pseudo-orthonormal basis {˜qY1,˜qY2,˜qY3} with ˜qY3 timelike:

    (gY5):

    [˜qY1,˜qY2]=0,[˜qY1,˜qY3]=α˜qY1+β˜qY2,[˜qY2,˜qY3]=γ˜qY1+δ˜qY2,α+δ0,αγ+βδ=0.

    (gY6):

    [˜qY1,˜qY2]=α˜qY2+β˜qY3,[˜qY1,˜qY3]=γ˜qY2+δ˜qY3,[˜qY2,˜qY3]=0,α+δ0,αγβδ=0.

    (gY7):

    [˜qY1,˜qY2]=α˜qY1β˜qY2β˜qY3,[˜qY1,˜qY3]=α˜qY1+β˜qY2+β˜qY3,
    [˜qY2,˜qY3]=γ˜qY1+δ˜qY2+δ˜qY3,α+δ0,αγ=0.

    Definition 2.3. (Gi,gY) is called ASS of the first kind related with YC when it satisfies

    RicY=(sYλ0+c)Id+D, (2.8)

    which c is an actual number, λ0 is a real-valued constant, as well D is derivation for gY, which can be

    D[UY,VY]=[DUY,VY]+[UY,DVY], (2.9)

    for UY,VYgY.

    In this section, we aim to obtain the formulas for YC as well their corresponding curvatures in seven LLGs. Using the Ricci operator and defining LLG associated to YC, we are able to fully classify three-dimensional LLG that admit ASS as the first kind associated with YC.

    In the subsection, we present the LA for G1 that satisfies the following condition

    [˜qY1,˜qY2]=α˜qY1β˜qY3,[˜qY1,˜qY3]=α˜qY1β˜qY2,[˜qY2,˜qY3]=β˜qY1+α˜qY2+α˜qY3,α0,

    the basis vectors ˜qY1, ˜qY2 and ˜qY3 form a pseudo-orthonormal basis where ˜qY3 is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in G1 with Lorentzian metric can be derived.

    Lemma 3.1 ([10,30]). The LCC for G1 can be given as

    L˜qY1˜qY1=α˜qY2α˜qY3, L˜qY1˜qY2=α˜qY1β2˜qY3, L˜qY1˜qY3=α˜qY1β2˜qY2,L˜qY2˜qY1=β2˜qY3, L˜qY2˜qY2=α˜qY3, L˜qY2˜qY3=β2˜qY1+α˜qY2,L˜qY3˜qY1=β2˜qY2, L˜qY3˜qY2=β2˜qY1α˜qY3, L˜qY3˜qY3=α˜qY2.

    Lemma 3.2. For G1, the following equalities hold

    L˜qY1(J)˜qY1=2α˜qY3, L˜qY1(J)˜qY2=β˜qY3, L˜qY1(J)˜qY3=2α˜qY1+β˜qY2,L˜qY2(J)˜qY1=β˜qY3, L˜qY2(J)˜qY2=2α˜qY3, L˜qY2(J)˜qY3=β˜qY12˜qY2,L˜qY3(J)˜qY1=0, L˜qY3(J)˜qY2=2α˜qY3, L˜qY3(J)˜qY3=2α˜qY2.

    Based on (2.1), as well as Lemmas 3.1 and 3.2, one can derive the subsequent lemma.

    Lemma 3.3. The YC for G1 can be given as

    Y˜qY1˜qY1=α˜qY2, Y˜qY1˜qY2=α˜qY1β˜qY3, Y˜qY1˜qY3=0,Y˜qY2˜qY1=β˜qY3, Y˜qY2˜qY2=0, Y˜qY2˜qY3=α˜qY3,Y˜qY3˜qY1=α˜qY1+β˜qY2, Y˜qY3˜qY2=β˜qY1α˜qY2, Y˜qY3˜qY3=0.

    Based on (2.2), as well as Lemma 3.3, one can derive the subsequent lemma.

    Lemma 3.4. The curvature RY for (G1,gY) can be given as

    RY(˜qY1,˜qY2)˜qY1=αβ˜qY1+(α2+β2)˜qY2, RY(˜qY1,˜qY2)˜qY2=(α2+β2)˜qY1αβ˜qY2+αβ˜qY3,RY(˜qY1,˜qY2)˜qY3=0, RY(˜qY1,˜qY3)˜qY1=3α2˜qY2, RY(˜qY1,˜qY3)˜qY2=α2˜qY1,RY(˜qY1,˜qY3)˜qY3=αβ˜qY3, RY(˜qY2,˜qY3)˜qY1=α2˜qY1, RY(˜qY2,˜qY3)˜qY2=α2˜qY2,RY(˜qY2,˜qY3)˜qY3=α2˜qY3.

    Using Lemmas 3.3 and 3.4, the following theorem regarding the ASS of the first kind in the first LG with Lorentzian metric can be established.

    Theorem 3.5. (G1,gY,J) is ASS of the first kind related to the YC if it satisfies β=c=0, α0. And specifically

    RicY(˜qY1˜qY2˜qY3)=(α2000α2α2000)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(α2+2α2λ0000α2+2α2λ0α2002α2λ0)(˜qY1˜qY2˜qY3).

    Proof. According to (2.3), we have

    ρY(˜qY1,˜qY1)=α2β2, ρY(˜qY1,˜qY2)=αβ, ρY(˜qY1,˜qY3)=αβ,ρY(˜qY2,˜qY1)=αβ, ρY(˜qY2,˜qY2)=(α2+β2), ρY(˜qY2,˜qY3)=α2,ρY(˜qY3,˜qY1)=0, ρY(˜qY3,˜qY2)=0, ρY(˜qY3,˜qY3)=0.

    By (2.4), the Ricci operator can be expressed as

    RicY(˜qY1˜qY2˜qY3)=(α2β2αβαβαβα2β2α2000)(˜qY1˜qY2˜qY3).

    As a result, the scalar curvature can be obtained as sY=2α22β2. If (G1,gY,J) is ASS of the first kind related to the YC, and by RicY=(sYλ0+c)Id+D, we can get

    {D˜qY1=[α2β2+(2α2+2β2)λ0c]˜qY1+αβ˜qY2+αβ˜qY3,D˜qY2=αβ˜qY1+[α2β2+(2α2+2β2)λ0c]˜qY2α2˜qY3,D˜qY3=[(2α2+2β2)λ0c]˜qY3.

    Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies

    {2α3λ02αβ2+2αβ2λ0αc=0,α2β=0,β3α2β=0,2β3λ02α2β+2α2βλ0βc=0. (3.1)

    Considering that α0, by solving the first and second equations in (3.1) leads to the conclusion that β=0 and c=0. Thus we get Theorem 3.5.

    In the subsection, we present the LA for G2 that satisfies the following condition

    [˜qY1,˜qY2]=γ˜qY2β˜qY3,[˜qY1,˜qY3]=β˜qY2γ˜qY3,[˜qY2,˜qY3]=α˜qY1,γ0,

    the basis vectors ˜qY1, ˜qY2 and ˜qY3 form a pseudo-orthonormal basis where ˜qY3 is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in G2 with Lorentzian metric can be derived.

    Lemma 3.6 ([10,30]). The LCC for G2 can be given as

    L˜qY1˜qY1=0, L˜qY1˜qY2=(α2β)˜qY3, L˜qY1˜qY3=(α2β)˜qY2,L˜qY2˜qY1=γ˜qY2+α2˜qY3, L˜qY2˜qY2=γ˜qY1, L˜qY2˜qY3=α2˜qY1,L˜qY3˜qY1=α2˜qY2+γ˜qY3, L˜qY3˜qY2=α2˜qY1, L˜qY3˜qY3=γ˜qY1.

    Lemma 3.7. For G2, the following equalities hold

    L˜qY1(J)˜qY1=0, L˜qY1(J)˜qY2=(α2β)˜qY3, L˜qY1(J)˜qY3=(α2β)˜qY2,L˜qY2(J)˜qY1=α˜qY3, L˜qY2(J)˜qY2=0, L˜qY2(J)˜qY3=α˜qY1,L˜qY3(J)˜qY1=2γ˜qY3, L˜qY3(J)˜qY2=0, L˜qY3(J)˜qY3=2γ˜qY1.

    Based on (2.1), as well as Lemmas 3.6 and 3.7, one can derive the subsequent lemma.

    Lemma 3.8. The YC for G2 can be given as

    Y˜qY1˜qY1=0, Y˜qY1˜qY2=β˜qY3, Y˜qY1˜qY3=2β˜qY2γ˜qY3,Y˜qY2˜qY1=γ˜qY2+β˜qY3, Y˜qY2˜qY2=γ˜qY1, Y˜qY2˜qY3=0,Y˜qY3˜qY1=β˜qY2, Y˜qY3˜qY2=α˜qY1, Y˜qY3˜qY3=0.

    Based on (2.2), as well as Lemma 3.8, one can derive the subsequent lemma.

    Lemma 3.9. The curvature RY for (G2,gY) can be given as

    RY(˜qY1,˜qY2)˜qY1=(γ2β2)˜qY2βγ˜qY3, RY(˜qY1,˜qY2)˜qY2=(γ2+αβ)˜qY1,RY(˜qY1,˜qY2)˜qY3=2βγ˜qY1, RY(˜qY1,˜qY3)˜qY1=0, RY(˜qY1,˜qY3)˜qY2=(βγαγ)˜qY1,RY(˜qY1,˜qY3)˜qY3=2αβ˜qY1, RY(˜qY2,˜qY3)˜qY1=(βγαγ)˜qY1,RY(˜qY2,˜qY3)˜qY2=βγ˜qY2+αβ˜qY3, RY(˜qY2,˜qY3)˜qY3=2αβ˜qY2+αγ˜qY3.

    Using Lemmas 3.8 and 3.9, the following theorem regarding the ASS of the first kind in the second LG with Lorentzian metric can be established.

    Theorem 3.10. (G2,gY,J) is ASS of the first kind related to YC if it satisfies α=β=0, γ0, c=γ2(2λ01). And specifically

    RicY(˜qY1˜qY2˜qY3)=(γ2000γ20000)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(00000000γ2)(˜qY1˜qY2˜qY3).

    Proof. According to (2.3), we have

    ρY(˜qY1,˜qY1)=β2γ2, ρY(˜qY1,˜qY2)=0, ρY(˜qY1,˜qY3)=0,ρY(˜qY2,˜qY1)=0, ρY(˜qY2,˜qY2)=γ22αβ, ρY(˜qY2,˜qY3)=2βγαγ,ρY(˜qY3,˜qY3)=0, ρY(˜qY3,˜qY2)=αγ, ρY(˜qY3,˜qY3)=0.

    By (2.4), the Ricci operator can be expressed as

    RicY(˜qY1˜qY2˜qY3)=(β2γ2000γ22αβαγ2βγ0αγ0)(˜qY1˜qY2˜qY3).

    As a result, the scalar curvature can be obtained as sY=β22γ22αβ. If (G2,gY,J) is ASS of the first kind related to the YC, and by RicY=(sYλ0+c)Id+D, we can get

    {D˜qY1=(β2γ2β2λ0+2γ2λ0+2αβλ0c)˜qY1,D˜qY2=(γ22αββ2λ0+2γ2λ0+2αβλ0c)˜qY2+(αγ2βγ)˜qY3,D˜qY3=αγ˜qY2+(β2λ0+2γ2λ0+2αβλ0c)˜qY3.

    Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies

    {β3β3λ0+2αγ26βγ22αβ2+2βγ2λ0+2αβ2λ0βc=0,β3β3λ0+2αγ2+2αβ2+2βγ2λ0+2αβ2λ0βc=0,γ32γ3λ03β2γ+β2γλ0+2αβγ2αβγλ0+γc=0,αβ22α2βαβ2λ0+2αγ2λ0+2α2βλ0αc=0. (3.2)

    By solving the first and second equations of (3.2) imply that

    2αβ2+3βγ2=0.

    As γ0, it follows that β must be zero. On this basis, the second equation of (3.2) reduces to

    2αγ2=0,

    we have α=0. In this case, the third equation of (3.2) can be simplified to

    γ32γ3λ0+γc=0,

    then we obtain c=γ2(2λ01). Thus we get Theorem 3.10.

    In the subsection, we present the LA for G3 that satisfies the following condition

    [˜qY1,˜qY2]=γ˜qY3,[˜qY1,˜qY3]=β˜qY2,[˜qY2,˜qY3]=α˜qY1,

    the basis vectors ˜qY1, ˜qY2 and ˜qY3 form a pseudo-orthonormal basis where ˜qY3 is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in G3 with Lorentzian metric can be derived.

    Lemma 3.11 ([10,30]). The LCC for G3 can be given as

    L˜qY1˜qY1=0, L˜qY1˜qY2=αβγ2˜qY3, L˜qY1˜qY3=αβγ2˜qY2,L˜qY2˜qY1=αβ+γ2˜qY3, L˜qY2˜qY2=0, L˜qY2˜qY3=αβ+γ2˜qY1,L˜qY3˜qY1=α+βγ2˜qY2, L˜qY3˜qY2=α+βγ2˜qY1, L˜qY3˜qY3=0.

    Lemma 3.12. For G3, the following equalities hold

    L˜qY1(J)˜qY1=0, L˜qY1(J)˜qY2=(αβγ)˜qY3, L˜qY1(J)˜qY3=(αβγ)˜qY2,L˜qY2(J)˜qY1=(αβ+γ)˜qY3, L˜qY2(J)˜qY2=0, L˜qY2(J)˜qY3=(αβ+γ)˜qY1,L˜qY3(J)˜qY1=0, L˜qY3(J)˜qY2=0, L˜qY3(J)˜qY3=0.

    Based on (2.1), as well as Lemmas 3.11 and 3.12, one can derive the subsequent lemma.

    Lemma 3.13. The YC for G3 can be given as

    Y˜qY1˜qY1=0, Y˜qY1˜qY2=γ˜qY3, Y˜qY1˜qY3=0,Y˜qY2˜qY1=γ˜qY3, Y˜qY2˜qY2=0, Y˜qY2˜qY3=γ˜qY1,Y˜qY3˜qY1=β˜qY2, Y˜qY3˜qY2=α˜qY1, Y˜qY3˜qY3=0.

    Based on (2.2), as well as Lemma 3.13, one can derive the subsequent lemma.

    Lemma 3.14. The curvature RY for (G3,gY) can be given as

    RY(˜qY1,˜qY2)˜qY1=βγ˜qY2, RY(˜qY1,˜qY2)˜qY2=(γ2+αγ)˜qY1, RY(˜qY1,˜qY2)˜qY3=0,RY(˜qY1,˜qY3)˜qY1=0, RY(˜qY1,˜qY3)˜qY2=0, RY(˜qY1,˜qY3)˜qY3=βγ˜qY1,RY(˜qY2,˜qY3)˜qY1=0, RY(˜qY2,˜qY3)˜qY2=0, RY(˜qY2,˜qY3)˜qY3=βγ˜qY2.

    Using Lemmas 3.13 and 3.14, the following theorem regarding the ASS of the first kind in the third LG with Lorentzian metric can be established.

    Theorem 3.15. (G3,gY,J) is ASS of the first kind related to YC if it satisfies

    (1)α=β=γ=0. And specifically

    RicY(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(c000c000c)(˜qY1˜qY2˜qY3).

    (2)α=β=0, γ0, c=γ2λ0γ2. And specifically

    RicY(˜qY1˜qY2˜qY3)=(0000γ20000)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(γ20000000γ2)(˜qY1˜qY2˜qY3).

    (3)γ=c=0. And specifically

    RicY(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3).

    (4)β=0, α0, γ0, γ3γ3λ0α2γ+α2γλ0+γcαc=0. And specifically

    RicY(˜qY1˜qY2˜qY3)=(0000γ2αγ0000)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(γ2λ0+αγλ0c000γ2+γ2λ0αγ+αγλ0c000γ2λ0+αγλ0c)(˜qY1˜qY2˜qY3).

    (5)α=0, β0, γ0, γ3γ3λ02βγ2λ0+β2γβ2γλ0+γcβc=0. And specifically

    RicY(˜qY1˜qY2˜qY3)=(βγ000γ20000)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(γ2λ0βγ+βγλ0c000γ2+γ2λ0+βγλ0c000γ2λ0+βγλ0c)(˜qY1˜qY2˜qY3).

    Proof. According to (2.3), we have

    ρY(˜qY1,˜qY1)=βγ, ρY(˜qY1,˜qY2)=0, ρY(˜qY1,˜qY3)=0,ρY(˜qY2,˜qY1)=0, ρY(˜qY2,˜qY2)=γ2αγ, ρY(˜qY2,˜qY3)=0,ρY(˜qY3,˜qY1)=0, ρY(˜qY3,˜qY2)=0, ρY(˜qY3,˜qY3)=0.

    By (2.4), the Ricci operator can be expressed as

    RicY(˜qY1˜qY2˜qY3)=(βγ000γ2αγ0000)(˜qY1˜qY2˜qY3).

    As a result, the scalar curvature can be obtained as sY=(γ2+αγ+βγ). If (G3,gY,J) is ASS of the first kind related to the YC, and by RicY=(sYλ0+c)Id+D, we can get

    {D˜qY1=(γ2λ0βγ+αγλ0+βγλ0c)˜qY1,D˜qY2=(γ2+γ2λ0αγ+αγλ0+βγλ0c)˜qY2,D˜qY3=(γ2λ0+αγλ0+βγλ0c)˜qY3.

    Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies

    {γ3γ3λ0+βγ2+αγ2αγ2λ0βγ2λ0+γc=0,β2γβγ2βγ2λ0β2γλ0αβγαβγλ0βc=0,αγ2+α2γαγ2λ0α2γλ0αβγαβγλ0+αc=0. (3.3)

    Assuming that γ=0, we get

    {βc=0,αc=0.

    If β=0, we obtain two cases (1) and (2) of Theorem 3.15 holds. If β0, for the case (3) of Theorem 3.15 holds. Next assuming that γ0, If β=0, and (3.3) can be simplified to

    {γ3γ3λ0+αγ2αγ2λ0+γc=0,αγ2+α2γαγ2λ0α2γλ0+αc=0.

    We get two cases (3) and (4) of Theorem 3.15 holds. If β0 and α=0, then a direct calculation reveals that (3.3) reduces to

    {γ3γ3λ0+βγ2βγ2λ0+γc=0,β2γβγ2βγ2λ0β2γλ0βc=0.

    We have case (5) of Theorem 3.15 holds. Thus we get Theorem 3.15.

    In the subsection, we present the LA for G4 that satisfies the following condition

    [˜qY1,˜qY2]=˜qY2+(2ηβ)˜qY2,η=±1,[˜qY1,˜qY3]=β˜qY2+˜qY3,[˜qY2,˜qY3]=α˜qY1,

    the basis vectors ˜qY1, ˜qY2 and ˜qY3 form a pseudo-orthonormal basis where ˜qY3is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in G4 with Lorentzian metric can be derived.

    Lemma 3.16 ([10,30]). The LCC for G4 can be given as

    L˜qY1˜qY1=0, L˜qY1˜qY2=(α2+ηβ)˜qY3, L˜qY1˜qY3=(α2+ηβ)˜qY2,L˜qY2˜qY1=˜qY2+(α2η)˜qY3, L˜qY2˜qY2=˜qY1, L˜qY2˜qY3=(α2η)˜qY1,L˜qY3˜qY1=(α2+η)˜qY2˜qY3, L˜qY3˜qY2=(α2+η)˜qY1, L˜qY3˜qY3=˜qY1.

    Lemma 3.17. For G4, the following equalities hold

    L˜qY1(J)˜qY1=0, L˜qY1(J)˜qY2=(α+2η2β)˜qY3, L˜qY1(J)˜qY3=(α+2η2β)˜qY2,L˜qY2(J)˜qY1=(α2η)˜qY3, L˜qY2(J)˜qY2=0, L˜qY2(J)˜qY3=(α2η)˜qY1,L˜qY3(J)˜qY1=2˜qY3, L˜qY3(J)˜qY2=0, L˜qY3(J)˜qY3=2˜qY1.

    Based on (2.1), as well as Lemmas 3.16 and 3.17, one can derive the subsequent lemma.

    Lemma 3.18. The YC for G4 can be given as

    Y˜qY1˜qY1=0, Y˜qY1˜qY2=(2ηβ)˜qY3, Y˜qY1˜qY3=˜qY3,Y˜qY2˜qY1=˜qY2+(β2η)˜qY3, Y˜qY2˜qY2=˜qY1, Y˜qY2˜qY3=0,Y˜qY3˜qY1=β˜qY2, Y˜qY3˜qY2=α˜qY1, Y˜qY3˜qY3=0.

    Based on (2.2), as well as Lemma 3.18, one can derive the subsequent lemma.

    Lemma 3.19. The curvature RY for (G4,gY) can be given as

    RY(˜qY1,˜qY2)˜qY1=(β22βη+1)˜qY2, RY(˜qY1,˜qY2)˜qY2=(2αηαβ1)˜qY1,RY(˜qY1,˜qY2)˜qY3=0, RY(˜qY1,˜qY3)˜qY1=0, RY(˜qY1,˜qY3)˜qY2=(αβ)˜qY1, RY(˜qY1,˜qY3)˜qY3=0,RY(˜qY2,˜qY3)˜qY1=(αβ)˜qY1, RY(˜qY2,˜qY3)˜qY2=(βα)˜qY2, RY(˜qY2,˜qY3)˜qY3=α˜qY3.

    Using Lemmas 3.18 and 3.19, the following theorem regarding the ASS of the first kind in the fourth LG with Lorentzian metric can be established.

    Theorem 3.20. The LG G4 cannot be ASS of a first kind related to the YC.

    Proof. According to (2.3), we have

    ρY(˜qY1,˜qY1)=2βηβ21, ρY(˜qY1,˜qY2)=0, ρY(˜qY1,˜qY3)=0,ρY(˜qY2,˜qY1)=0, ρY(˜qY2,˜qY2)=2αηαβ1, ρY(˜qY2,˜qY3)=α,ρY(˜qY3,˜qY1)=0, ρY(˜qY3,˜qY2)=0, ρY(˜qY3,˜qY3)=0.

    By (2.4), the Ricci operator can be expressed as

    RicY(˜qY1˜qY2˜qY3)=(β2+2βη10002αηαβ1α000)(˜qY1˜qY2˜qY3).

    As a result, the scalar curvature can be obtained as sY=β2+2αη+2βηαβ2.If (G4,gY,J) is ASS of the first kind related to the YC, and by RicY=(sYλ0+c)Id+D, we can get

    {D˜qY1=(β2+β2λ0+2βη2αηλ02βηλ0+αβλ0+2λ01c)˜qY1,D˜qY2=(β2λ0+2αηαβ2αηλ02βηλ0+αβλ0+2λ01c)˜qY2α˜qY3,D˜qY3=(β2λ02αηλ02βηλ0+αβλ0+2λ0c)˜qY3.

    Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies

    {2α+(2ηβ)(β2β2λ02αη2βη+αβ+2αηλ0+2βηλ0αβλ02λ0+2+c)=0,β3β3λ0αβ22β2η+2β2ηλ0αβ2λ0+2αβη+2αβηλ02βλ0+βc=0,β2λ0β2+2βηαβ2αηλ02βηλ0+αβλ0+2λ01c=0,2α2ηα2β+αβ22α2ηλ0+α2βλ0+αβ2λ02αβη2αβηλ0+2αλ0αc=0. (3.4)

    By the first equation of (3.4), we assume that

    α=0,β=2η.

    On this basis, by the second equation of (3.4), we have c=2λ0. By the third equation of (3.4), we get c=2λ01, and there is a contradiction. One can prove Theorem 3.20.

    In the subsection, we present the LA for G5 that satisfies the following condition

    [˜qY1,˜qY2]=0,[˜qY1,˜qY3]=α˜qY1+β˜qY2,[˜qY2,˜qY3]=γ˜qY1+δ˜qY2,α+δ0,αγ+βδ=0,

    the basis vectors ˜qY1, ˜qY2 and ˜qY3 form a pseudo-orthonormal basis where ˜qY3 is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in G5 with Lorentzian metric can be derived.

    Lemma 3.21 ([10,30]). The LCC for G5 can be given as

    L˜qY1˜qY1=α˜qY3, L˜qY1˜qY2=β+γ2˜qY3, L˜qY1˜qY3=α˜qY1+β+γ2˜qY2,L˜qY2˜qY1=β+γ2˜qY3, L˜qY2˜qY2=δ˜qY3, L˜qY2˜qY3=β+γ2˜qY1+δ˜qY2,L˜qY3˜qY1=βγ2˜qY2, L˜qY3˜qY2=βγ2˜qY1, L˜qY3˜qY3=0.

    Lemma 3.22. For G5, the following equalities hold

    L˜qY1(J)˜qY1=2α˜qY3, L˜qY1(J)˜qY2=(β+γ)˜qY3, L˜qY1(J)˜qY3=2α˜qY1(β+γ)˜qY2,L˜qY2(J)˜qY1=(β+γ)˜qY3, L˜qY2(J)˜qY2=2δ˜qY3, L˜qY2(J)˜qY3=(β+γ)˜qY12δ˜qY2,L˜qY3(J)˜qY1=0, L˜qY3(J)˜qY2=0, L˜qY3(J)˜qY3=0.

    Based on (2.1), as well as Lemmas 3.21 and 3.22, one can derive the subsequent lemma.

    Lemma 3.23. The YC for G5 can be given as

    Y˜qY1˜qY1=0, Y˜qY1˜qY2=0, Y˜qY1˜qY3=0,Y˜qY2˜qY1=0, Y˜qY2˜qY2=0, Y˜qY2˜qY3=0,Y˜qY3˜qY1=α˜qY1+(β+γ)˜qY2, Y˜qY3˜qY2=γ˜qY1δ˜qY2, Y˜qY3˜qY3=0.

    Based on (2.2), as well as Lemma 3.23, one can derive the subsequent lemma.

    Lemma 3.24. The curvature RY for (G5,gY) can be given as

    RY(˜qY1,˜qY2)˜qYj=RY(˜qY1,˜qY3)˜qYj=RY(˜qY2,˜qY3)˜qYj=0,

    where 1j3.

    Using Lemmas 3.23 and 3.24, the following theorem regarding the ASS of the first kind in the fifth LG with Lorentzian metric can be established..

    Theorem 3.25. (G5,gY,J) is ASS of the first kind related to YC if it satisfies c=0. And specifically

    RicY(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3).

    Proof. According to (2.3), we have

    ρY(˜qY1,˜qYj)=ρY(˜qY2,˜qYj)=ρY(˜qY3,˜qYj)=0,

    where 1j3.

    By (2.4), the Ricci operator can be expressed as

    RicY(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3).

    As a result, the scalar curvature can be obtained as sY=0. If (G5,gY,J) is ASS of the first kind related to the YC, and by RicY=(sYλ0+c)Id+D, we can get

    {D˜qY1=c˜qY1,D˜qY2=c˜qY2,D˜qY3=c˜qY3.

    Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies

    {αc=0,βc=0,γc=0,δc=0. (3.5)

    Since α+δ0 and αγ+βδ=0, by solving (3.5), we have c=0. Thus we get Theorem 3.25.

    In the subsection, we present the LA for G6 that satisfies the following condition

    [˜qY1,˜qY2]=α˜qY2+β˜qY3,[˜qY1,˜qY3]=γ˜qY2+δ˜qY3,[˜qY2,˜qY3]=0,α+δ0,αγβδ=0.

    The basis vectors ˜qY1, ˜qY2 and ˜qY3 form a pseudo-orthonormal basis where ˜qY3 is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in G6 with Lorentzian metric can be derived.

    Lemma 3.26 ([10,30]). The LCC for G6 can be given as

    L˜qY1˜qY1=0, L˜qY1˜qY2=β+γ2˜qY3, L˜qY1˜qY3=β+γ2˜qY2,L˜qY2˜qY1=α˜qY2βγ2˜qY3, L˜qY2˜qY2=α˜qY1, L˜qY2˜qY3=βγ2˜qY1,L˜qY3˜qY1=βγ2˜qY2δ˜qY3, L˜qY3˜qY2=βγ2˜qY1, L˜qY3˜qY3=δ˜qY1.

    Lemma 3.27. For G6, the following equalities hold

    L˜qY1(J)˜qY1=0, L˜qY1(J)˜qY2=(β+γ)˜qY3, L˜qY1(J)˜qY3=(β+γ)˜qY2,L˜qY2(J)˜qY1=(βγ)˜qY3, L˜qY2(J)˜qY2=0, L˜qY2(J)˜qY3=(βγ)˜qY1,L˜qY3(J)˜qY1=2δ˜qY3, L˜qY3(J)˜qY2=0, L˜qY3(J)˜qY3=2δ˜qY1.

    Based on (2.1), as well as Lemmas 3.26 and 3.27, one can derive the subsequent lemma.

    Lemma 3.28. The YC for G6 can be given as

    Y˜qY1˜qY1=0, Y˜qY1˜qY2=β˜qY3, Y˜qY1˜qY3=δ˜qY3,Y˜qY2˜qY1=α˜qY2β˜qY3, Y˜qY2˜qY2=α˜qY1, Y˜qY2˜qY3=0,Y˜qY3˜qY1=γ˜qY2, Y˜qY3˜qY2=0, Y˜qY3˜qY3=0.

    Based on (2.2), as well as Lemma 3.28, one can derive the subsequent lemma.

    Lemma 3.29. The curvature RY for (G6,gY) can be given as

    RY(˜qY1,˜qY2)˜qY1=(βγ+α2)˜qY2βδ˜qY3, RY(˜qY1,˜qY2)˜qY2=α2˜qY1, RY(˜qY1,˜qY2)˜qY3=0,RY(˜qY1,˜qY3)˜qY1=(αγ+δγ)˜qY2, RY(˜qY1,˜qY3)˜qY2=αγ˜qY1, RY(˜qY1,˜qY3)˜qY3=0,RY(˜qY2,˜qY3)˜qY1=αγ˜qY1, RY(˜qY2,˜qY3)˜qY2=αγ˜qY2, RY(˜qY2,˜qY3)˜qY3=0.

    Using Lemmas 3.28 and 3.29, the following theorem regarding the ASS of the first kind in the sixth LG with Lorentzian metric can be established.

    Theorem 3.30. (G6,gY,J) is ASS of the first kind related to YC if it satisfies

    (1)α=β=c=0, δ0. And specifically

    RicY(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3).

    (2)α0, β=γ=0, α+δ0, c=2α2λ0α2. And specifically

    RicY(˜qY1˜qY2˜qY3)=(α2000α20000)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(00000000α2)(˜qY1˜qY2˜qY3).

    Proof. According to (2.3), we have

    ρY(˜qY1,˜qY1)=(βγ+α2), ρY(˜qY1,˜qY2)=0, ρY(˜qY1,˜qY3)=0,ρY(˜qY2,˜qY1)=0, ρY(˜qY2,˜qY2)=α2, ρY(˜qY2,˜qY3)=0,ρY(˜qY3,˜qY1)=0, ρY(˜qY3,˜qY2)=0, ρY(˜qY3,˜qY3)=0.

    By (2.4), the Ricci operator can be expressed as

    RicY(˜qY1˜qY2˜qY3)=((α2+βγ)000α20000)(˜qY1˜qY2˜qY3).

    As a result, the scalar curvature can be obtained as sY=2α2βγ. If (G6,gY,J) is ASS of the first kind related to the YC, and by RicY=(sYλ0+c)Id+D, we can get

    {D˜qY1=(2α2λ0α2βγ+βγλ0c)˜qY1,D˜qY2=(2α2λ0α2+βγλ0c)˜qY2,D˜qY3=(2α2λ0+βγλ0c)˜qY3.

    Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies

    {2α3λ0α3αβγ+αβγλ0αc=0,β2γλ0β2γ2α2β+2α2βλ0βc=0,βγ2+βγ2λ0+2α2γλ0γc=0,2α2δλ0α2δβδγ+βγδλ0δc=0. (3.6)

    Because α+δ0 as well αγβδ=0, we suppose first that α=β=0, δ0. On this basis, the fourth equation of (3.6) can be simplified to

    δc=0,

    we get c=0, for the case (1) of Theorem 3.30 holds. Suppose second that γ=0, α0, α+δ0, on this basis, the first and fourth equations of (3.6) reduces to

    α2+2α2λ0c=0,

    and the second equation of (3.6) can be simplified to

    β(2α2λ02α2c)=0,

    we have α2β=0, thus β=0, for the case (2) of Theorem 3.30 holds. It turns out Theorem 3.30.

    In the subsection, we present the LA for G7 that satisfies the following condition

    [˜qY1,˜qY2]=α˜qY1β˜qY2β˜qY3,[˜qY1,˜qY3]=α˜qY1+β˜qY2+β˜qY3,
    [˜qY2,˜qY3]=γ˜qY1+δ˜qY2+δ˜qY3,α+δ0,αγ=0,

    the basis vectors ˜qY1, ˜qY2 and ˜qY3 form a pseudo-orthonormal basis where ˜qY3 is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in G7 with Lorentzian metric can be derived.

    Lemma 3.31 ([10,30]). The LCC for G7 can be given as

    L˜qY1˜qY1=α˜qY2+α˜qY3, L˜qY1˜qY2=α˜qY1+γ2˜qY3, L˜qY1˜qY3=α˜qY1+γ2˜qY2,L˜qY2˜qY1=β˜qY2+(β+γ2)˜qY3, L˜qY2˜qY2=β˜qY1+δ˜qY3, L˜qY2˜qY3=(β+γ2)˜qY1+δ˜qY2,L˜qY3˜qY1=(βγ2)˜qY2β˜qY3, L˜qY3˜qY2=(βγ2)˜qY1δ˜qY3, L˜qY3˜qY3=β˜qY1δ˜qY2.

    Lemma 3.32. For G7, the following equalities hold

    L˜qY1(J)˜qY1=2α˜qY3, L˜qY1(J)˜qY2=γ˜qY3, L˜qY1(J)˜qY3=2α˜qY1γ˜qY2,L˜qY2(J)˜qY1=(2β+γ)˜qY3, L˜qY2(J)˜qY2=2δ˜qY3, L˜qY2(J)˜qY3=(2β+γ)˜qY12δ˜qY2,L˜qY3(J)˜qY1=2β˜qY3, L˜qY3(J)˜qY2=2δ˜qY3, L˜qY3(J)˜qY3=2β˜qY1+2δ˜qY2.

    Based on (2.1), as well as Lemmas 3.31 and 3.32, one can derive the subsequent lemma.

    Lemma 3.33. The YC for G7 can be given as

    Y˜qY1˜qY1=α˜qY2, Y˜qY1˜qY2=α˜qY1β˜qY3, Y˜qY1˜qY3=β˜qY3,Y˜qY2˜qY1=β˜qY2+β˜qY3, Y˜qY2˜qY2=β˜qY1, Y˜qY2˜qY3=δ˜qY3,Y˜qY3˜qY1=α˜qY1β˜qY2, Y˜qY3˜qY2=γ˜qY1δ˜qY2, Y˜qY3˜qY3=0.

    Based on (2.2), as well as Lemma 3.33, one can derive the subsequent lemma.

    Lemma 3.34. The curvature RY for (G7,gY) can be given as

    RY(˜qY1,˜qY2)˜qY1=αβ˜qY1+α2˜qY2+β˜qY3,RY(˜qY1,˜qY2)˜qY2=(α2+β2+βγ)˜qY1βδ˜qY2+βδ˜qY3,RY(˜qY1,˜qY2)˜qY3=(βδ+αβ)˜qY3, RY(˜qY1,˜qY3)˜qY1=(2αβ+αγ)˜qY1+(αδ2α2)˜qY2,RY(˜qY1,˜qY3)˜qY2=(β2+βγ+αδ)˜qY1+(αβαγ+βδ)˜qY2+(βδ+αβ)˜qY3,RY(˜qY1,˜qY3)˜qY3=(αβ+βδ)˜qY3,RY(˜qY2,˜qY3)˜qY1=(β2+βγ+αδ)˜qY1+(βδαβαγ)˜qY2(αβ+βδ)˜qY3,RY(˜qY2,˜qY3)˜qY2=(2βδαβ+αγ+γδ)˜qY1+(δβγβ2)˜qY2,RY(˜qY2,˜qY3)˜qY3=(βγ+δ2)˜qY3.

    The following theorem regarding the ASS of the first kind in the seventh LG with Lorentzian metric can be established.

    Theorem 3.35. (G7,gY,J) is ASS of the first kind related to YC if it satisfies

    (1)α=β=γ=0, δ0, δ=1,c=1. And specifically

    RicY(˜qY1˜qY2˜qY3)=(00000δ20δ0)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(c000cδ20δc)(˜qY1˜qY2˜qY3).

    (2)α=β=c=0, δ0, γ0, δ=1. And specifically

    RicY(˜qY1˜qY2˜qY3)=(00000δ20δ0)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(00000δ20δ0)(˜qY1˜qY2˜qY3).

    (3)α0, β=γ=0, α+δ0, α=λδ, δ=1λ2λ1, c=1λ2+2λ2λ0(λ2λ1)2, λ0. And specifically

    RicY(˜qY1˜qY2˜qY3)=(α2000α2δ20αδ+δ0)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(δ2000δ2δ20αδ+δα2δ2)(˜qY1˜qY2˜qY3).

    Proof. According to (2.3), we have

    ρY(˜qY1,˜qY1)=α2, ρY(˜qY1,˜qY2)=αβ, ρY(˜qY1,˜qY3)=αβ+βδ,ρY(˜qY2,˜qY1)=βδ, ρY(˜qY2,˜qY2)=α2β2βγ, ρY(˜qY2,˜qY3)=βγ+δ2,ρY(˜qY3,˜qY1)=αβ+βδ, ρY(˜qY3,˜qY2)=αδ+δ, ρY(˜qY3,˜qY3)=0.

    By (2.4), the Ricci operator can be expressed as

    RicY(˜qY1˜qY2˜qY3)=(α2αβ(αβ+βδ)βδα2β2βγ(δ2+βγ)αβ+βδαδ+δ0)(˜qY1˜qY2˜qY3).

    As a result, the scalar curvature can be obtained as sY=2α2β2βγ. If (G7,gY,J) is ASS of the first kind related to the YC, and by RicY=(sYλ0+c)Id+D, we can get

    {D˜qY1=(α2+2α2λ0+β2λ0+βγλ0c)˜qY1αβ˜qY2(αβ+βδ)˜qY3,D˜qY2=βδ˜qY1+(α2β2+2α2λ0+β2λ0βγ+βγλ0c)˜qY2(δ2+βγ)˜qY3,D˜qY3=(αβ+βδ)˜qY1+(αδ+δ)˜qY2+(2α2λ0+β2λ0+βγλ0c)˜qY3.

    Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies

    {α32α3λ0+2αβ2αδ2+2β2δαβ2λ0+αβγ+βδγαβγλ0+αc=0,β3λ0+2α2βλ0+β2γλ0+β2γ2αβδβδβc=0,β3β3λ0+α2ββ2γβδ22α2βλ0β2γλ0+βc=0,2α3λ0α2δ2β2δαβ2+αβ2λ0αδαβγ+αβγλ0αc=0,β3+β3λ0+α2β+β2γ+2α2βλ0+β2γλ03αβδ2βδβc=0,β3λ0+β2γ+βδ2+2α2βλ0+β2γλ0αβδβδβc=0,α2ββ2γβγ22βδ2+2α2γλ0+β2γλ0+βγ2λ0+αβδγc=0,αβ2+2β2δαδ2δ2+2α2δλ0+β2δλ0+αβγ+βδγλ0δc=0,δ3+αβ2+β2δα2δ+2α2δλ0+β2δλ0+αβγ+βγδ+βγδλ0δc=0. (3.7)

    Because α+δ0 as well αγ=0. Let's first suppose that α=0. On this basis, (3.7) can be simplified to

    {2β2δ+βδγ=0,β3λ0+β2γλ0+β2γβδβc=0,β3β3λ0β2γβδ2β2γλ0+βc=0,β2δ=0,β3+β3λ0+β2γ+β2γλ02βδβc=0,β3λ0+β2γ+βδ2+β2γλ0βδβc=0,β2γ+βγ2+2βδ2β2γλ0βγ2λ0+γc=0,2β2δδ2+β2δλ0+βδγλ0δc=0,δ3+β2δ+β2δλ0+βγδ+βγδλ0δc=0.

    If γ0 and δ0, we get case (2) of Theorem 3.35 holds. If γ=0 as well δ0, on this basis, we calculate that

    {β2δ=0,β3λ0βδβc=0,β3β3λ0βδ2+βc=0,β3+β3λ02βδβc=0,β3λ0+βδ2βδβc=0,βδ2=0,2β2δδ2+β2δλ0δc=0,δ3+β2δ+β2δλ0δc=0.

    we obtain case (1) of Theorem 3.35 holds. Assume second that α0, α+δ0 and γ=0. In this case, (3.7) reduces to

    {α32α3λ0+2αβ2αδ2+2β2δαβ2λ0+αc=0,β3λ0+2α2βλ02αβδβδβc=0,β3β3λ0+α2ββδ22α2βλ0+βc=0,2α3λ0α2δ2β2δαβ2+αβ2λ0αδαc=0,β3+β3λ0+α2β+2α2βλ03αβδ2βδβc=0,β3λ0+βδ2+2α2βλ0αβδβδβc=0,α2β2βδ2+αβδ=0,αβ2+2β2δαδ2δ2+2α2δλ0+β2δλ0δc=0,δ3+αβ2+β2δα2δ+2α2δλ0+β2δλ0δc=0.

    Next suppose that β=0, we have

    {α32α3λ0αδ2+αc=0,2α3λ0α2δαδαc=0,αδ2+δ22α2δλ0+δc=0,δ3α2δ+2α2δλ0δc=0.

    Then we get

    α3δ32αδ2δ2αδ=0.

    Let α=λδ, λ0, it becomes

    (λ32λ1)δ3(λ+1)δ2=0,

    for the cases (3) of Theorem 3.35 holds. Thus it turns out Theorem 3.35.

    In the section, we use the soliton equation in an effort to finish a categorization about three-dimensional LLG that support ASS of the second kind associated with YC.

    Let

    ˜ρY(UY,VY)=ρY(UY,VY)+ρY(VY,UY)2, (4.1)

    and

    ˜ρY(UY,VY)=gY(~RicY(UY),VY). (4.2)

    Similar to the formulae (2.6), we have

    ˜SY(˜qYi,˜qYj)=˜ρY(˜qYi,˜qYj)sYλ0gY(˜qYi,˜qYj), (4.3)

    where λ0 is a real number. Refer to [29], we can get

    sY=˜ρY(˜qY1,˜qY1)+˜ρY(˜qY2,˜qY2)˜ρY(˜qY3,˜qY3). (4.4)

    for vector fields UY, VY.

    Definition 4.1. (Gi,gY) is called ASS of the second kind related with YC when it satisfies

    ~RicY=(sYλ0+c)Id+D, (4.5)

    which c is an actual number, λ0 is a real-valued constant, as well D is derivation for gY, which can be

    D[UY,VY]=[DUY,VY]+[UY,DVY], (4.6)

    for UY, VYgY.

    Theorem 4.2. (G1,gY,J) is ASS of the second kind related to YC if it satisfies α0, β=0, α222α2λ0+c=0. And specifically

    ~RicY(˜qY1˜qY2˜qY3)=(α2000α2α220α220)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(α22000α22α220α22α22)(˜qY1˜qY2˜qY3).

    Proof. For (G1,Y), according to (4.1), we have

    ˜ρY(˜qY1,˜qY1)=(α2+β2), ˜ρY(˜qY1,˜qY2)=αβ, ˜ρY(˜qY1,˜qY3)=αβ2,˜ρY(˜qY2,˜qY2)=(α2+β2), ˜ρY(˜qY2,˜qY3)=α22, ˜ρY(˜qY3,˜qY3)=0. (4.7)

    By (4.2), the Ricci operator can be expressed as

    ~RicY(˜qY1˜qY2˜qY3)=(α2β2αβαβ2αβα2β2α22αβ2α220)(˜qY1˜qY2˜qY3).

    As a result, the scalar curvature can be obtained as sY=2α22β2. If (G1,gY,J) is ASS of the second kind related to the YC, and by ~RicY=(sYλ0+c)Id+D, we can get

    {D˜qY1=[α2β2+(2α2+2β2)λ0c]˜qY1+αβ˜qY2+αβ2˜qY3,D˜qY2=αβ˜qY1+[α2β2+(2α2+2β2)λ0c]˜qY2α22˜qY3,D˜qY3=αβ2˜qY1+α22˜qY2+[(2α2+2β2)λ0c]˜qY3.

    Therefore, by (4.6) and ASS of the second kind related to the YC can be established if it satisfies

    {2α3λ0α322αβ2+2αβ2λ0αc=0,α2β=0,β3=0,2β3λ02α2β+2α2βλ0βc=0,2β3λ0α2β+2α2βλ0βc=0. (4.8)

    Since α0, by solving the second and third equations of (4.8) imply that β=0. In this case, the first equation of (4.8) can be simplified to

    2α3λ0α32αc=0,

    we have α222α2λ0+c=0. Thus we get Theorem 4.2.

    Theorem 4.3. (G2,gY,J) is ASS of the second kind related to YC if it satisfies α=β=0, γ0, c=γ2(2λ01). And specifically

    ~RicY(˜qY1˜qY2˜qY3)=(γ2000γ20000)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(00000000γ2)(˜qY1˜qY2˜qY3).

    Proof. For (G2,Y), according to (4.1), we can get

    ˜ρY(˜qY1,˜qY1)=β2γ2, ˜ρY(˜qY1,˜qY2)=0, ˜ρY(˜qY1,˜qY3)=0,˜ρY(˜qY2,˜qY2)=γ22αβ, ˜ρY(˜qY2,˜qY3)=βγαγ, ˜ρY(˜qY3,˜qY3)=0.

    By (4.2), the Ricci operator can be expressed as

    ~RicY(˜qY1˜qY2˜qY3)=(β2γ2000γ22αβαγβγ0βγαγ0)(˜qY1˜qY2˜qY3).

    As a result, the scalar curvature can be obtained as sY=β22γ22αβ. If (G2,gY,J) is ASS of the second kind related to the YC, and by ~RicY=(sYλ0+c)Id+D, we can get

    {D˜qY1=(β2γ2β2λ0+2γ2λ0+2αβλ0c)˜qY1,D˜qY2=(γ22αββ2λ0+2γ2λ0+2αβλ0c)˜qY2+(αγβγ)˜qY3,D˜qY3=(βγαγ)˜qY2+(β2λ0+2γ2λ0+2αβλ0c)˜qY3.

    Therefore, by (4.6) and ASS of the second kind related to the YC can be established if it satisfies

    {β3β3λ0+2αγ24βγ22αβ2+2βγ2λ0+2αβ2λ0βc=0,β3β3λ02βγ2+2αγ2+2αβ2+2βγ2λ0+2αβ2λ0βc=0,γ32γ3λ03β2γ+β2γλ0+2αβγ2αβγλ0+γc=0,αβ22α2βαβ2λ0+2αγ2λ0+2α2βλ0αc=0. (4.9)

    By solving the first and second equations of (4.9) imply that

    2αβ2+βγ2=0.

    Since γ0, we have β=0. In this case, the first equation of (4.9) reduces to

    2αγ2=0,

    we get α=0. In this case, the third equation of (4.9) can be simplified to

    γ32γ3λ0+γc=0,

    then we have c=γ2(2λ01). Thus we get Theorem 4.3.

    Theorem 4.4. (G3,gY,J) is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.

    Proof. For (G3,Y), according to (4.1), we have

    ˜ρY(˜qY1,˜qY1)=βγ, ˜ρY(˜qY1,˜qY2)=0, ˜ρY(˜qY1,˜qY3)=0,˜ρY(˜qY2,˜qY2)=γ2αγ, ˜ρY(˜qY2,˜qY3)=0, ˜ρY(˜qY3,˜qY3)=0.

    By (4.2), the Ricci operator can be expressed as

    ~RicY(˜qY1˜qY2˜qY3)=(βγ000γ2αγ0000)(˜qY1˜qY2˜qY3).

    Since ρY(˜qYi,˜qYj)=ρY(˜qYj,˜qYi), then ˜ρY(˜qYi,˜qYj)=ρY(˜qYi,˜qYj). So (G3,gY,J) is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.

    Theorem 4.5. The LG G4 cannot be ASS of a second kind related to the YC.

    Proof. For (G4,Y), according to (4.1), we can get

    ˜ρY(˜qY1,˜qY1)=2βηβ21, ˜ρY(˜qY1,˜qY2)=0, ˜ρY(˜qY1,˜qY3)=0,˜ρY(˜qY2,˜qY2)=2αηαβ1, ˜ρY(˜qY2,˜qY3)=α2, ˜ρY(˜qY3,˜qY3)=0.

    By (4.2), the Ricci operator can be expressed as

    ~RicY(˜qY1˜qY2˜qY3)=(β2+2βη10002αηαβ1α20α20)(˜qY1˜qY2˜qY3).

    As a result, the scalar curvature can be obtained as sY=β2+2αη+2βηαβ2. If (G4,gY,J) is ASS of the second kind related to the YC, and by ~RicY=(sYλ0+c)Id+D, we can get

    {D˜qY1=(β2+β2λ0+2βη2αηλ02βηλ0+αβλ0+2λ01c)˜qY1,D˜qY2=(β2λ0+2αηαβ2αηλ02βηλ0+αβλ0+2λ01c)˜qY2α2˜qY3,D˜qY3=α2˜qY2+(β2λ02αηλ02βηλ0+αβλ0+2λ0c)˜qY3.

    Therefore, by (4.6) and ASS of the second kind related to the YC can be established if it satisfies

    {α+(2ηβ)(β2β2λ02αη2βη+αβ+2αηλ0+2βηλ0αβλ02λ0+2+c)=0,β3β3λ0αβ22β2η+2β2ηλ0αβ2λ0+2αβη+2αβηλ02βλ0α+βc=0,β2λ0β2+αη+2βηαβ2αηλ02βηλ0+αβλ0+2λ01c=0,2α2ηα2β+αβ22α2ηλ0+α2βλ0+αβ2λ02αβη2αβηλ0+2αλ0αc=0. (4.10)

    By the first equation of (4.10), we assume that

    α=0,β=2η.

    On this basis, by the second equation of (4.10), we get c=2λ0. By the third equation of (4.10), we have c=2λ01, and there is a contradiction. One can prove Theorem 4.5.

    Theorem 4.6. (G5,gY,J) is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.

    Proof. For (G5,Y), according to (4.1), we can get

    ˜ρY(˜qY1,˜qY1)=˜ρY(˜qY1,˜qY2)=˜ρY(˜qY1,˜qY3)=0,˜ρY(˜qY2,˜qY2)=˜ρY(˜qY2,˜qY3)=˜ρY(˜qY3,˜qY3)=0.

    By (4.2), the Ricci operator can be expressed as

    ~RicY(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3).

    Since ρY(˜qYi,˜qYj)=ρY(˜qYj,˜qYi)=0, then ˜ρY(˜qYi,˜qYj)=ρY(˜qYi,˜qYj)=0. So (G5,gY,J) is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.

    Theorem 4.7. (G6,gY,J) is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.

    Proof. For (G6,Y), according to (4.1), we have

    ˜ρY(˜qY1,˜qY1)=(βγ+α2), ˜ρY(˜qY1,˜qY2)=0, ˜ρY(˜qY1,˜qY3)=0,˜ρY(˜qY2,˜qY2)=α2, ˜ρY(˜qY2,˜qY3)=0, ˜ρY(˜qY3,˜qY3)=0.

    By (4.2), the Ricci operator can be expressed as

    ~RicY(˜qY1˜qY2˜qY3)=(α2βγ000α20000)(˜qY1˜qY2˜qY3).

    Since ρY(˜qYi,˜qYj)=ρY(˜qYj,˜qYi), then ˜ρY(˜qYi,˜qYj)=ρY(˜qYi,˜qYj). So (G6,gY,J) is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.

    Theorem 4.8. (G7,gY,J) is ASS of the second kind related to YC if it satisfies α=β=γ=c=0, δ0. And specifically

    ~RicY(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3),
    D(˜qY1˜qY2˜qY3)=(00000δ2+δ20δ2+δ20)(˜qY1˜qY2˜qY3).

    Proof. For (G7,Y), according to (4.1), we can get

    ˜ρY(˜qY1,˜qY1)=α2, ˜ρY(˜qY1,˜qY2)=βδαβ2, ˜ρY(˜qY1,˜qY3)=αβ+βδ,˜ρY(˜qY2,˜qY2)=α2β2βγ, ˜ρY(˜qY2,˜qY3)=δ2+δ+αδ+βγ2, ˜ρY(˜qY3,˜qY3)=0.

    According to (4.2), the Ricci operator can be expressed as

    ~RicY(˜qY1˜qY2˜qY3)=(α2βδαβ2(αβ+βδ)βδαβ2α2β2βγδ2+δ+αδ+βγ2αβ+βδδ2+δ+αδ+βγ20)(˜qY1˜qY2˜qY3).

    As a result, the scalar curvature can be obtained as sY=2α2β2βγ. If (G7,gY,J) is ASS of the second kind related to the YC, and by ~RicY=(sYλ0+c)Id+D, we can get

    {D˜qY1=(α2+2α2λ0+β2λ0+βγλ0c)˜qY1+βδαβ2˜qY2(αβ+βδ)˜qY3,D˜qY2=βδαβ2˜qY1+(α2β2+2α2λ0+β2λ0βγ+βγλ0c)˜qY2δ2+δ+αδ+βγ2˜qY3,D˜qY3=(αβ+βδ)˜qY1+δ2+δ+αδ+βγ2˜qY2+(2α2λ0+β2λ0+βγλ0c)˜qY3.

    Therefore, by (4.6) and ASS of the second kind related to the YC can be established if it satisfies

    {α32α3λ0+32αβ212αδ212α2δ+32β2δαβ2λ012αδ+32αβγ+βδγαβγλ0+αc=0,β3λ0+2α2βλ0+β2γλ012α2ββδ232αβδβc=0,β3β3λ0+α2β2α2βλ0β2γλ0αβδβδ+βc=0,2α3λ012α2δ12αδ232β2δ12αβ2+αβ2λ012αδαβγ+12βδγ+αβγλ0αc=0,β3+β3λ0+12α2β12δ2β+2α2βλ0+β2γλ02αβδβc=0,β3λ0+12βδ2+2α2βλ0+β2γλ0+12αβδβc=0,12α2ββ2γβγ232βδ2+2α2γλ0+β2γλ0+βγ2λ0+αβδγc=0,12δ312αβ232β2δ+12αδ2+12δ22α2δλ0β2δλ0+βγδ12αβγβδγλ0+δc=0,12δ3+12αβ2+12β2δα2δ+12δ2+2α2δλ0+β2δλ0+βδ+12αδ+αβγ12βγδ+βγδλ0δc=0. (4.11)

    Because α+δ=0 as well αγ=0. Let's first suppose α=0. On this basis, (4.11) reduces to

    {32β2δ+βδγ=0,β3λ0+β2γλ0βδ2βc=0,β3β3λ0β2γλ0βδ+βc=0,32β2δ12βδγ=0,β3+β3λ012δ2β+β2γλ0βc=0,β3λ0+12βδ2+β2γλ0βc=0,β2γ+βγ2+32βδ2β2γλ0βγ2λ0+γc=0,12δ332β2δ+12δ2β2δλ0+βγδβδγλ0+δc=0,12δ3+12β2δ+12δ2+β2δλ0+βδ12βγδ+βγδλ0δc=0. (4.12)

    If γ0 as well δ0, on this basis, the first and fourth equations of (4.12) can be simplified to

    βδγ=0,

    we get β=0. The seventh equation of (4.12) reduces to

    γc=0,

    we obtain c=0. The eighth and ninth equations of (4.12) can be simplified to

    δ3δ22δc=0,

    we have c=12(δ2+δ), and there is a contradiction. If γ=0 as well δ0, on this basis, we calculate that

    {32β2δ=0,β3λ0βδ2βc=0,β3β3λ0βδ+βc=0,β3+β3λ012δ2ββc=0,β3λ0+12βδ2βc=0,32βδ2=0,12δ332β2δ+12δ2β2δλ0+δc=0,12δ3+12β2δ+12δ2+β2δλ0+βδδc=0. (4.13)

    By solving (4.13), we obtain β=c=0. Suppose second that α0, α+δ0 as well γ=0. In this case, (4.11) can be simplified to

    {α32α3λ0+32αβ212αδ212α2δ+32β2δαβ2λ012αδ+αc=0,β3λ0+2α2βλ012α2ββδ232αβδβc=0,β3β3λ0+α2β2α2βλ0αβδβδ+βc=0,2α3λ012α2δ12αδ232β2δ12αβ2+αβ2λ012αδαc=0,β3+β3λ0+12α2β12δ2β+2α2βλ02αβδβc=0,β3λ0+12βδ2+2α2βλ0+12αβδβc=0,12α2β32βδ2+αβδ=0,12δ312αβ232β2δ+12αδ2+12δ22α2δλ0β2δλ0+δc=0,12δ3+12αβ2+12β2δα2δ+12δ2+2α2δλ0+β2δλ0+βδ+12αδδc=0.

    Next suppose that β=0, we have

    {α32α3λ012αδ212α2δ12αδ+αc=0,2α3λ012α2δ12αδ212αδαc=0,12δ3+12αδ2+12δ22α2δλ0+δc=0,12δ3α2δ+12δ2+2α2δλ0+12αδδc=0. (4.14)

    Then we get

    α3+δ312αδ2+δ2α2δ12αδ=0.

    Let α=λδ, λ0, it becomes

    (λ3λ212λ+1)δ3+(112λ)δ2=0,

    we have δ=λ22λ32λ2λ+2. For α=λδ, (4.14) now reduces to

    {λ2δ212δ212δ+12λδ22λ2δ2λ0+c=0,12δ2+12δ+12λδ22λ2δ2λ0+c=0,λ2δ212δ212δ12λδ2λ2δ2λ0+c=0.

    A simple computation demonstrates that the result is δ=1, then we get λ=1, α=1. In this case, we have c=32+2λ0 and c=12+2λ0, so there is a contradiction. Thus it turns out Theorem 4.8.

    We focus on the existence conditions of ASS related to YC in the context of three-dimensional LLG. We classify those ASS in three-dimensional LLG. The major results demonstrate that ASS related to YC are present in G1, G2, G3, G5, G6 and G7, while they are not identifiable in G4. Based on this research, we will explore gradient Schouten solitons associated with YC using the theories in [31,32,33].

    During writing this work, the authors confirm that they are not using any AI techniques.

    The study was funded by the Special Fund for Scientific and Technological Innovation of Graduate Students in Mudanjiang Normal University (Grant No. kjcx2022-018mdjnu), the "Four New" Special Project of in Mudanjiang Normal University(Grant No. 22-XJ22024) and the Project of Science and Technology of Mudanjiang Normal University(Grant No. GP2022006).

    The authors declare that there are no conflicts of interest.



    [1] S. Zhang, T. C. Xia, A generalized new auxiliary equation method and its applications to nonlinear partial differential equations, Phys. Lett. A, 363 (2007), 356–360. doi: 10.1016/j.physleta.2006.11.035
    [2] Sirendaoreji, S. Jiong, Auxiliary equation method for solving nonlinear partial differential equations, Phys. Lett. A, 309 (2003), 387–396. doi: 10.1016/S0375-9601(03)00196-8
    [3] M. L. Wang, Y. B. Zhou, Z. B. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A, 216 (1996), 67–75. doi: 10.1016/0375-9601(96)00283-6
    [4] M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A, 213 (1996), 279–287. doi: 10.1016/0375-9601(96)00103-X
    [5] X. H. Wu, J. H. He, EXP-function method and its application to nonlinear equations, Chaos Solitons Fract., 38 (2008), 903–910. doi: 10.1016/j.chaos.2007.01.024
    [6] X. H. Wu, J. H. He, Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method, Comput. Math. Appl., 54 (2007), 966–986. doi: 10.1016/j.camwa.2006.12.041
    [7] H. A. Abdusalam, On an improved complex tanh-function method, Int. J. Nonlin. Sci. Num., 6 (2005), 99–106.
    [8] H. C. Hu, X. Y. Tang, S. Y. Lou, Q. P. Liu, Variable separation solutions obtained from Darboux transformations for the asymmetric Nizhnik-Novikov-Veselov system. Chaos Solitons Fract., 22 (2004), 327–334.
    [9] S. B. Leble, N. V. Ustinov, Darboux transforms, deep reductions and solitons, J. Phys. A: Math. Gen, 26 (1993), 5007–5016. doi: 10.1088/0305-4470/26/19/029
    [10] J. Lee, R. Sakthivel, New exact travelling wave solutions of bidirectional wave equations, Pramana-J. Phys., 76 (2011), 819–829. doi: 10.1007/s12043-011-0105-4
    [11] A. Bekir, O. Unsal, Analytic treatment of nonlinear evolution equations using first integral method, Pramana-J. phys., 79 (2012), 3–17. doi: 10.1007/s12043-012-0282-9
    [12] S. Abbasbandy, A. Shirzadi, The first integral method for modified Benjamin-Bona-Mahony equation, Commun. Nonlinear Sci., 15 (2010), 1759–1764. doi: 10.1016/j.cnsns.2009.08.003
    [13] E. J. Parkes, B. R. Duffy, P. C. Abbott, The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations, Phys. Lett. A, 295 (2002), 280–286. doi: 10.1016/S0375-9601(02)00180-9
    [14] S. K. Liu, Z. T. Fu, S. D. Liu, Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289 (2001), 69–74. doi: 10.1016/S0375-9601(01)00580-1
    [15] A. J. M. Jawad, M. D. Petkovic, A. Biswas, Modified simple equation method for nonlinear evolution equations, Appl. Math. Comput., 217 (2010), 869–877.
    [16] K. Khan, M. A. Akbar, Exact and solitary wave solutions for the Tzitzeica-Dodd-Bullough and the modified KdV-Zakharov-Kuznetsov equations using the modified simple equation method, Ain Shams Eng. J., 4 (2013), 903–909. doi: 10.1016/j.asej.2013.01.010
    [17] K. Khan, M. A. Akbar, Exact solutions of the (2+1)-dimensional cubic Klein-Gordon equation and the (3+1)-dimensional Zakharov-Kuznetsov equation Using the modified simple equation method, J. Assoc. Arab Uni. Basic Appl. Sci., 15 (2014), 74–81.
    [18] K. Khan, M. A. Akbar, Application of exp(-φ(ξ))-expansion method to find the exact solutions of modified Benjamin-Bona- Mahony equation, World Appl. Sci. J., 24 (2013), 1373–1377.
    [19] M. L. Wang, X. Z. Li, J. L. Zhang, The GG-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2008), 417–423. doi: 10.1016/j.physleta.2007.07.051
    [20] H. Kim, R. Sakthivel, New exact travelling wave solutions of some nonlinear higher dimensional physical models, Rep. Math. Phys., 70 (2012), 39–50. doi: 10.1016/S0034-4877(13)60012-9
    [21] K. Khan, M. A. Akbar, Traveling wave solutions of nonlinear evolution equations via the enhanced (G/G)-expansion method, J. Egypt. Math. Soc., 22 (2014), 220–226. doi: 10.1016/j.joems.2013.07.009
    [22] M. E. Islam, K. Khan, M. A. Akbar, R. Islam, Traveling wave solutions of nonlinear evolution equations via enhanced (G/G)- expansion method, GANIT J. Bangladesh Math. Soc., 33, (2013), 83–92.
    [23] H. Naher, F. A. Abdullah, New generalized and improved (G/G)-expansion method for nonlinear evolution equations in mathematical physics, J. Egypt. Math. Soc., 22 (2014), 390–395. doi: 10.1016/j.joems.2013.11.008
    [24] H. Naher, F. A. Abdullah, A. Bekir, Abundant traveling wave solutions of the compound KdV-Burgers equation via the improved (G/G)-expansion method, AIP Adv., 2 (2012), 042163. doi: 10.1063/1.4769751
    [25] H. Naher, F. A. Abdullah, Some new traveling solutions of the nonlinear reaction diffusion equation by using the improved (G/G)-expansion method, Math. Probl. Eng., 2012 (2012), 871724.
    [26] H. Naher, F. A. Abdullah, M. A. Akbar, Generalized and improved (G/G)-expansion method for (3+ 1)-dimensional modified KdV-Zakharov-Kuznetsev equation, PloS One, 8 (2013), e64618. doi: 10.1371/journal.pone.0064618
    [27] Y. Molliq R, M. S. M. Noorani, I. Hashim, Variational iteration method for fractional heat-and wave-like equations, Nonlinear Anal-Real., 10 (2009), 1854–1869. doi: 10.1016/j.nonrwa.2008.02.026
    [28] S. T. Mohiud-Din, M. A. Noor, Homotopy perturbation method for solving fourth-order boundary value problems, Math. Probl. Eng., 2007 (2007), 098602.
    [29] S. T. Mohyud-Din, M. A. Noor, Homotopy perturbation method for solving partial differential equations, Z. Naturforsch, 64a (2009), 157–170.
    [30] S. T. Mohyud-Din, A. Yildrim, S. Sariaydin, Approximate series solutions of the viscous Cahn-Hilliard equation via the homotopy perturbation method, World Appl. Sci. J., 11 (2010), 813–818.
    [31] C. Changbum, R. Sakthivel, Homotopy perturbation technique for solving two-point boundary value problems-comparison with other methods, Comput. Phys. Commun., 181 (2010), 1021–1024. doi: 10.1016/j.cpc.2010.02.007
    [32] R. Sakthivel, C. Changbum, L. Jonu, New travelling wave solutions of Burgers equation with finite transport memory, Z. Naturforsch, 65 (2010), 633–640. doi: 10.1515/zna-2010-8-903
    [33] Y. M. Zhao, F-expansion method and its application for finding new exact solutions to the Kudryashov-Sinelshchikov equation, J. Appl. Math., 2013 (2013), 895760.
    [34] H. W. Hua, A generalized extended F-expansion method and its application in (2+1)-dimensional dispersive long wave equation, Commun. Theor. Phys., 46 (2006), 580. doi: 10.1088/0253-6102/46/4/002
    [35] M. S. Islam, k. Khan, M. A. Akbar, A. Mastroberardino, A note on improved F-expansion method combined with Riccati equation applied to nonlinear evolution equations, R. Soc. Open Sci., 1, (2014), 140038.
    [36] W. J. Liu, Y. N. Zhu, M. L. Liu, B. Wen, S. B. Fang, H. Teng, et al., Optical properties and applications for MoS2-Sb2Te3-MoS2 heterostructure materials, Photonics Res., 6 (2018), 220–227. doi: 10.1364/PRJ.6.000220
    [37] W. J. Liu, L. H. Pang, H. N. Han, M. L. Liu, M. Lei, S. B. Fang, et al., Tungsten disulfide saturable absorbers for 67 fs mode-locked erbium-doped fiber lasers, Opt. Express, 25 (2017), 2950–2959. doi: 10.1364/OE.25.002950
    [38] N. Raza, S. Arshed, A. Javid, Optical solitons and stability analysis for the generalized second-order nonlinear Schrodinger equation in an optical fiber, Int. J. Nonlin. Sci. Num., 21 (2020), 855–863. doi: 10.1515/ijnsns-2019-0287
    [39] W. J. Liu, W. T. Yu, C. Y. Yang, M. L. Liu, Y. J. Zhang, M. Lei, Analytic solutions for the generalized complex Ginzburg-Landau equation in fiber lasers, Nonlinear Dyn., 89 (2017), 2933–2939. doi: 10.1007/s11071-017-3636-5
    [40] X. Y. Fan, T. Q. Qu, S. C. Huang, X. X. Chen, M. H. Cao, Q. Zhou, et al., Analytic study on the influences of higher-order effects on optical solitons in fiber laser, Optik, 186 (2019), 326–331. doi: 10.1016/j.ijleo.2019.04.102
    [41] Y. Khan, Fractal modification of complex Ginzburg-Landau model arising in the oscillating phenomena, Results Phys., 18 (2020), 103324. doi: 10.1016/j.rinp.2020.103324
    [42] M. Arshad, A. R. Seadawy, D. C. Lu, Exact bright-dark solitary wave solutions of the higher-order cubic-quintic nonlinear Schrodinger equation and its stability, Optik, 138 (2017), 40–49. doi: 10.1016/j.ijleo.2017.03.005
    [43] A. R. Seadawy, M. Arshad, D. C. Lu, Modulation stability analysis and solitary wave solutions of nonlinear higher-order Schrodinger dynamical equation with second-order spatiotemporal dispersion, Indian. J. Phys., 93 (2019), 1041–1049. doi: 10.1007/s12648-018-01361-y
    [44] K. Porsezian, A. K. Shafeeque Ali, A. I. Maimistov, Modulation instability in two-dimensional waveguide arrays with alternating signs of refractive index, J. Opt. Soc. Am. B, 35 (2018), 2057–2064. doi: 10.1364/JOSAB.35.002057
    [45] A. Hasegawa, F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers, Appl. Phys. Lett., 23 (1973), 142. doi: 10.1063/1.1654836
    [46] A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow, N. Akhmediev, Infinite hierarchy of nonlinear Schrodinger equations and their solutions, Phys. Rev. E, 93 (2016), 012206. doi: 10.1103/PhysRevE.93.012206
    [47] A. Ankiewicz, N. Akhmediev, Higher-order integrable evolution equation and its soliton solutions, Phys. Lett. A, 378 (2014), 358–361. doi: 10.1016/j.physleta.2013.11.031
    [48] M. A. Banaja, S. A. Alkhateeb, A. A. Alshaery, E. M. Hilal, A. H. Bhrawy, L. Moraru, et al., Optical solitons in dual-core couplers, Wulfenia J., 21 (2014), 366–380.
    [49] A. H. Bhrawy, A. A. Alshaery, E. M. Hilal, M. Savescu, D. Milovic, K. R. Khang, et al., Optical solitons in birefringent fibers with spatio-temporal dispersion, Optik, 125 (2014), 4935–4944. doi: 10.1016/j.ijleo.2014.04.025
    [50] T. A. Sulaiman, T. Akturk, H. Bulut, H. M. Baskonus, Investigation of various soliton solutions to the Heisenberg ferromagnetic spin chain equation, J. Electromagnet Wave., 32 (2017), 1093–1105.
    [51] M. A. Banaja, A. A. Al Qarni, H. O. Bakodah, Q. Zhou, S. P. Moshokoa, A. Biswas, The investigate of optical solitons in cascaded system by improved adomian decomposition scheme, Optik, 130 (2017), 1107–1114. doi: 10.1016/j.ijleo.2016.11.125
    [52] A. R. Seadawy, D. C. Lu, Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Schrodinger equation and its stability, Results Phys., 7 (2017), 43–48. doi: 10.1016/j.rinp.2016.11.038
    [53] A. Biswas, S. Arshed, Optical solitons in presence of higher order dispersions and absence of self-phase modulation, Optik, 174 (2018), 452–459. doi: 10.1016/j.ijleo.2018.08.037
    [54] M. Kaplan, A. Bekir, A. Akbulut, A generalized Kudryashov method to some nonlinear evolution equations in mathematical physics, Nonlinear Dyn., 85 (2016), 2843–2850. doi: 10.1007/s11071-016-2867-1
    [55] A. Biswas, M. Ekici, A. Sonmezoglu, R. T. Alqahtanib, Sub-pico-second chirped optical solitons in mono-mode fibers with Kaup-Newell equation by extended trial function method, Optik, 168 (2018), 208–216. doi: 10.1016/j.ijleo.2018.04.069
    [56] H. Ur Rehman, N. Ullah, M. A. Imran, Optical solitons of Biswas-Arshed equation in birefringent fibers using extended direct algebraic method, Optik, 226 (2021), 165378. doi: 10.1016/j.ijleo.2020.165378
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