Research article Special Issues

Propagation patterns of dromion and other solitons in nonlinear Phi-Four (ϕ4) equation

  • The Phi-Four (also embodied as ϕ4) equation (PFE) is one of the most significant models in nonlinear physics, that emerges in particle physics, condensed matter physics and cosmic theory. In this study, propagating soliton solutions for the PFE were obtained by employing the extended direct algebraic method (EDAM). This transformational method reformulated the model into an assortment of nonlinear algebraic equations using a series-form solution. These equations were then solved with the aid of Maple software, producing a large number of soliton solutions. New families of soliton solutions, including exponential, rational, hyperbolic, and trigonometric functions, are included in these solutions. Using 3D, 2D, and contour graphs, the shape, amplitude, and propagation behaviour of some solitons were visualized which revealed the existence of kink, shock, bright-dark, hump, lump-type, dromion, and periodic solitons in the context of PFE. The study was groundbreaking as it extended the suggested strategy to the PFE that was being aimed at, yielding a significant amount of soliton wave solutions while providing new insights into the behavioral characteristics of soliton. This approach surpassed previous approaches by offering a systematic approach to solving nonlinear problems in analogous challenging situations. Furthermore, the results also showed that the suggested method worked well for building families of propagating soliton solutions for intricate models such as the PFE.

    Citation: Mohammed Aldandani, Abdulhadi A. Altherwi, Mastoor M. Abushaega. Propagation patterns of dromion and other solitons in nonlinear Phi-Four (ϕ4) equation[J]. AIMS Mathematics, 2024, 9(7): 19786-19811. doi: 10.3934/math.2024966

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  • The Phi-Four (also embodied as ϕ4) equation (PFE) is one of the most significant models in nonlinear physics, that emerges in particle physics, condensed matter physics and cosmic theory. In this study, propagating soliton solutions for the PFE were obtained by employing the extended direct algebraic method (EDAM). This transformational method reformulated the model into an assortment of nonlinear algebraic equations using a series-form solution. These equations were then solved with the aid of Maple software, producing a large number of soliton solutions. New families of soliton solutions, including exponential, rational, hyperbolic, and trigonometric functions, are included in these solutions. Using 3D, 2D, and contour graphs, the shape, amplitude, and propagation behaviour of some solitons were visualized which revealed the existence of kink, shock, bright-dark, hump, lump-type, dromion, and periodic solitons in the context of PFE. The study was groundbreaking as it extended the suggested strategy to the PFE that was being aimed at, yielding a significant amount of soliton wave solutions while providing new insights into the behavioral characteristics of soliton. This approach surpassed previous approaches by offering a systematic approach to solving nonlinear problems in analogous challenging situations. Furthermore, the results also showed that the suggested method worked well for building families of propagating soliton solutions for intricate models such as the PFE.



    The concept of lightlike submanifolds in geometry was initially established and expounded upon in a work produced by Duggal and Bejancu [1]. A nondegenerate screen distribution was employed in order to produce a nonintersecting lightlike transversal vector bundle of the tangent bundle. They defined the CR-lightlike submanifold as a generalization of lightlike real hypersurfaces of indefinite Kaehler manifolds and showed that CR-lightlike submanifolds do not contain invariant and totally real lightlike submanifolds. Further, they defined and studied GCR-lightlike submanifolds of Kaehler manifolds as an umbrella of invariant submanifolds, screen real submanifolds, and CR-lightlike and SCR-lightlike submanifolds in [2,3], respectively. Subsequently, B. Sahin and R. Gunes investigated geodesic property of CR-lightlike submanifolds [4] and the integrability of distributions in CR-lightlike submanifolds [5]. In the year 2010, Duggal and Sahin published a book [6]pertaining to the field of differential geometry, specifically focusing on the study of lightlike submanifolds. This book provides a comprehensive examination of recent advancements in lightlike geometry, encompassing novel geometric findings, accompanied by rigorous proofs, and exploring their practical implications in the field of mathematical physics. The investigation of the geometric properties of lightlike hypersurfaces and lightlike submanifolds has been the subject of research in several studies (see [7,8,9,10,11,12,13,14]).

    Crasmareanu and Hretcanu[15] created a special example of polynomial structure [16] on a differentiable manifold, and it is known as the golden structure (¯M,g). Hretcanu C. E. [17] explored Riemannian submanifolds with the golden structure. M. Ahmad and M. A. Qayyoom studied geometrical properties of Riemannian submanifolds with golden structure [18,19,20,21] and metallic structure [22,23]. The integrability of golden structures was examined by A. Gizer et al. [24]. Lightlike hypersurfaces of a golden semi-Riemannian manifold was investigated by N. Poyraz and E. Yasar [25]. The golden structure was also explored in the studies [26,27,28,29].

    In this research, we investigate the CR-lightlike submanifolds of a golden semi-Riemannian manifold, drawing inspiration from the aforementioned studies. This paper has the following outlines: Some preliminaries of CR-lightlike submanifolds are defined in Section 2. We establish a number of properties of CR-lightlike submanifolds on golden semi-Riemannian manifolds in Section 3. In Section 4, we look into several CR-lightlike submanifolds characteristics that are totally umbilical. We provide a complex illustration of CR-lightlike submanifolds of a golden semi-Riemannian manifold in the final section.

    Assume that (¯,g) is a semi-Riemannian manifold with (k+j)-dimension, k,j1, and g as a semi-Riemannian metric on ¯. We suppose that ¯ is not a Riemannian manifold and the symbol q stands for the constant index of g.

    [15] Let ¯ be endowed with a tensor field ψ of type (1,1) such that

    ψ2=ψ+I, (2.1)

    where I represents the identity transformation on Γ(Υ¯). The structure ψ is referred to as a golden structure. A metric g is considered ψ-compatible if

    g(ψγ,ζ)=g(γ,ψζ) (2.2)

    for all γ, ζ vector fields on Γ(Υ¯), then (¯,g,ψ) is called a golden Riemannian manifold. If we substitute ψγ into γ in (2.2), then from (2.1) we have

    g(ψγ,ψζ)=g(ψγ,ζ)+g(γ,ζ). (2.3)

    for any γ,ζΓ(Υ¯).

    If (¯,g,ψ) is a golden Riemannian manifold and ψ is parallel with regard to the Levi-Civita connection ¯ on ¯:

    ¯ψ=0, (2.4)

    then (¯,g,ψ) is referred to as a semi-Riemannian manifold with locally golden properties.

    The golden structure is the particular case of metallic structure [22,23] with p=1, q=1 defined by

    ψ2=pψ+qI,

    where p and q are positive integers.

    [1] Consider the case where is a lightlike submanifold of k of ¯. There is the radical distribution, or Rad(Υ), on that applies to this situation such that Rad(Υ)=ΥΥ, p. Since RadΥ has rank r0, is referred to as an r-lightlike submanifold of ¯. Assume that is a submanifold of that is r-lightlike. A screen distribution is what we refer to as the complementary distribution of a Rad distribution on Υ, then

    Υ=RadΥS(Υ).

    As S(Υ) is a nondegenerate vector sub-bundle of Υ¯|, we have

    Υ¯|=S(Υ)S(Υ),

    where S(Υ) consists of the orthogonal vector sub-bundle that is complementary to S(Υ) in Υ¯|. S(Υ),S(Υ) is an orthogonal direct decomposition, and they are nondegenerate.

    S(Υ)=S(Υ)S(Υ).

    Let the vector bundle

    tr(Υ)=ltr(Υ)S(Υ).

    Thus,

    Υ¯=Υtr(Υ)=S(Υ)S(Υ)(Rad(Υ)ltr(Υ).

    Assume that the Levi-Civita connection is ¯ on ¯. We have

    ¯γζ=γζ+h(γ,ζ),γ,ζΓ(Υ) (2.5)

    and

    ¯γζ=Ahζ+γh,γΓ(Υ)andhΓ(tr(Υ)), (2.6)

    where {γζ,Ahγ} and {h(γ,ζ),γh} belongs to Γ(Υ) and Γ(tr(Υ)), respectively.

    Using projection L:tr(Υ)ltr(Υ), and S:tr(Υ)S(Υ), we have

    ¯γζ=γζ+hl(γ,ζ)+hs(γ,ζ), (2.7)
    ¯γ=Aγ+lγ+λs(γ,), (2.8)

    and

    ¯γχ=Aχγ+sγ+λl(γ,χ) (2.9)

    for any γ,ζΓ(Υ),Γ(ltr(Υ)), and χΓ(S(Υ)), where hl(γ,ζ)=Lh(γ,ζ),hs(γ,ζ)=Sh(γ,ζ),lγ,λl(γ,χ)Γ(ltr(T)),sγλs(γ,)Γ(S(Υ)), and γζ,Aγ,AχγΓ(Υ).

    The projection morphism of Υ on the screen is represented by P, and we take the distribution into consideration.

    γPζ=γPζ+h(γ,Pζ),γξ=Aξγ+tγξ, (2.10)

    where γ,ζΓ(Υ),ξΓ(Rad(Υ)).

    Thus, we have the subsequent equation.

    g(h(γ,Pζ),)=g(Aγ,Pζ), (2.11)

    Consider that ¯ is a metric connection. We get

    (γg)(ζ,η)=g(hl(γ,ζ),η)+g(hl(γ,ζη),ζ). (2.12)

    Using the characteristics of a linear connection, we can obtain

    (γhl)(ζ,η)=lγ(hl(ζ,η))hl(¯γζ,η)hl(ζ,¯γη), (2.13)
    (γhs)(ζ,η)=sγ(hs(ζ,η))hs(¯γζ,η)hs(ζ,¯γη). (2.14)

    Based on the description of a CR-lightlike submanifold in [4], we have

    Υ=λλ,

    where λ=Rad(Υ)ψRad(Υ)λ0.

    S and Q stand for the projection on λ and λ, respectively, then

    ψγ=fγ+wγ

    for γ,ζΓ(Υ), where fγ=ψSγ and wγ=ψQγ.

    On the other hand, we have

    ψζ=Bζ+Cζ

    for any ζΓ(tr(Υ)), BζΓ(Υ) and CζΓ(tr(Υ)), unless 1 and 2 are denoted as ψL1 and ψL2, respectively.

    Lemma 2.1. Assume that the screen distribution is totally geodesic and that is a CR-lightlike submanifold of the golden semi-Riemannian manifold, then γζΓ(S(ΥN)), where γ,ζΓ(S(Υ)).

    Proof. For γ,ζΓ(S(Υ)),

    g(γζ,)=g(¯γζh(γ,ζ),)=g(ζ,¯γ).

    Using (2.8),

    g(γζ,)=g(ζ,Aγ+γ)=g(ζ,Aγ).

    Using (2.11),

    g(γζ,)=g(h(γ,ζ),).

    Since screen distribution is totally geodesic, h(γ,ζ)=0,

    g(¯γζ,)=0.

    Using Lemma 1.2 in [1] p.g. 142, we have

    γζΓ(S(Υ)),

    where γ,ζΓ(S(Υ)).

    Theorem 2.2. Assume that is a locally golden semi-Riemannian manifold ¯ with CR-lightlike properties, then γψγ=ψγγ for γΓ(λ0).

    Proof. Assume that γ,ζΓ(λ0). Using (2.5), we have

    g(γψγ,ζ)=g(¯γψγh(γ,ψγ),ζ)g(γψγ,ζ)=g(ψ(¯γγ),ζ)g(γψγ,ζ)=g(ψ(γγ),ζ),g(γψγψ(γγ),ζ)=0.

    Nondegeneracy of λ0 implies

    γψγ=ψ(γγ),

    where γΓ(λ0).

    Definition 3.1. [4] A CR-lightlike submanifold of a golden semi-Riemannian manifold is mixed geodesic if h satisfies

    h(γ,α)=0,

    where h stands for second fundamental form, γΓ(λ), and αΓ(λ).

    For γ,ζΓ(λ) and α,βΓ(λ) if

    h(γ,ζ)=0

    and

    h(α,β)=0,

    then it is known as λ-geodesic and λ-geodesic, respectively.

    Theorem 3.2. Assume is a CR-lightlike submanifold of ¯, which is a golden semi-Riemannian manifold. is totally geodesic if

    (Lg)(γ,ζ)=0

    and

    (Lχg)(γ,ζ)=0

    for α,βΓ(Υ),ξΓ(Rad(Υ)), and χΓ(S(Υ)).

    Proof. Since is totally geodesic, then

    h(γ,ζ)=0

    for γ,ζΓ(Υ).

    We know that h(γ,ζ)=0 if

    g(h(γ,ζ),ξ)=0

    and

    g(h(γ,ζ),χ)=0.
    g(h(γ,ζ),ξ)=g(¯γζγζ,ξ)=g(ζ,[γ,ξ]+¯ξγ=g(ζ,[γ,ξ])+g(γ,[ξ,ζ])+g(¯ζξ,γ)=(Lξg)(γ,ζ)+g(¯ζξ,γ)=(Lξg)(γ,ζ)g(ξ,h(γ,ζ)))2g(h(γ,ζ)=(Lξg)(γ,ζ).

    Since g(h(γ,ζ),ξ)=0, we have

    (Lξg)(γ,ζ)=0.

    Similarly,

     g(h(γ,ζ),χ)=g(¯γζγζ,χ)=g(ζ,[γ,χ])+g(γ,[χ,ζ])+g(¯ζχ,γ)=(Lχg)(γ,ζ)+g(¯ζχ,γ)2g(h(γ,ζ),χ)=(Lχg)(γ,ζ).

    Since g(h(γ,ζ),χ)=0, we get

    (Lχg)(γ,ζ)=0

    for χΓ(S(Υ)).

    Lemma 3.3. Assume that ¯ is a golden semi-Riemannian manifold whose submanifold is CR-lightlike, then

    g(h(γ,ζ),χ)=g(Aχγ,ζ)

    for γΓ(λ),ζΓ(λ) and χΓ(S(Υ)).

    Proof. Using (2.5), we get

    g(h(γ,ζ),χ)=g(¯γζγζ,χ)=g(ζ,¯γχ).

    From (2.9), it follows that

    g(h(γ,ζ),χ)=g(ζ,Aχγ+sγχ+λs(γ,χ))=g(ζ,Aχγ)g(ζ,sγχ)g(ζ,λs(γ,χ))g(h(γ,ζ),χ)=g(ζ,Aχγ),

    where γΓ(λ),ζΓ(λ),χΓ(S(Υ)).

    Theorem 3.4. Assume that is a CR-lightlike submanifold of the golden semi-Riemannian manifold and ¯ is mixed geodesic if

    AξγΓ(λ0ψL1)

    and

    AχγΓ(λ0Rad(Υ)ψL1)

    for γΓ(λ),ξΓ(Rad(Υ)), and χΓ(S(Υ)).

    Proof. For γΓ(λ),ζΓ(λ), and χΓ(S(Υ)), we get

    Using (2.5),

    g(h(γ,ζ),ξ)=g(¯γζγζ,ξ)=g(ζ,¯γξ).

    Again using (2.5), we obtain

    g(h(γ,ζ),ξ)=g(ζ,γξ+h(γ,ξ))=g(ζ,γξ).

    Using (2.10), we have

    g(h(γ,ζ),ξ)=g(ζ,Aξγ+tγξ)g(ζ,Aξγ)=0.

    Since the CR-lightlike submanifold is mixed geodesic, we have

    g(h(γ,ζ),ξ)=0
    g(ζ,Aξγ)=0
    AξγΓ(λ0ψL1),

    where γΓ(λ),ζΓ(λ).

    From (2.5), we get

    g(h(γ,ζ),χ)=g(¯γζγζ,χ)=g(ζ,¯γχ).

    From (2.9), we get

    g(h(γ,ζ),χ)=g(ζ,Aχγ+sγχ+λl(γ,χ))g(h(γ,ζ),χ)=g(ζ,Aχγ).

    Since, is mixed geodesic, then g(h(γ,ζ),χ)=0

    g(ζ,Aχγ)=0.
    AχγΓ(λ0Rad(Υ)ψ1).

    Theorem 3.5. Suppose that is a CR-lightlike submanifold of a golden semi-Riemannian manifold ¯, then is λ-geodesic if Aχη and Aξη have no component in 2ψRad(Υ) for ηΓ(λ),ξΓ(Rad(Υ)), and χΓ(S(Υ)).

    Proof. From (2.5), we obtain

    g(h(η,β),χ)=g(¯ηβγζ,χ)=¯g(γζ,χ),

    where χ,βΓ(λ).

    Using (2.9), we have

    g(h(η,β),χ)=g(β,Aχη+sη+λl(η,χ))g(h(η,β),χ)=g(β,Aχη). (3.1)

    Since is λ-geodesic, then g(h(η,β),χ)=0.

    From (3.1), we get

    g(β,Aχη)=0.

    Now,

    g(h(η,β),ξ)=g(¯ηβηβ,ξ)=g(¯ηβ,ξ)=g(β,¯ηξ).

    From (2.10), we get

    g(h(η,β),ξ)=g(η,Aξη+tηξ)g(h(η,β),ξ)=g(Aξβ,η).

    Since is λ- geodesic, then

    g(h(η,β),ξ)=0
    g(Aξβ,η)=0.

    Thus, Aχη and Aξη have no component in M2ψRad(Υ).

    Lemma 3.6. Assume that ¯ is a golden semi-Riemannian manifold that has a CR-lightlike submanifold . Due to the distribution's integrability, the following criteria hold.

    (ⅰ) ψg(λl(ψγ,χ),ζ)g(λl(γ,χ),ψζ)=g(Aχψγ,ζ)g(Aχγ,ψζ),

    (ⅱ) g(λl(ψγ),ξ)=g(Aχγ,ψξ),

    (ⅲ) g(λl(γ,χ),ξ)=g(Aχψγ,ψξ)g(Aχγ,ψξ),

    where γ,ζΓ(Υ),ξΓ(Rad(Υ)), and χΓ(S(Υ)).

    Proof. From Eq (2.9), we obtain

    g(λl(ψγ,χ),ζ)=g(¯ψγχ+Aχψγsψγχ,ζ)=g(χ,¯ψγζ)+g(Aχψγ,ζ).

    Using (2.5), we get

    g(λl(ψγ,χ),ζ)=g(χ,ψγζ+h(ψγ,ζ))+g(Aχψγ,ζ)=g(χ,h(γ,ψζ))+g(Aχψγ,ζ).

    Again, using (2.5), we get

    g(λl(ψγ,χ),ζ)=g(χ,¯γψζγψζ)+g(Aχψγ,ζ)=g(¯γχ,ψζ)+g(Aχψγ,ζ).

    Using (2.9), we have

    g(λl(ψγ,χ),ζ)=g(Aχγ+sγχ+λl(γ,χ),ψζ)+g(λl(ψγ,χ),ζ)g(λl(γ,χ),ψζ)=g(Aχψγ,ζ)g(Aχγ,ψζ).

    (ⅱ) Using (2.9), we have

    g(λl(ψγ,χ),ξ)=g(Aχψγsψγχ+ψγχ,ξ)=g(Aχψγ,ξ)g(χ,¯ψγξ).

    Using (2.10), we get

    g(λl(ψγ,χ),ξ)=g(Aχψγ,ξ)+g(χ,Aξψγ)g(χ,tψγ,ξ)g(λl(ψγ),ξ)=g(Aχγ,ψξ).

    (ⅲ) Replacing ζ by ψξ in (ⅰ), we have

    ψg(λl(ψγ,χ),ψξ)g(λl(γ,χ),ψ2ξ)=g(Aχψγ,ψξ)g(Aχγ,ψ2ξ).

    Using Definition 2.1 in [18] p.g. 9, we get

    ψg(λl(ψγ,χ),ψξ)g(λl(γ,χ),(ψ+I)ξ)=g(Aχψγ,ψξ)g(Aχγ,(ψ+I)ξ)ψg(λl(ψγ,χ),ψξ)g(λl(γ,χ),ψξ)g(λl(γ,χ),ξ)=g(Aχψγ,ψξ)g(Aχγ,ψξ)g(Aχγ,ξ).g(λl(γ,χ),ξ)=g(Aχψγ,ψξ)g(Aχγ,ψξ).

    Definition 4.1. [12] A CR-lightlike submanifold of a golden semi-Riemannian manifold is totally umbilical if there is a smooth transversal vector field Htr Γ(Υ) that satisfies

    h(χ,η)=Hg(χ,η),

    where h is stands for second fundamental form and χ, η Γ(Υ).

    Theorem 4.2. Assume that the screen distribution is totally geodesic and that is a totally umbilical CR-lightlike submanifold of the golden semi-Riemannian manifold ¯, then

    Aψηχ=Aψχη,χ,ηΓλ.

    Proof. Given that ¯ is a golden semi-Riemannian manifold,

    ψ¯ηχ=¯ηψχ.

    Using (2.5) and (2.6), we have

    ψ(ηχ)+ψ(h(η,χ))=Aψχη+tηψχ. (4.1)

    Interchanging η and χ, we obtain

    ψ(χη)+ψ(h(χ,η))=Aψηχ+tχψη. (4.2)

    Subtracting Eqs (4.1) and (4.2), we get

    ψ(ηχχη)tηψχ+tχψη=AψηχAψχη. (4.3)

    Taking the inner product with γΓ(λ0) in (4.3), we have

    g(ψ(χη,γ)g(ψ(χη,γ)=g(Aψηχ,γ)g(Aψχη,γ).g(AψηχAψχη,γ)=g(χη,ψγ)g(χη,ψγ). (4.4)

    Now,

    g(χη,ψγ)=g(¯χηh(χ,η),ψγ)g(χη,ψγ)=g(η,(¯χψ)γψ(¯χγ)).

    Since ψ is parallel to ¯, i.e., ¯γψ=0,

    g(χη,ψγ)=ψ(¯χγ)).

    Using (2.7), we have

    g(χη,ψγ)=g(ψη,χγ+hs(χ,γ)+hl(χ,γ))g(χη,ψγ)=g(ψη,χγ)g(ψη,hs(χ,γ))g(ψη,hl(χ,γ)). (4.5)

    Since is a totally umbilical CR-lightlike submanifold and the screen distribution is totally geodesic,

    hs(χ,γ)=Hsg(χ,γ)=0

    and

    hl(χ,γ)=Hlg(χ,γ)=0,

    where χΓ(λ) and γΓ(λ0).

    From (4.5), we have

    g(χη,ψγ)=g(ψη,χγ).

    From Lemma 2.1, we get

    g(χη,ψγ)=0.

    Similarly,

    g(ηχ,ψγ)=0

    Using (4.4), we have

    g(AψηχAψχη,γ)=0.

    Since λ0 is nondegenerate,

    AψηχAψχη=0
    Aψηχ=Aψχη.

    Theorem 4.3. Let be the totally umbilical CR-lightlike submanifold of the golden semi-Riemannian manifold ¯. Consequently, 's sectional curvature, which is CR-lightlike, vanishes, resulting in ¯K(π)=0, for the entire CR-lightlike section π.

    Proof. We know that is a totally umbilical CR-lightlike submanifold of ¯, then from (2.13) and (2.14),

    (γhl)(ζ,ω)=g(ζ,ω)lγHlHl{(γg)(ζ,ω)}, (4.6)
    (γhs)(ζ,ω)=g(ζ,ω)sγHsHs{(γg)(ζ,ω)} (4.7)

    for a CR-lightlike section π=γω,γΓ(λ0),ωΓ(λ).

    From (2.12), we have (Ug)(ζ,ω)=0. Therefore, from (4.6) and (4.7), we get

    (γhl)(ζ,ω)=g(ζ,ω)lγHl, (4.8)
    (γhs)(ζ,ω)=g(ζ,ω)sγHs. (4.9)

    Now, from (4.8) and (4.9), we get

    {¯R(γ,ζ)ω}tr=g(ζ,ω)lγHlg(γ,ω)lζHl+g(ζ,ω)λl(γ,Hs)g(γ,ω)λl(ζ,Hs)+g(ζ,ω)sγHsg(γ,ω)sζHs+g(ζ,ω)λs(γ,Hl)g(γ,ω)λs(ζ,Hl). (4.10)

    For any βΓ(tr(Υ)), from Equation (4.10), we get

    ¯R(γ,ζ,ω,β)=g(ζ,ω)g(lγHl,β)g(γ,ω)g(lζHl,β)+g(ζ,ω)g(λl(γ,Hs),ζ)g(γ,ω)g(λl(ζ,Hs),β)+g(ζ,ω)g(sγHs,β)g(γ,ω)g(sζHs,β)+g(ζ,ω)g(λs(γ,Hl),β)g(γ,ω)g(λs(ζ,Hl,β).
    R(γ,ω,ψγ,ψω)=g(ω,ψγ)g(lγHl,ψω)g(γ,ψγ)g(lωHl,ψω)+g(ω,ψγ)g(λl(γ,Hs),ψω)g(γ,ψγ)g(λl(ω,Hs),ψω)+g(ω,ψγ)g(sγHs,ψω)g(γ,ψγ)g(sωHs,ψω)+g(ω,ψγ)g(λs(γ,Hl),ψω)g(γ,ψγ)g(λs(ω,Hl,ψU).

    For any unit vectors γΓ(λ) and ωΓ(λ), we have

    ¯R(γ,ω,ψγ,ψω)=¯R(γ,ω,γ,ω)=0.

    We have

    K(γ)=KN(γζ)=g(¯R(γ,ζ)ζ,γ),

    where

    ¯R(γ,ω,γ,ω)=g(¯R(γ,ω)γ,ω)

    or

    ¯R(γ,ω,ψγ,ψω)=g(¯R(γ,ω)ψγ,ψω)

    i.e.,

    ¯K(π)=0

    for all CR-sections π.

    Example 5.1. We consider a semi-Riemannian manifold R62 and a submanifold of co-dimension 2 in R62, given by equations

    υ5=υ1cosαυ2sinαυ3z4tanα,
    υ6=υ1sinαυ2cosαυ3υ4,

    where αR{π2+kπ; kz}. The structure on R62 is defined by

    ψ(υ1,υ2,υ3,υ4,υ5,υ6)=(¯ϕ υ1,¯ϕυ2,ϕυ3,ϕυ4,ϕυ5,ϕυ6).

    Now,

    ψ2(υ1,υ2,υ3,υ4,υ5,υ6)=((¯ϕ+1) υ1,(¯ϕ+1)υ2,(ϕ+1)υ3,(ϕ+1)υ4,
    (ϕ+1)υ5,(ϕ+1)υ6)
    ψ2=ψ+I.

    It follows that (R62,ψ) is a golden semi-Reimannian manifold.

    The tangent bundle Υ is spanned by

    Z0=sinα υ5cosα υ6ϕ υ2,
    Z1=ϕ sinα υ5ϕ cosα υ6+ υ2,
    Z2=υ5¯ϕ sinα υ2+υ1,
    Z3=¯ϕ cosα υ2+υ4+iυ6.

    Thus, is a 1-lightlike submanifold of R62 with RadΥ=Span{X0}. Using golden structure of R62, we obtain that X1=ψ(X0). Thus, ψ(RadΥ) is a distribution on . Hence, the is a CR-lightlike submanifold.

    In general relativity, particularly in the context of the black hole theory, lightlike geometry finds its uses. An investigation is made into the geometry of the golden semi-Riemannian manifolds that are CR-lightlike in nature. There are many intriguing findings on completely umbilical and completely geodesic CR-lightlike submanifolds that are examined. We present a required condition for a CR-lightlike submanifold to be completely geodesic. Moreover, it is demonstrated that the sectional curvature K of an entirely umbilical CR-lightlike submanifold of a golden semi-Riemannian manifold ¯ disappears.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The present work (manuscript number IU/R&D/2022-MCN0001708) received financial assistance from Integral University in Lucknow, India as a part of the seed money project IUL/IIRC/SMP/2021/010. All of the authors would like to express their gratitude to the university for this support. The authors are highly grateful to editors and referees for their valuable comments and suggestions for improving the paper. The present manuscript represents the corrected version of preprint 10.48550/arXiv.2210.10445. The revised version incorporates the identities of all those who have made contributions, taking into account their respective skills and understanding.

    Authors have no conflict of interests.



    [1] W. Gao, H. Rezazadeh, Z. Pinar, H. Baskonus, S. Sarwar, G. Yel, Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique, Opt. Quant. Electron., 52 (2020), 52. http://dx.doi.org/10.1007/s11082-019-2162-8 doi: 10.1007/s11082-019-2162-8
    [2] M. Khater, Solitary wave solutions for the generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony nonlinear evolution equation, Global J. Sci. Front. Res. Phys. Space Sci., 16 (2016), 37–41.
    [3] H. Bulut, T. Sulaiman, H. Baskonus, H. Rezazadeh, M. Eslami, M. Mirzazadeh, Optical solitons and other solutions to the conformable space time fractional Fokas Lenells equation, Optik, 172 (2018), 20–27. http://dx.doi.org/10.1016/j.ijleo.2018.06.108 doi: 10.1016/j.ijleo.2018.06.108
    [4] S. Vlase, M. Marin, A. Öchsner, M. Scutaru, Motion equation for a flexible one-dimensional element used in the dynamical analysis of a multibody system, Continuum Mech. Thermodyn., 31 (2019), 715–724. http://dx.doi.org/10.1007/s00161-018-0722-y doi: 10.1007/s00161-018-0722-y
    [5] M. Khater, A. Seadawy, D. Lu, Bifurcations of solitary wave solutions for (two and three)-dimensional nonlinear partial differential equation in quantum and magnetized plasma by using two different methods, Results Phys., 9 (2018), 142–150. http://dx.doi.org/10.1016/j.rinp.2018.02.010 doi: 10.1016/j.rinp.2018.02.010
    [6] V. Senthil Kumar, H. Rezazadeh, M. Eslami, F. Izadi, M. Osman, Jacobi elliptic function expansion method for solving KdV equation with conformable derivative and dual-power law nonlinearity, Int. J. Appl. Comput. Math., 5 (2019), 127. http://dx.doi.org/10.1007/s40819-019-0710-3 doi: 10.1007/s40819-019-0710-3
    [7] M. Khater, A. Seadawy, D. Lu, Dispersive solitary wave solutions of new coupled Konno-Oono, Higgs field and Maccari equations and their applications, J. King Saud Univ. Sci., 30 (2018), 417–423. http://dx.doi.org/10.1016/j.jksus.2017.11.003 doi: 10.1016/j.jksus.2017.11.003
    [8] M. Ghasemi, High order approximations using spline-based differential quadrature method: implementation to the multi-dimensional PDEs, Appl. Math. Model., 46 (2017), 63–80. http://dx.doi.org/10.1016/j.apm.2017.01.052 doi: 10.1016/j.apm.2017.01.052
    [9] S. Noor, W. Albalawi, R. Shah, M. Mossa Al-Sawalha, S. Ismaeel, S. El-Tantawy, On the approximations to fractional nonlinear damped Burger's-type equations that arise in fluids and plasmas using Aboodh residual power series and Aboodh transform iteration methods, Front. Phys., 12 (2024), 1374481. http://dx.doi.org/10.3389/fphy.2024.1374481 doi: 10.3389/fphy.2024.1374481
    [10] N. Perrone, R. Kao, A general finite difference method for arbitrary meshes, Comput. Struct., 5 (1975), 45–57. http://dx.doi.org/10.1016/0045-7949(75)90018-8 doi: 10.1016/0045-7949(75)90018-8
    [11] M. Abdou, A. Soliman, New applications of variational iteration method, Physica D, 211 (2005), 1–8. http://dx.doi.org/10.1016/j.physd.2005.08.002 doi: 10.1016/j.physd.2005.08.002
    [12] M. Hammad, R. Shah, B. Alotaibi, M. Alotiby, C. Tiofack, A. Alrowaily, et al., On the modified versions of GG-expansion technique for analyzing the fractional coupled Higgs system, AIP Adv., 13 (2023), 105131. http://dx.doi.org/10.1063/5.0167916 doi: 10.1063/5.0167916
    [13] E. Yusufoğlu, A. Bekir, Solitons and periodic solutions of coupled nonlinear evolution equations by using the sine cosine method, Int. J. Comput. Math., 83 (2006), 915–924. http://dx.doi.org/10.1080/00207160601138756 doi: 10.1080/00207160601138756
    [14] Y. Chen, B. Li, H. Zhang, Generalized Riccati equation expansion method and its application to the Bogoyavlenskii's generalized breaking soliton equation, Chinese Phys., 12 (2003), 940. http://dx.doi.org/10.1088/1009-1963/12/9/303 doi: 10.1088/1009-1963/12/9/303
    [15] H. Liu, T. Zhang, A note on the improved tan(ϕ(ξ)/2)-expansion method, Optik, 131 (2017), 273–278. http://dx.doi.org/10.1016/j.ijleo.2016.11.029 doi: 10.1016/j.ijleo.2016.11.029
    [16] M. Guo, H. Dong, J. Liu, H. Yang, The time-fractional mZK equation for gravity solitary waves and solutions using sech-tanh and radial basic function method, Nonlinear Anal.-Model., 24 (2018), 1–19. http://dx.doi.org/10.15388/NA.2019.1.1 doi: 10.15388/NA.2019.1.1
    [17] M. Kaplan, A. Bekir, A. Akbulut, E. Aksoy, The modified simple equation method for nonlinear fractional differential equations, Rom. J. Phys., 60 (2015), 1374–1383.
    [18] K. L. Wang, K. J. Wang, C. He, Physical insight of local fractional calculus and its application to fractional Kdv-Burgers-Kuramoto equation, Fractals, 27 (2019), 1950122. http://dx.doi.org/10.1142/S0218348X19501226 doi: 10.1142/S0218348X19501226
    [19] K. L. Wang, K. J. Wang, A modification of the reduced differential transform method for fractional calculus, Therm. Sci., 22 (2018), 1871–1875. http://dx.doi.org/10.2298/TSCI1804871W doi: 10.2298/TSCI1804871W
    [20] K. J. Wang, On a high-pass filter described by local fractional derivative, Fractals, 28 (2020), 2050031. http://dx.doi.org/10.1142/S0218348X20500310 doi: 10.1142/S0218348X20500310
    [21] R. Ali, Z. Zhang, H. Ahmad, Exploring soliton solutions in nonlinear spatiotemporal fractional quantum mechanics equations: an analytical study, Opt. Quant. Electron., 56 (2024), 838. http://dx.doi.org/10.1007/s11082-024-06370-2 doi: 10.1007/s11082-024-06370-2
    [22] A. Iftikhar, A. Ghafoor, T. Zubair, S. Firdous, S. Mohyud-Din, (G'/G, 1/G)-expansion method for traveling wave solutions of (2+1) dimensional generalized KdV, Sin Gordon and Landau-Ginzburg-Higgs equations, Sci. Res. Essays, 8 (2013), 1349–1359. http://dx.doi.org/10.5897/SRE2013.5555 doi: 10.5897/SRE2013.5555
    [23] R. Ali, S. Barak, A. Altalbe, Analytical study of soliton dynamics in the realm of fractional extended shallow water wave equations, Phys. Scr., 99 (2024), 065235. http://dx.doi.org/10.1088/1402-4896/ad4784 doi: 10.1088/1402-4896/ad4784
    [24] M. Bhatti, D. Lu, An application of Nwogu Boussinesq model to analyze the head-on collision process between hydroelastic solitary waves, Open Phys., 17 (2019), 177–191. http://dx.doi.org/10.1515/phys-2019-0018 doi: 10.1515/phys-2019-0018
    [25] S. Behera, N. Aljahdaly, Nonlinear evolution equations and their traveling wave solutions in fluid media by modified analytical method, Pramana, 97 (2023), 130. http://dx.doi.org/10.1007/s12043-023-02602-4 doi: 10.1007/s12043-023-02602-4
    [26] H. Khan, S. Barak, P. Kumam, M. Arif, Analytical solutions of fractional Klein-Gordon and gas dynamics equations, via the (G'/G)-expansion method, Symmetry, 11 (2019), 566. http://dx.doi.org/10.3390/sym11040566 doi: 10.3390/sym11040566
    [27] J. He, X. Wu, Exp-function method for nonlinear wave equations, Chaos Soliton. Fract., 30 (2006), 700–708. http://dx.doi.org/10.1016/j.chaos.2006.03.020 doi: 10.1016/j.chaos.2006.03.020
    [28] A. Alharbi, M. Almatrafi, Riccati-Bernoulli sub-ODE approach on the partial differential equations and applications, Int. J. Math. Comput. Sci., 15 (2020), 367–388.
    [29] W. Thadee, A. Chankaew, S. Phoosree, Effects of wave solutions on shallow-water equation, optical-fibre equation and electric-circuit equation, Maejo Int. J. Sci. Tech., 16 (2022), 262–274.
    [30] J. Alzaidy, Fractional sub-equation method and its applications to the space-time fractional differential equations in mathematical physics, British Journal of Mathematics and Computer Science, 3 (2013), 153–163. http://dx.doi.org/10.9734/BJMCS/2013/2908 doi: 10.9734/BJMCS/2013/2908
    [31] M. Cinar, A. Secer, M. Ozisik, M. Bayram, Derivation of optical solitons of dimensionless Fokas-Lenells equation with perturbation term using Sardar sub-equation method, Opt. Quant. Electron., 54 (2022), 402. http://dx.doi.org/10.1007/s11082-022-03819-0 doi: 10.1007/s11082-022-03819-0
    [32] K. J. Wang, F. Shi, Multi-soliton solutions and soliton molecules of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation for the incompressible fluid, EPL, 145 (2024), 42001. http://dx.doi.org/10.1209/0295-5075/ad219d doi: 10.1209/0295-5075/ad219d
    [33] M. Alqhtani, K. Saad, R. Shah, W. Hamanah, Discovering novel soliton solutions for (3+1)-modified fractional Zakharov-Kuznetsov equation in electrical engineering through an analytical approach, Opt. Quant. Electron., 55 (2023), 1149. http://dx.doi.org/10.1007/s11082-023-05407-2 doi: 10.1007/s11082-023-05407-2
    [34] H. Yasmin, N. Aljahdaly, A. Saeed, R. Shah, Probing families of optical soliton solutions in fractional perturbed Radhakrishnan Kundu Lakshmanan model with improved versions of extended direct algebraic method, Fractal Fract., 7 (2023), 512. http://dx.doi.org/10.3390/fractalfract7070512 doi: 10.3390/fractalfract7070512
    [35] M. Mossa Al-Sawalha, H. Yasmin, R. Shah, A. Ganie, K. Moaddy, Unraveling the dynamics of singular stochastic solitons in stochastic fractional Kuramoto-Sivashinsky equation, Fractal Fract., 7 (2023), 753. http://dx.doi.org/10.3390/fractalfract7100753 doi: 10.3390/fractalfract7100753
    [36] H. Yasmin, N. Aljahdaly, A. Saeed, R. Shah, Investigating families of soliton solutions for the complex structured coupled fractional Biswas-Arshed model in birefringent fibers using a novel analytical technique, Fractal Fract., 7 (2023), 491. http://dx.doi.org/10.3390/fractalfract7070491 doi: 10.3390/fractalfract7070491
    [37] W. Gao, P. Veeresha, D. Prakasha, H. Baskonus, G. Yel, New numerical results for the time-fractional Phi-four equation using a novel analytical approach, Symmetry, 12 (2020), 478. http://dx.doi.org/10.3390/sym12030478 doi: 10.3390/sym12030478
    [38] H. Rezazadeh, H. Tariq, M. Eslami, M. Mirzazadeh, Q. Zhou, New exact solutions of nonlinear conformable time-fractional Phi-4 equation, Chinese J. Phys., 56 (2018), 2805–2816. http://dx.doi.org/10.1016/j.cjph.2018.08.001 doi: 10.1016/j.cjph.2018.08.001
    [39] M. Khater, A. Mousa, M. El-Shorbagy, R. Attia, Analytical and semi-analytical solutions for Phi-four equation through three recent schemes, Results Phys., 22 (2021), 103954. http://dx.doi.org/10.1016/j.rinp.2021.103954 doi: 10.1016/j.rinp.2021.103954
    [40] Z. Li, T. Han, C. Huang, Bifurcation and new exact traveling wave solutions for time-space fractional Phi-4 equation, AIP Adv., 10 (2020), 115113. http://dx.doi.org/10.1063/5.0029159 doi: 10.1063/5.0029159
    [41] S. Bibi, N. Ahmed, U. Khan, S. Mohyud-Din, Auxiliary equation method for ill-posed Boussinesq equation, Phys. Scr., 94 (2019), 085213. http://dx.doi.org/10.1088/1402-4896/ab1951 doi: 10.1088/1402-4896/ab1951
    [42] M. Abdelrahman, H. Alkhidhr, Closed-form solutions to the conformable space-time fractional simplified MCH equation and time fractional Phi-4 equation, Results Phys., 18 (2020), 103294. http://dx.doi.org/10.1016/j.rinp.2020.103294 doi: 10.1016/j.rinp.2020.103294
    [43] F. Mahmud, M. Samsuzzoha, M. Ali Akbar, The generalized Kudryashov method to obtain exact traveling wave solutions of the PHI-four equation and the Fisher equation, Results Phys., 7 (2017), 4296–4302. http://dx.doi.org/10.1016/j.rinp.2017.10.049 doi: 10.1016/j.rinp.2017.10.049
    [44] M. Younis, A. Zafar, The modified simple equation method for solving nonlinear Phi-Four equation, International Journal of Innovation and Applied Studies, 2 (2013), 661–664.
    [45] P. Sunthrayuth, N. Aljahdaly, A. Ali, R. Shah, I. Mahariq, A. Tchalla, ϕ-Haar wavelet operational matrix method for fractional relaxation-oscillation equations containing ϕ-Caputo fractional derivative, J. Funct. Space., 2021 (2021), 7117064. http://dx.doi.org/10.1155/2021/7117064 doi: 10.1155/2021/7117064
    [46] S. Noor, H. Alyousef, A. Shafee, R. Shah, S. El-Tantawy, A novel analytical technique for analyzing the (3+1)-dimensional fractional calogero-bogoyavlenskii-schiff equation: investigating solitary/shock waves and many others physical phenomena, Phys. Scr., 99 (2024), 065257. http://dx.doi.org/10.1088/1402-4896/ad49d9 doi: 10.1088/1402-4896/ad49d9
    [47] S. Noor, A. Alshehry, A. Shafee, R. Shah, Families of propagating soliton solutions for (3+1)-fractional Wazwaz-BenjaminBona-Mahony equation through a novel modification of modified extended direct algebraic method, Phys. Scr., 99 (2024), 045230. http://dx.doi.org/10.1088/1402-4896/ad23b0 doi: 10.1088/1402-4896/ad23b0
    [48] H. Yasmin, A. Alshehry, A. Ganie, A. Mahnashi, R. Shah, Perturbed Gerdjikov-Ivanov equation: soliton solutions via Backlund transformation, Optik, 298 (2024), 171576. http://dx.doi.org/10.1016/j.ijleo.2023.171576 doi: 10.1016/j.ijleo.2023.171576
    [49] S. El-Tantawy, H. Alyousef, R. Matoog, R. Shah, On the optical soliton solutions to the fractional complex structured (1+1)-dimensional perturbed gerdjikov-ivanov equation, Phys. Scr., 99 (2024), 035249. http://dx.doi.org/10.1088/1402-4896/ad241b doi: 10.1088/1402-4896/ad241b
    [50] S. Alshammari, K. Moaddy, R. Shah, M. Alshammari, Z. Alsheekhhussain, M. Mossa Al-sawalha, et al., Analysis of solitary wave solutions in the fractional-order Kundu-Eckhaus system, Sci. Rep., 14 (2024), 3688. http://dx.doi.org/10.1038/s41598-024-53330-7 doi: 10.1038/s41598-024-53330-7
    [51] H. Yasmin, A. Alshehry, A. Ganie, A. Shafee, R. Shah, Noise effect on soliton phenomena in fractional stochastic Kraenkel-Manna-Merle system arising in ferromagnetic materials, Sci. Rep., 14 (2024), 1810. http://dx.doi.org/10.1038/s41598-024-52211-3 doi: 10.1038/s41598-024-52211-3
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