Research article

A basis construction for free arrangements between Linial arrangements and Shi arrangements

  • A central arrangement A was termed free if the module of A-derivations was a free module. The combinatorial structure of arrangements was heavily influenced by the freeness. Yet, there has been scarce exploration into the construction of their bases. In this paper, we constructed the explicit bases for a class of free arrangements that positioned between the cone of Linial arrangements and Shi arrangements.

    Citation: Meihui Jiang, Ruimei Gao. A basis construction for free arrangements between Linial arrangements and Shi arrangements[J]. AIMS Mathematics, 2024, 9(12): 34827-34837. doi: 10.3934/math.20241658

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  • A central arrangement A was termed free if the module of A-derivations was a free module. The combinatorial structure of arrangements was heavily influenced by the freeness. Yet, there has been scarce exploration into the construction of their bases. In this paper, we constructed the explicit bases for a class of free arrangements that positioned between the cone of Linial arrangements and Shi arrangements.



    Differential equations are very useful in many fields of science, including applied sciences and mathematical physics. Partial differential equations (PDEs) are widely employed in engineering to understand the behavior of physical systems through mathematical models. Many efficient approaches for identifying the solutions of nonlinear PDEs have been developed in recent years, such as: Hirotäs bilinear method [1], exp(Φ(ξ))-expansion method [2,3], generalized exponential rational function method [4], auxiliary equation method [5], first integral method [6], homotopy analysis method [7,8], tanh-method [9], transformed rational function method [10], residual power series method [11,12], extended direct algebraic method [13] and many other powerful mathematical techniques. Nonlinear PDEs are studied extensively because they help to understand the propagation of waves in many areas of mathematical physics, fluid mechanics and electromagnetic theory [14,15]. In addition, soliton solutions also have an effective contribution in fields of engineering and nonlinear optics [2,16,18,19,20,21].

    The traveling wave solutions of nonlinear PDEs equation are essential to explore and interpret various real life physical phenomena. The significance of the traveling wave solutions of nonlinear evolution equations has motivated many researches to investigate exact traveling wave solutions using effective and reliable mathematical techniques. The traveling wave solutions include solitary and other kinds of wave solutions. In particular, solitons are of great significance due to their useful applications in various areas of science and engineering [22,23].

    This study aims to investigate soliton and other traveling wave solutions of Gardner equation, which is an integrable nonlinear partial differential. The Gardner equation was originally proposed by Clifford Gardner in 1968 [24]. This equation is frequently referred to as combined Korteweg-de Vries-modified Korteweg-de Vries (KdV-mKdV) equation since it can be generalized to Korteweg-de Vries (KdV) equation. Gardner's equation has a wide range of applications in research, including quantum field theory and hydrodynamics [25,26,27].

    In this work, solitary wave solutions of Gardner equation (GE) are retrieved by utilizing the improved tan(Ω(Υ)2)-expansion method. This technique is a recently developed direct technique which provides a variety of traveling wave solutions for a wide class of nonlinear evolution equations [28,29,30,31].

    A soliton is an autonomous wave that diffuses while maintaining its shape and velocity. The nonlinear integrable KdV equation can be written, as

    rt+lrrx+mrxxx=0, (1.1)

    where r(x,t) in Eq (1.1) is the appropriate field variable and l, m are real constants, also x is representing the spatial variable and t is indicating the temporal variable. The solitary waves are generated due to nonlinear term rrx and the linear dispersion rxxx. The Gardener equation with constant coefficients [32,33] is considered in the form

    rt6(r+δ2r2)rx+rxxx=0, (1.2)

    where δ is a nonzero constant. Eq (1.2) is also called the combined KdV-mKdV equation. A higher order nonlinear term was added to the Eq (1.1) to generate the Gardner equation. Like KdV equation, Equation(1.2) is also an integrable equation.

    Step 2.1. The nonlinear partial differential equation (NLPDE) for r(x,t) is considered in the form

    Ω(r,rt,rx,rtt,rxt,rxx,)=0. (2.1)

    By the aid of transformation Υ=κ(xϖt), Eq (2.1) can be converted into an ordinary differential equation, as

    Γ(r,r,ϖr,r,ϖ2r)=0, (2.2)

    where ϖ is to be evaluated later and Υ is a wave variable.

    Step 2.2. The solitary wave solution of Eq (2.2) is supposed, as

    R(Υ)=P(Φ)=pj=0Gj[q+tan(Ω(Υ)2)]j+pj=1Hj[q+tan(Ω(Υ)2)]j, (2.3)

    where Gj(0jp) and Hj(1jp) are constants to be determined later. Also, Gp0,Hp0 and Ω=Ω(Υ) satisfy the ordinary differential equation (ODE),

    Ω(Υ)=q0sin(Ω(Υ))+q1cos(Ω(Υ))+q2. (2.4)

    Following are the special wave solutions for Eq (2.4).

    Family 2.1. For q20+q21q22<0 and q1q20,

    Ω(Υ)=2arctan[q0q1q2q22q21q20q1q2tan(q22q21q202(Υ+K))].

    Family 2.2. For q20+q21q22>0 and q1q20,

    Ω(Υ)=2arctan[q0q1q2q21+q20q22q1q2tanh(q21+q20q222(Υ+K))].

    Family 2.3. For q20+q21q22>0, q10 and q2=0,

    Ω(Υ)=2arctan[q0q1q21+q20q1tanh(q21+q202(Υ+K))].

    Family 2.4. For q20+q21q22<0, q20 and q1=0,

    Ω(Υ)=2arctan[q0q2+q22q20q2tan(q22q202(Υ+K))].

    Family 2.5. For q20+q21q22>0, q1q20 and q0=0,

    Ω(Υ)=2arctan[q1+q2q1q2tanh(q21q222(Υ+K))].

    Family 2.6. For q0=0 and q2=0,

    Ω(Υ)=arctan[e2q1(Υ+K)1e2q1(Υ+K)+1,2eq1(Υ+K)e2q1(Υ+K)+1].

    Family 2.7. For q1=0 and q2=0,

    Ω(Υ)=arctan[2eq0(Υ+K)e2q0(Υ+K)+1,e2q0(Υ+K)1e2q0(Υ+K)+1].

    Family 2.8. For q20+q21=q22,

    Ω(Υ)=2arctan[(q1+q2)(q0(Υ+K)+2)q20(Υ+K)].

    Family 2.9. For q0=q1=q2=il0,

    Ω(Υ)=2arctan[eil0(Υ+K)1].

    Family 2.10. For q0=q2=il0 and q1=il0,

    Ω(Υ)=2arctan[eil0(Υ+K)1+eil0(Υ+K)].

    Family 2.11. For q2=q0,

    Ω(Υ)=2arctan[(q0+q1)eq1(Υ+K)1(q0q1)eq1(Υ+K)1].

    Family 2.12. For q0=q2,

    Ω(Υ)=2arctan[(q1+q2)eq1(Υ+K)+1(q1q2)eq1(Υ+K)1].

    Family 2.13. For q2=q0,

    Ω(Υ)=2arctan[eq1(Υ+K)+q1q0eq1(Υ+K)q1q0].

    Family 2.14. For q1=q2,

    Ω(Υ)=2arctan[q0eq0(Υ+K)q2eq0(Υ+K)].

    Family 2.15. For q1=0, q0=q2,

    Ω(Υ)=2arctan[q2(Υ+K)+2q2(Υ+K)].

    Family 2.16. For q0=0 and q1=q2,

    Ω(Υ)=2arctan[q2(Υ+K)].

    Family 2.17. For q0=0, q1=q2,

    Ω(Υ)=2arctan[1q2(Υ+K)].

    Family 2.18. For q0=0 and q1=0,

    Ω(Υ)=q2Υ+K,

    where q0,q1, q2 and G0,Gj,Hj(j=1,2,....p) are to be evaluated. Homogeneous balance principle is used to find the value of p by considering highest order derivatives and highest non-linear terms occurring in Eq (2.2). If p is not an integer, then suitable transformation is implemented.

    Step 2.3. Once the value of p is obtained, Eq (2.3) is substituted into Eq (2.2). By gathering the coefficients of tan(Ω(Υ)2)j, cot(Ω(Υ)2)j (j=0,1,2,...) and setting each coefficient equal to zero, a set of algebraic equations for G0,Gj,Hj(j=1,2,...p), q0,q1,q2 and q can be obtained.

    Step 2.4. The set of over determined equations are solved and the values of G0,G1,H1,...,Gp,Hp,ϖ and q are substituted in Eq (2.3).

    Consider the integrable nonlinear Gardner equation given by Eq (1.2). Substituting the wave transformation,

    r(x,t)=R(Υ),Υ=κ(xϖt), (3.1)

    into Eq (1.2) yields an ordinary differential equation, as

    ϖR+3R2+2δ2R3κ2R=0. (3.2)

    Implementing the homogeneous balance principle the value of positive integer is obtained, as p=1. The trial solution becomes

    R(Υ)=G0+G1[q+tan(Ω(Υ)2)]+H1[q+tan(Ω(Υ)2)]1. (3.3)

    Substituting Eq (3.3) and Eq (2.4) into Eq (3.2), the following set of algebraic equations can be derived for q0,q1,q2,κ,ϖ, G0,G1 and H1 by collecting the terms with the same order of tan(Ω(Υ)2) and setting every coefficient of all the polynomials equal to zero.

    (tan(Ω(Υ)2))0=4δ2H13κ2H1q122κ2H1q1q2κ2H1q22,(tan(Ω(Υ)2))1=12δ2G0H123κ2H1q0l13κ2H1q0q2+6H12,(tan(Ω(Υ)2))2=12δ2G02H1+12δ2G1H122κ2H1q02+κ2H1q12κ2H1q22+2ϖH1+12G0H1,(tan(Ω(Υ)2))3=4δ2G03+24δ2G0G1H1κ2G1q0q1κ2G1q0q2+κ2H1q0q1κ2H1q0q2+2ϖG0+6G02+12G1H1,(tan(Ω(Υ)2))4=12δ2G02G1+12δ2G12H12κ2G1q02+κ2G1q12κ2G1q22+2ϖG1+12G0G1,(tan(Ω(Υ)2))5=12δ2G0G12+3κ2G1q0l13κ2G1q0q2+6G12,(tan(Ω(Υ)2))6=4δ2G13κ2G1q12+2κ2G1q1q2κ2G1q22.

    Following are the solutions obtained by solving the system of algebraic equation.

    Set 3.1. κ=κ,ϖ=1δ2,G0=12δ2,G1=±142q1κδ4δ2κ2q121δ2,H1=±142q1κδ+4δ2κ2q121δ2,q0=0,q1=q1,q2=124δ2κ2q121δκ,

    R(Υ)=G0+G1[tan(Ω(Υ)2)]+H1[tan(Ω(Υ)2)]1, (3.4)

    where q0, q1, q2 are arbitrary constants.

    Using Eq (3.4) and Families 2.2, 2.5 and 2.18 respectively, yields the following solutions:

    R1(Υ)=12δ2±142q1κδ4δ2κ2q121δ2[q0q1q2+q21+q20q22q1q2tanh(q21+q20q222(Υ+K))]±142q1κδ+4δ2κ2q121δ2[q0q1q2+q21+q20q22q1q2tanh(q21+q20q222(Υ+K))]1, (3.5)
    R2(Υ)=12δ2±142q1κδ4δ2κ2q121δ2[q1+q2q1q2tanh(q21q222(Υ+K))]±142q1κδ+4δ2κ2q121δ2[q1+q2q1q2tanh(q21q222(Υ+K))]1, (3.6)
    R3(Υ)=12δ2±142q1κδ4δ2κ2q121δ2[tan(12arctan[Υq2+K])]±142q1κδ+4δ2κ2q121δ2[tan(12arctan[Υq2+K])]1. (3.7)

    Set 3.2. κ=κ,ϖ=1δ2,G0=121+1+(q12+q22)δ2κ2δ2,G1=0,H1=12κ(q1+q2)δ, q0=1+(q12+q22)δ2κ2κδ,q1=q1,q2=q2,

    R(Υ)=G0+H1[tan(Ω(Υ)2)]1, (3.8)

    where q0, q1, q2 are arbitrary constants.

    The following solutions are determined by using Eq (3.8) and Families 2.2, 2.3, 2.6, 2.7 and 2.11–2.14, respectively.

    R4(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1+q2)δ[q0q1q2+q21+q20q22q1q2tanh(q21+q20q222(Υ+K))]1, (3.9)
    R5(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1+q2)δ[q0q1+q21+q20q1tanh(q21+q202(Υ+K))]1, (3.10)
    R6(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1+q2)δ[tan12(arctan[e2q1(Υ+K)1e2q1(Υ+K)+1,2eq1(Υ+K)e2q1(Υ+K)+1])]1, (3.11)
    R7(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1+q2)δ[tan(12arctan[2eq0(Υ+K)e2q0(Υ+K)+1,e2q0(Υ+K)1e2q0(Υ+K)+1])]1, (3.12)
    R8(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1+q2)δ[(q0+q1)eq1(Υ+K)1(q0q1)eq1(Υ+K)1]1, (3.13)
    R9(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1+q2)δ[(q1+q2)eq1(Υ+K)+1(q1q2)eq1(Υ+K)1]1, (3.14)
    R10(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1+q2)δ[eq1(Υ+K)+q1q0eq1(Υ+K)q1q0]1, (3.15)
    R11(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1+q2)δ[q0eq0(Υ+K)q2eq0(Υ+K)1]1. (3.16)

    Set 3.3. κ=κ,ϖ=1δ2,G0=121+1+(q12+q22)δ2κ2δ2,G1=12κ(q1q2)δ,H1=0, q0=1+(q12+q22)δ2κ2κδ,q1=q1,q2=q2,

    R(Υ)=G0+G1[tan(Ω(Υ)2)], (3.17)

    where q0, q1, q2 are arbitrary constants.

    Using Eq (3.17) and Families 2.2, 2.3, 2.6, 2.7, 2.11–2.14, respectively, yields the following solution:

    R12(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1q2)δ[q0q1q2+q21+q20q22q1q2tanh(q21+q20q222(Υ+K))], (3.18)
    R13(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1q2)δ[q0q1+q21+q20q1tanh(q21+q202(Υ+K))], (3.19)
    R14(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1q2)δ[tan12(arctan[e2q1(Υ+K)1e2q1(Υ+K)+1,2eq1(Υ+K)e2q1(Υ+K)+1])], (3.20)
    R15(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1q2)δ[tan(12arctan[2eq0(Υ+K)e2q0(Υ+K)+1,e2q0(Υ+K)1e2q0(Υ+K)+1])], (3.21)
    R16(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1q2)δ[(q0+q1)eq1(Υ+K)1(q0q1)eq1(Υ+K)1], (3.22)
    R17(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1q2)δ[(q1+q2)eq1(Υ+K)+1(q1q2)eq1(Υ+K)1], (3.23)
    R18(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1q2)δ[eq1(Υ+K)+q1q0eq1(Υ+K)q1q0], (3.24)
    R19(Υ)=121+1+(q12+q22)δ2κ2δ212κ(q1q2)δ[q0eq0(Υ+K)q2eq0(Υ+K)1]. (3.25)

    Set 3.4. κ=κ,ϖ=1+(q12q22)κ2δ2δ2,G0=1δ2,G1=12κ(q1q2)δ,H1=12κ(q1+q2)δ,q0=1κδ,q1=q1,q2=q2,

    R(Υ)=G0+G1[tan(Ω(Υ)2)]+H1[tan(Ω(Υ)2)]1, (3.26)

    where q0, q1, q2 are arbitrary constants.

    Using Eq (3.26) and Families 2.1–2.4, 2.8–2.10, 2.13–2.15, respectively, gives the following solutions:

    R20(Υ)=1δ212κ(q1q2)δ[q0q1q2q02q12+q22q1q2tan(q02q12+q222(Υ+K))]12κ(q1+q2)δ[q0q1q2q02q12+q22q1q2tan(q02q12+q222(Υ+K))]1, (3.27)
    R21(Υ)=1δ212κ(q1q2)δ[q0q1q2+q21+q20q22q1q2tanh(q21+q20q222(Υ+K))]12κ(q1+q2)δ[q0q1q2+q21+q20q22q1q2tanh(q21+q20q222(Υ+K))]1, (3.28)
    R22(Υ)=1δ212κ(q1q2)δ[q0q1+q21+q20q1tanh(q21+q202(Υ+K))]12κ(q1+q2)δ[q0q1+q21+q20q1tanh(q21+q202(Υ+K))]1, (3.29)
    R23(Υ)=1δ212κ(q1q2)δ[q0q2+q22q20q2tan(q22q202(Υ+K))]12κ(q1+q2)δ[q0q2+q22q20q2tan(q22q202(Υ+K))]1, (3.30)
    R24(Υ)=1δ212κ(q1q2)δ[(q1+q2)(q0(Υ+K)+2)q20(Υ+K)]12κ(q1+q2)δ[(q1+q2)(q0(Υ+K)+2)q20(Υ+K)]1, (3.31)
    R25(Υ)=1δ212κ(q1q2)δ[eϑq0(Υ+K)1]12κ(q1+q2)δ[eϑq0(Υ+K)1]1, (3.32)
    R26(Υ)=1δ212κ(q1q2)δ[eϑq0(Υ+K)1+eϑq0(Υ+K)]12κ(q1+q2)δ[eϑq0(Υ+K)1+eϑq0(Υ+K)]1, (3.33)
    R27(Υ)=1δ212κ(q1q2)δ[eq1(Υ+K)+q1q0eq1(Υ+K)q1q0]12κ(q1+q2)δ[eq1(Υ+K)+q1q0eq1(Υ+K)q1q0]1, (3.34)
    R28(Υ)=1δ212κ(q1q2)δ[q0eq0(Υ+K)q2eq0(Υ+K)1]12κ(q1+q2)δ[q0eq0(Υ+K)q2eq0(Υ+K)1]1, (3.35)
    R29(Υ)=1δ212κ(q1q2)δ[q2(Υ+K)+2q2(Υ+K)]12κ(q1+q2)δ[q2(Υ+K)+2q2(Υ+K)]1. (3.36)

    Set 3.5. κ=κ,ϖ=2(1+(q12q22)κ2δ2)(12+(q12q22)κ2δ2)δ2,G0=q12κ2q22κ2,G1=12κ(q1q2)δ,H1=12κ(q1+q2)δ,q0=1+(2q122q22)δ2κ2κδ,q1=q1,q2=q2,

    R(Υ)=G0+G1[tan(Ω(Υ)2)]+H1[tan(Ω(Υ)2)]1, (3.37)

    where q0, q1, q2 are arbitrary constants.

    Using Eq (3.37) and Families 2.1–2.3, 2.9, 2.10, 2.13 and 2.14, respectively, yields the following solutions:

    R30(Υ)=q12κ2q22κ212κ(q1q2)δ[q0q1q2q02q12+q22q1q2tan(q02q12+q222(Υ+K))]12κ(q1+q2)δ[q0q1q2q02q12+q22q1q2tan(q02q12+q222(Υ+K))]1, (3.38)
    R31(Υ)=q12κ2q22κ212κ(q1q2)δ[q0q1q2+q21+q20q22q1q2tanh(q21+q20q222(Υ+K))]12κ(q1+q2)δ[q0q1q2+q21+q20q22q1q2tanh(q21+q20q222(Υ+K))]1, (3.39)
    R32(Υ)=q12κ2q22κ212κ(q1q2)δ[q0q1+q21+q20q1tanh(q21+q202(Υ+K))]12κ(q1+q2)δ[q0q1+q21+q20q1tanh(q21+q202(Υ+K))]1, (3.40)
    R33(Υ)=q12κ2q22κ212κ(q1q2)δ[eϑq0(Υ+K)1]12κ(q1+q2)δ[eϑq0(Υ+K)1]1, (3.41)
    R34(Υ)=q12κ2q22κ212κ(q1q2)δ[eϑq0(Υ+K)1+eϑq0(Υ+K)]12κ(q1+q2)δ[eϑq0(Υ+K)1+eϑq0(Υ+K)]1, (3.42)
    R35(Υ)=q12κ2q22κ212κ(q1q2)δ[eq1(Υ+K)+q1q0eq1(Υ+K)q1q0]12κ(q1+q2)δ[eq1(Υ+K)+q1q0eq1(Υ+K)q1q0]1, (3.43)
    R36(Υ)=q12κ2q22κ212κ(q1q2)δ[q0eq0(Υ+K)q2eq0(Υ+K)1]12κ(q1+q2)δ[q0eq0(Υ+K)q2eq0(Υ+K)1]1, (3.44)

    where Υ=κ(xϖt).

    Some of the obtained soliton solutions are graphically represented in this section. Kink solitons, dark-bright solitons, bright solitons, singular solitons and periodic wave solutions are retrieved.

    The 3D-graph and contour plot for the solution R4(Υ) are shown in Figure 1. The solution R4(Υ) is derived from Family 2.2 of solution Set 2.2 as defined by Eq (3.9). The obtained graphs show a kink soliton solution. Kink soliton is a type of solitary wave that ascend or descend from one asymptotic state to another. The contour graph is also included along with surface graph to illustrate the wave structure corresponding to the obtained solution.

    Figure 1.  This figure demonstrates the 3D graph and corresponding contour of Eq (3.9) at κ=1, δ=1, q0=2.82, q1=3, q2=4, G0=0.914, G1=0, H1=1, K=0.5, ϖ=1.

    The graphical illustration of R23(Υ) is presented in Figure 2. The solution R23(Υ) is given by Equation (3.30) using the values of Set 2.4 for Family 2.4. The graphs in Figure 2 show a bright soliton. The surface graph shows a localized intensity peak above the continuous wave background which means that there is a temporary increase in the wave amplitude.

    Figure 2.  This figure demonstrates the 3D graph and corresponding contour of Eq (3.30) at κ=1, δ=2, q0=0.5, q1=0, q2=0.5, G0=0.25, G1=0.125, H1=0.125, K=1, ϖ=0, ϑ=1.

    Figure 3 shows the 3D plot and the corresponding contour plot of R20(Υ) given by Eq (3.27). The graph of R12(Υ) given by Eq (3.18) is illustrated in Figure 4. Figure 5 provides the graphical illustration of R30(Υ) given by Eq (3.38). Figure 6 shows the graph of a dark-bright soliton which is graphical illustration of the solution R3(Υ) expressed by Eq (3.7). Similarly, Figures 79 presents the graphical illustrations for the solutions presented by Eq (3.35), Eq (3.6) and Eq (3.32), respectively.

    Figure 3.  This figure demonstrates the 3D graph and corresponding contour of Eq (3.27) at κ=1, δ=1, q0=1, q1=1, q2=3, G0=1, G1=2, H1=1,K=1, ϖ=7.
    Figure 4.  This figure demonstrates the 3D graph and corresponding contour of Eq (3.18) at κ=1, δ=1, q0=2.828, q1=3, q2=4, G0=0.914, G1=3.5, H1=0, K=1, ϖ=1.
    Figure 5.  This figure demonstrates the 3D graph and corresponding contour of Eq (3.38) at κ=1, δ=3, q0=0.206, q1=0, q2=0.3, G0=0.09, G1=0.05, H1=0.05, K=1, ϖ=0.055, ϑ=1.
    Figure 6.  This figure demonstrates the 3D graph and corresponding contour of Eq (3.7) at κ=1, δ=1, q0=0, q1=0, q2=0.5I, G0=0.5, G1=0.25I, H1=0.25I, K=1, ϖ=1.
    Figure 7.  This figure demonstrates the 3D graph and corresponding contour of Eq (3.35) at κ=1,δ=3,q0=0.33,q1=3,q2=3,G0=0.11,G1=1,H1=0,K=1,ϖ=0.11,ϑ=1.
    Figure 8.  This figure demonstrates the 3D graph and corresponding contour of Eq (3.6) at κ=1, δ=1, q0=0, q1=2.5, q2=2.449, G0=0.5, G1=0.025,  H1=2.474, K=1, ϖ=1.
    Figure 9.  This figure demonstrates the 3D graph and corresponding contour of Eq (3.32) at κ=1,δ=1,q0=1,q1=1,q2=1,G0=1,G1=0,H1=1,K=1,ϖ=1,ϑ=1.

    It can be easily observed that improved tan(Ω(Υ)2) technique is a spectacular technique as compared to many other direct techniques as it gave abundant soliton solutions. Using this technique, kink, singular, bright and dark-bright soliton solutions have been retrieved in this paper. This method is clearly more powerful than many other methods, such as: the tanh-method [34], the GG expansion method [35] and the generalized exponential rational function method, the Jacobi elliptic solution method [36], because the improved tan(Ω(Υ)2) method retrieved many more new solutions than the previously mentioned techniques.

    In this study, the soliton and other solitary wave solutions of the constant-coefficient Gardner equation are investigated using tan(Ω(Υ)2)-expansion method. A variety of precise closed form traveling wave solutions have been constructed including bright solitons, dark-bright solitons, kink solitons and periodic wave solutions. Some of the obtained solutions are illustrated using graphical simulations for suitable choice of parameters. The wave profile corresponding to the obtained solutions is depicted through 3D-surface graphs and corresponding 2D-contour plots. Comparison of the obtained results with those available in the literature depict the efficacy and productivity of the improved tan(Ω(Υ)2) technique. Mathematical computations and simulations were obtained using Maple software. The reported results may be helpful in further explorations of the nonlinear physical problems governed by the Gardner equation in fluid dynamics, plasma physics and other fields. The improved tan(Ω(Υ)2) technique will be useful for the analytic study of a large class of nonlinear PDEs that are widely used in engineering, physics and other sciences.

    The authors declare no conflicts of interest.



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