Two new Painlevé integrable KdV–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation and new negative-order KdV-CBS equation

In this work, we develop two new (3+1)-dimensional KdV–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation and (3+1)-dimensional negative-order KdV-CBS (nKdV-nCBS) equation. The newly developed equations pass the Painlevé integrability test via examining the compatibility conditions for each developed model. We examine the dispersion relation and derive multiple soliton solutions for each new equation.

in the context of (2+1)-and (3+1)-dimensional equations. The field of integrable equations is an active area of research because it describes the real features and reveals the scientific nature of the nonlinearity in science areas. The concept of multiple soliton solutions is one important feature of completely integrable equations.
It is the aim of this work to extend the KdV-CBS Eq. (3) to a new (3+1)-dimensional KdV-CBS equation given as where α, β, γ , a, b, and c are arbitrary constants. This equation is obtained by adding the z-component of the in addition to three linear terms, namely v x , v y , and v z . Equation (4) can be reduced to other integrable nonlinear equations with distinct significant physical features.
We draw the attention that the KdV equation and the CBS equation can be obtained by using the KdV recursion operator In other words, we obtain the KdV and the CBS equations and respectively. Verosky [4] extended the Olver work in [3] and admitted the use of the negative direction to obtain a sequence of equations of increasingly negative orders. Verosky [4] elaborated that the hierarchy of evolution equations given as and for the construction of the standard KdV and CBS equations, respectively, can be used in the negative order hierarchy in the form and or equivalently and where the power of goes to the opposite direction for the KdV and the CBS equations, respectively. Using the negative order hierarchy (12) and (13), we obtain the integrable negative-order KdV (nKdV) equation and the integrable CBS (nCBS) equation given as abd or equivalently abd where we used v = u x . In [1][2][3], we proved that these two Eqs. (16) and (17) nicely pass the Painlevé integrability test.
In the present work, we follow the sense of combining the KdV and the CBS equations as explored earlier to establish a new combination of the negative-order KdV Eq. (16) and the negative-order CBS Eq. (17); hence, we establish a new (3+1)-dimensional model that will be called the negative-order KdV-CBS equation (nKdV-nCBS). It is obvious that for μ = 0 and ν = 0, Eq. (18) will be reduced to the negative-order KdV equation (16). However, λ = 0 and ν = 0, Eq. (18) will be reduced to the negative-order CBS equation (17). Our aim for this work first is to apply the Painlevé test to examine the integrability feature via determining the compatibility conditions for these two newly developed equations KdV-CBS Eq. (4) and the (nKdV-nCBS) Eq. (18). We next plan to derive N -soliton solutions for each developed equation. The simplified Hirota's method is a powerful technique due to its ease of use and does not require the use of the bilinear forms.

(3+1)-dimensional KdV-CBS equation
As stated earlier, we first employ the Painlevé analysis to confirm the integrability of the (3+1)-dimensional KdV-CBS Eq. (4). We next continue to derive the multiple soliton solutions for this equation.

Painlevé analysis
To emphasize the integrability of the developed (3+1)dimensional KdV-CBS Eq. (4), where it is assumed to have a solution as a Laurent expansion about a singular manifold ψ = ψ(x, y, z, t) as where u k (x, y, z, t),, k = 0, 1, 2, ..., are functions of x, y, z, and t. We follow the analysis we presented in our work in [2] to get a characteristic equation for resonances with one branch with four resonances at k = −1, 1, 4, and 6. Proceeding as in [2], we observed explicit expressions for u 2 , u 3 , and u 5 , where we found that u 1 , u 4 , u 6 turn out to be arbitrary functions and also compatibility conditions, for k = 1, 4, 6, are satisfied identically which implies that Eq. (4) passes the Painlevé test for complete integrability. Note that integrability of Eq. (4) is confirmed without any restriction on the parameters, α, β, γ , a, b, and c and therefore does not depend on these parameters.

Multiple soliton solutions for the KdV-CBS equation
In this section, we employ the simplified Hirota's method to determine the multiple soliton solutions for the new (3+1)-dimensional KdV-CBS equation that reads or equivalently obtained upon using the potential v(x, y, z, t) = u x (x, y, z, t). Substituting into the linear terms of (21), and solving the resulting equation for dispersion relation c i we obtain and hence the wave variable θ i becomes t. (25) We next substitute u(x, y, z, t) = 2(ln ( f (x, y, z, t))) x , (26) where the auxiliary function is given by Recall from (22) that v(x, t) = u x (x, t).
We proceed as before to formally derive the three soliton solutions.

Multiple soliton solutions for the nKdV-nCBS equation
In this section, we employ the simplified Hirota's method to determine the multiple soliton solutions for the new (3+1)-dimensional nKdV-nCBS equation that reads u xt + u xxxy + 4u x u xy + 2u xx u y +λu xx + μu xy + νu xz = 0, Substituting into the linear terms of (33), and solving the resulting equation for dispersion relation c i we obtain and hence the wave variable θ i becomes We next substitute u(x, y, z, t) = 2(ln ( f (x, y, z, t))) x , where the auxiliary function is given by f (x, y, z, t) = 1 + e k 1 x+r 1 y+s 1 z−(k 2 1 r 1 +λk 1 +μr 1 +νs 1 ) t .
To derive the two-soliton solutions, we substitute into Eq. (37), and then into (33), where θ 1 and θ 2 are given in (36), we find that the phase shift a 12 is given by and hence To determine the two-soliton solutions explicitly, we substitute (40) into (37).
We proceed as before to formally derive the three soliton solutions.

Discussion
In this work, we examined two new (3+1)-dimensional KdV-CBS equation and the negative-order of the KdV-CBS equation. We showed that both equations pass nicely the Painlevé test without any restriction on the compatibility conditions or the parameters involved in each equation. We noticed that the dispersion relations of the two models are distinct, whereas the phase shifts remain the same for the two developed models. The phase shift retained the KdV-type phase shift. Both equations were handled by using a simplified form of the Hirota method. Multiple soliton solutions for each equation were formally derived.