Based on the Hirota’s bilinear Equation (5), the N-order soliton solutions of (3 + 1)-dimensional vc-YTSF Equation (2) can be obtained,

$f=f_N=\underset{\mu =0,1}{{\displaystyle \sum }}\exp \left(\underset{i=1}{\overset{N}{{\displaystyle \sum }}}{\mu }_i{\eta }_i+\underset{i<j}{\overset{\left(N\right)}{{\displaystyle \sum }}}{A}_{ij}{\mu }_i{\mu }_j\right),$(6)

where the ${{\displaystyle \sum }}_{\mu =0,1}$ represents summation over all possible combinations of ${\mu }_1=0,1,{\mu }_2=0,1,\cdots ,{\mu }_N=0,1$, and the ${{\displaystyle \sum }}_{i<j}^{\left(N\right)}$ is over all possible combinations of the N elements with the specific condition $i<j$.

By combining (6) and (5), we get the parameter relationships as

$\{\begin{array}{l}{\eta }_i=k_i(m_i\sigma +p_{i}y+w_i\left(t\right)+{\gamma }_i,\\ w_i\left(t\right)=-{\displaystyle \int \frac{\lambda k_i^2{m}_i^4+\alpha \left(t\right)p_i^2}{\beta \left(t\right)m_i}\text{d}t},\\ \text{exp}\left(A_{ij}\right)=\frac{\alpha \left(t\right){\left(m_i{p}_j-m_j{p}_i\right)}^2-3\lambda {\left(k_i{m}_i^2{m}_j-k_j{m}_j^2{m}_i\right)}^2}{\alpha \left(t\right){\left(m_i{p}_j-m_j{p}_i\right)}^2-3\lambda {\left(k_i{m}_i^2{m}_j+k_j{m}_j^2{m}_i\right)}^2}\end{array}$(7)

where $m_i\ne 0$ and $k_i,p_i,{\gamma }_i$ are some free parameters. Substituting (6) with (7) into (4), we can obtain the N-soliton solutions of (3 + 1)-dimensional vc-YTSF Equation (2).

By consulting references [22] - [24] and combining the equation, we choose the following four functions for $\alpha \left(t\right)$ and $\beta \left(t\right)$.

Case 1:

$\alpha \left(t\right)=3,\beta \left(t\right)=-4$(8)

Case 2:

$\alpha \left(t\right)=t,\beta \left(t\right)=1$(9)

Case 3:

$\alpha \left(t\right)=\frac{1}{2}\text{cos}\left(\frac{t}{2}\right),\beta \left(t\right)=1$(10)

Case 4:

$\alpha \left(t\right)=\frac{t}{2}\text{cos}\left(\frac{t}{2}\right),\beta \left(t\right)=1$(11)

Among the above four different sets of functions, there is one set of constant coefficients and the remaining three of variable coefficients. Through these four sets of functions, we will get one set of three-dimensional diagrams of constant coefficient functions and three sets of variable coefficient functions. Through the comparison of the diagrams, we can clearly see the difference in the exact solutions between the variable coefficient equation and the constant coefficient equation.

We give four different three-dimensional diagrams of 1-soliton solution in Figure 1 , with different parameter values as (a):

$\left(\lambda ,k_1,m_1,p_1,{\gamma }_1\right)=\left(-1,\frac{1}{5},\frac{2}{5},\frac{3}{2},1\right)$, (b): $\left(\lambda ,k_1,m_1,p_1,{\gamma }_1\right)=\left(-1,\frac{1}{4},\frac{4}{5},\frac{1}{4},1\right)$, (c), (d): $\left(\lambda ,k_1,m_1,p_1,{\gamma }_1\right)=\left(-1,\frac{1}{2},\frac{1}{4},\frac{1}{2},1\right)$. In Figure 2 , we obtain four different three-dimensional diagrams of 2-soliton solutions when the parameters are: (a): $\lambda =-1$, $k_1=-\frac{1}{4}$, $k_2=\frac{3}{2}$, $m_1=1$, $m_2=\frac{2}{5}$, $p_1=3$, $p_2=\frac{1}{5}$, ${\gamma }_1={\gamma }_2=1$; (b): $\lambda =-1$, $k_1=\frac{1}{4}$, $k_2=\frac{1}{2}$, $m_1=\frac{3}{5}$, $m_2=\frac{2}{5}$, $p_1=3$, $p_2=\frac{1}{5}$, ${\gamma }_1={\gamma }_2=1$; (c), (d): $\lambda =-1$, $k_1=\frac{1}{2}$, $k_2=\frac{3}{2}$, $m_1=3$, $m_2=\frac{2}{5}$, $p_1=\frac{5}{2}$, $p_2=\frac{1}{3}$, ${\gamma }_1={\gamma }_2=1$. Similarly, we get the diagrams of 3-soliton solutions in Figure 3 by taking the following parameters: (a): $\lambda =-1$, $k_1=\frac{1}{2}$, $k_2=k_3=\frac{3}{2}$, $m_1=\frac{1}{2}$, $m_2=\frac{3}{4}$, $m_3=\frac{2}{3}$, $p_1=\frac{3}{2}$, $p_2=\frac{1}{5}$, $p_3=\frac{1}{2}$, ${\gamma }_i=1\left(i=1,2,3\right)$, (b): $\lambda =-1$, $k_1=1$, $k_2=2$, $k_3=\frac{5}{2}$, $m_1=\frac{4}{5}$, $m_2=\frac{1}{2}$, $m_3=\frac{2}{3}$, $p_1=\frac{1}{4}$, $p_2=3$, $p_3=\frac{1}{2}$, ${\gamma }_i=1\left(i=1,2,3\right)$, (c), (d): $\lambda =-1$, $k_1=\frac{1}{2}$, $k_2=2$, $k_3=\frac{3}{2}$, $m_1=\frac{7}{2}$, $m_2=\frac{1}{4}$, $m_3=\frac{2}{3}$, $p_1=\frac{5}{2}$, $p_2=\frac{1}{5}$, $p_3=\frac{1}{2}$, ${\gamma }_i=1\left(i=1,2,3\right)$.

In this section, we first get the N-order rational solution from the N-order soliton solutions by using the long wave limit method, and then complex the parameters of the N-order rational solutions to obtain the lump solutions. In order to get the N-order rational solution of (3 + 1)-dimensional vc-YTSF Equation (2), we take ${\gamma }_i=\text{I}\pi$ in (6) and (7), then $f_N$ can be written as follows

$f_N=\underset{\mu =0,1}{{\displaystyle \sum }}\underset{i=1}{\overset{N}{{\displaystyle \prod }}}{\left(-1\right)}^{{\mu }_i}\exp \left({\mu }_i{\xi }_i\right)\underset{i<j}{\overset{\left(N\right)}{{\displaystyle \prod }}}\exp \left({\mu }_i{\mu }_j{A}_{ij}\right),$(12)

with ${\xi }_i=k_i\left(m_i\sigma +p_{i}y-{\displaystyle \int \frac{\lambda k_i^2{m}_i^4+\alpha \left(t\right)p_i^2}{\beta \left(t\right)m_i}\text{d}t}\right)$. Letting $k_i$ tend to 0 and taking $k_i/k_j=O\left(1\right)$ in (12) and (6), $f_N$ can be rewritten as

$f_N=\underset{\mu =0,1}{{\displaystyle \sum }}\underset{i=1}{\overset{N}{{\displaystyle \prod }}}{\left(-1\right)}^{{\mu }_i}\left(1+{\mu }_i{k}_i{\theta }_i\right)\underset{i<j}{\overset{\left(N\right)}{{\displaystyle \prod }}}\exp \left(1+{\mu }_i{\mu }_j{k}_i{k}_j{B}_{ij}\right)+O\left(k^{N+1}\right),$(13)

with

$\{\begin{array}{l}{\theta }_i=m_i\sigma +p_{i}y-{\displaystyle \int \frac{\alpha \left(t\right)p_i^2}{\beta \left(t\right)m_i}\text{d}t},\hfill \\ B_{ij}=\frac{12\lambda m_i^3{m}_j^3}{\alpha \left(t\right){\left(m_i{p}_j-m_j{p}_i\right)}^2}.\hfill \end{array}$(14)

Combining the above Equation (13) with the transformation u given by Equation (4), we can get the following form of rational solutions,

$\begin{array}{c}{f}_N=\underset{i=1}{\overset{N}{{\displaystyle \prod }}}{\theta }_i+\frac{1}{2}\underset{i,j}{\overset{\left(N\right)}{{\displaystyle \sum }}}{B}_{ij}\underset{r\ne i,j}{\overset{N}{{\displaystyle \prod }}}{\theta }_r++\frac{1}{2!2^2}\underset{i,j,p,s}{\overset{\left(N\right)}{{\displaystyle \sum }}}{B}_{ij}{B}_{ps}\underset{r\ne i,j,p,s}{\overset{N}{{\displaystyle \prod }}}{\theta }_r+\cdots \\ +\frac{1}{M!2^M}\underset{i,j,\cdots ,m,n}{\overset{\left(N\right)}{{\displaystyle \sum }}}\stackrel{M}{\stackrel{︷}{B_{ij}{B}_{kl}\cdots B_{mn}}}\underset{s\ne i,j,k,l,\cdots ,m,n}{\overset{N}{{\displaystyle \prod }}}{\theta }_s+\cdots ,\end{array}$(15)

where ${{\displaystyle \sum }}_{i,j,\cdots ,m,n}^{\left(N\right)}$ denotes the summation over all possible combinations of $i,j,\cdots ,m,n$. We can get the corresponding N-order rational solution of (3 + 1)-dimensional vc-YTSF Equation (2) by taking $N=1,2,\cdots$ in (15) and substituting it into (4).

For example, take $N=2$ in (15), then, we get the following form

$f_2={\theta }_1{\theta }_2+B_{12}.$(16)

The following form of 2-order rational solution can be obtained by bringing (14) and (16) into (4)

$u\left(x,y,t\right)=\frac{2m_1{\theta }_2+2m_2{\theta }_1}{{\theta }_1{\theta }_2+B_{12}}.$(17)

In order to obtain 1-order lump solution, first, we set $m_2=m_1{}^{\text{*}}=a_1-b_{1}I$, $p_2=p_1{}^{\text{*}}=c_1-d_{1}I$ in (16), so $f_2$ can be rewritten as

$\begin{array}{c}{\check{f}}_2={\left(a_1\sigma +c_{1}y+\frac{a_1\left(d_1^2-c_1^2\right)-2b_1{c}_1{d}_1}{a_1^2+b_1^2}{\displaystyle \int \frac{\alpha \left(t\right)}{\beta \left(t\right)}\text{d}t}\right)}^2\\ +{\left(b_1\sigma +d_{1}y+\frac{b_1\left(c_1^2-d_1^2\right)-2a_1{c}_1{d}_1}{a_1^2+b_1^2}{\displaystyle \int \frac{\alpha \left(t\right)}{\beta \left(t\right)}\text{d}t}\right)}^2\\ +\frac{-3\lambda {\left(a_1^2+b_1^2\right)}^3}{\alpha \left(t\right){\left(b_1{c}_1-a_1{d}_1\right)}^2}\\ \stackrel{\text{def}}{=}{\phi }^2+{\varphi }^2+\Omega .\end{array}$

Then, substitute the above formula into transformation (4), we can the following solution.

$u\left(x,y,t\right)=\frac{4\left(a_1\phi +b_1\varphi \right)}{{\phi }^2+{\varphi }^2+\text{Ω}}.$(18)

Similar to the calculation procedure for $N=2$, when $N=4,6,\cdots ,M$, we can get M-lump solutions. In Figures 4-7, we can obtain different diagrams of 1-lump solution with different $\alpha \left(t\right)$ and $\beta \left(t\right)$ by selecting appropriate parameters. We take $\lambda =-1$, $a_1=d_1=-\frac{1}{2}$, $b_1=\frac{5}{4}$, $c_1=2$ in (18), then we get the 1-lump solutions for constant coefficient (3 + 1)-YTSF equation in Figure 4 . Although lump solutions for constant coefficient equations have been widely studied [25] - [27] , Figure 4 is given as a special case. Of course, by taking the following three different parameters, we can obtain some diagrams of 1-lump solutions with different variable coefficients in Figures 5-7 . In Figure 5 , the parameters are $\lambda =-1$, $a_1=-\frac{1}{2}$, $b_1=\frac{5}{4}$, $c_1=\frac{5}{2}$, $d_1=-1$. The parameters in Figure 6 are $\lambda =-1$, $a_1=-\frac{3}{4}$, $b_1=\frac{3}{5}$, $c_1=2$, $d_1=1$. It can be seen from Figure 6 that

when $y=0$, the diagrams of 1-lump solution change dramatically, which is worth focusing on in our future research work. When the parameters are $\lambda =-1$, $a_1=-\frac{3}{4}$, $b_1=\frac{3}{5}$, $c_1=\frac{3}{2}$, $d_1=\frac{1}{2}$, we can get diagrams of 1-lump solution in Figure 7 .

To obtain the hybrid solution between 1-lump solution and 1-soliton, first, we take $N=4$, ${\gamma }_1={\gamma }_2=\text{I}\pi$ in (5) and let $k_1,k_2$ tend to 0 to get ${\tilde{f}}_3$ as follows.

${\tilde{f}}_3={\theta }_1{\theta }_2+B_{12}+\left(C_{13}{C}_{23}+C_{13}{\theta }_2+C_{23}{\theta }_1+{\theta }_1{\theta }_2+B_{12}\right){\text{e}}^{{\eta }_3},$(19)

where

$\{\begin{array}{l}{\theta }_i={\theta }_i=m_i\sigma +p_{i}y-{\displaystyle \int \frac{\alpha \left(t\right)p_i^2}{\beta \left(t\right)m_i}\text{d}t},\\ {\eta }_3=k_3\left(m_3\sigma +p_{3}y-{\displaystyle \int \frac{\lambda k_3^2{m}_3^4+\alpha \left(t\right)p_3^2}{\beta \left(t\right)m_3}\text{d}t}\right)+{\gamma }_3,\\ B_{12}=\frac{12\lambda m_1^3{m}_2^3}{\alpha \left(t\right){\left(m_1{p}_2-m_2{p}_1\right)}^2},\\ c_{ij}=\frac{12\lambda k_j{m}_i^3{m}_j^3}{-3\lambda k_j^2{m}_i^2{m}_j^4+\alpha \left(t\right){\left(m_i{p}_j-m_j{p}_i\right)}^2}\left(i=1,2;j=3\right).\end{array}$(20)

Then take $m_2=m_1{}^{\text{*}}=a_1-b_{1}I$, $q_2=q_1{}^{\text{*}}=c_1-d_{1}I$ in (19) and (20), and substitute it into (4), the final formula is the hybrid solution between 1-lump solution and 1-soliton (see Figures 8-11 ). By taking different and appropriate parameters, we give four different sets of diagrams for hybrid solution between 1-lump solution and 1-soliton, including one set of constant coefficients and three sets of variable coefficients. The values of the parameters in Figure 8 are $\lambda =-1$, $a_1=d_1=-\frac{1}{2}$, $b_1=\frac{5}{4}$, $c_1=\frac{5}{2}$, $k_1=\frac{1}{2}$, $m_1=\frac{4}{5}$, $p_1=\frac{3}{2}$, ${\gamma }_3=0$. When we take $\lambda =-1$, $a_1=-\frac{1}{2}$, $b_1=\frac{5}{4}$, $c_1=\frac{5}{2}$, $d_1=-1$, $k_1=\frac{1}{2}$, $m_1=\frac{4}{5}$, $p_1=-\frac{1}{4}$, ${\gamma }_3=0$, we can get the hybrid solution between 1-lump solution and 1-soliton in Figure 9 . We give the diagrams of the hybrid solution between 1-lump solution and 1-soliton in Figure 10 and Figure 11 , with different parameters as Figure 10 : $\left(\lambda ,a_1,b_1,c_1,d_1,k_1,m_1,p_1,{\gamma }_1\right)=\left(-1,-\frac{3}{4},\frac{3}{5},2,1,\frac{3}{2},\frac{1}{2},\frac{1}{2},0\right)$, Figure 11 : $\left(\lambda ,a_1,b_1,c_1,d_1,k_1,m_1,p_1,{\gamma }_1\right)=\left(-1,-\frac{3}{4},\frac{3}{5},\frac{3}{2},\frac{3}{4},\frac{1}{2},\frac{1}{4},\frac{1}{2},0\right)$.

Similar to the method of finding the hybrid solution between the 1-lump solution and 1-soliton, first, we take $N=4$, ${\gamma }_1={\gamma }_2=\text{I}\pi$ in (5) and let $k_1\to 0,k_2\to 0$ to get

$\begin{array}{c}{\tilde{f}}_4={\theta }_1{\theta }_2+B_{12}+\left(C_{13}{C}_{23}+C_{13}{\theta }_2+C_{23}{\theta }_1+{\theta }_1{\theta }_2+B_{12}\right){\text{e}}^{{\eta }_3}\\ +\left(C_{14}{C}_{24}+C_{14}{\theta }_2+C_{24}{\theta }_1+{\theta }_1{\theta }_2+B_{12}\right){\text{e}}^{{\eta }_4}\\ +({\theta }_1{\theta }_2+\left(C_{23}+C_{24}\right){\theta }_1+\left(C_{13}+C_{14}\right){\theta }_2+B_{12}\\ +C_{13}{C}_{23}+C_{14}{C}_{23}+C_{13}{C}_{24}+C_{14}{C}_{24}){\text{e}}^{{\eta }_3+{\eta }_4}\text{exp}\left(A_{34}\right),\end{array}$(21)

where ${\eta }_3$, $B_{12}$, $\text{exp}\left(A_{34}\right)$ have been mentioned previously, and

$\{\begin{array}{l}{\eta }_4=k_4\left(m_4\sigma +p_{4}y-{\displaystyle \int \frac{\lambda k_4^2{m}_4^4+\alpha \left(t\right)p_4^2}{\beta \left(t\right)m_4}\text{d}t}\right)+{\gamma }_4,\\ m_2=m_1{}^{\text{*}}=a_1-b_{1}I,q_2=q_1{}^{\text{*}}=c_1-d_{1}I,\\ c_{ij}=\frac{12\lambda k_j{m}_i^3{m}_j^3}{-3\lambda k_j^2{m}_i^2{m}_j^4+\alpha \left(t\right){\left(m_i{p}_j-m_j{p}_i\right)}^2}\left(i=1,2;j=3,4\right).\end{array}$(22)

Then, substituting (21) and (22) into (4), we get the hybrid solution between 1-lump solution and 2-soliton. We give the diagrams of the hybrid solution between 1-lump solution and 2-soliton in Figures 12-15 , with different parameters as Figure 12 :

$\left(\lambda ,a_1,b_1,c_1,d_1,k_3,k_4,m_3,m_4,p_3,p_4,{\gamma }_3,{\gamma }_4\right)=\left(-1,-\frac{1}{2},\frac{5}{4},\frac{5}{2},-\frac{1}{2},-\frac{3}{4},\frac{3}{2},1,\frac{2}{5},3,\frac{1}{5},0,0\right)$, Figure 13 : $\left(\lambda ,a_1,b_1,c_1,d_1,k_3,k_4,m_3,m_4,p_3,p_4,{\gamma }_3,{\gamma }_4\right)=\left(-1,-\frac{1}{2},\frac{5}{4},\frac{5}{2},-1,\frac{1}{4},\frac{1}{2},\frac{4}{5},\frac{3}{4},\frac{3}{2},\frac{1}{4},0,0\right)$, Figure 14 , Figure 15 : $\left(\lambda ,a_1,b_1,c_1,d_1,k_3,k_4,m_3,m_4,p_3,p_4,{\gamma }_3,{\gamma }_4\right)=\left(-1,-\frac{3}{4},\frac{3}{5},\frac{3}{2},\frac{3}{4},\frac{1}{2},\frac{3}{2},3,\frac{2}{5},\frac{5}{2},\frac{1}{3},0,0\right)$.