Unstable novel and accurate soliton wave solutions of the nonlinear biological population model

Figures & data

Table 1. Values of computational and approximate solutions for $x\in [0,30].$

Value of x	Computational sol.	Approximate sol.	|Error|
0	2	1.99999997266803	2.73319740173861E-08
1	2	1.9999889734929	0.0000110265071027182
2	2	1.99555130249855	0.00444869750140042
3	2	0.159088354811456	1.84091164516811
4	1.99999999175539	−646.666666666667	648.666666658422
5	1.99999667388789	0.105768212673022	1.89422846121487
6	1.99865859947813	−1.99456904594458	3.99322764542271
7	1.52318831191153	−1.99998653670116	3.52317484861269
8	−1.92805516015163	−1.99999996662781	0.0719448064761767
9	−1.99981840852519	−2	0.000181591392088176
10	−1.99999954985935	−2	4.50140443364333E-07
11	−1.99999999888421	−2	1.11578724037997E-09
12	−2	−2	2.76578759894619E-12
13	−2	−2	6.88338275267597E-15
14	−2	−2	−4.44089209850063E-16
15	−2	−2	4.44089209850063E-16
16	−2	−2	4.44089209850063E-16
17	−2	−2	0
18	−2	−2	0
19	−2	−2	−2.22044604925031E-16
20	−2	−2	0
21	−2	−2	−2.22044604925031E-16
22	−2	−2	0
23	−2	−2	0
24	−2	−2	−2.22044604925031E-16
25	−2	−2	0
26	−2	−2	−4.44089209850063E-16
27	−2	−2	0
28	−2	−2	0
29	−2	−2	4.44089209850063E-16
30	−2	−2	−6.66133814775094E-16

Figure 1. Distinct graphs of Eq. (4)(4) $${\mathcal{B}}_{\mathrm{I},1}(x,y,t)=-\frac{\sqrt{\mathcal{S}} \left(e^{i\mathrm{ }\sqrt{\delta }\mathrm{ }(c\mathrm{ }t+x+i\mathrm{ }y)}-i\right)}{e^{i\mathrm{ }\sqrt{\delta }\mathrm{ }(c\mathrm{ }t+x+i\mathrm{ }y)}+i},$$(4) in three different types (3 D, 2 D, contour) for its real, imaginary and absolute values.

Figure 2. Distinct graphs of Eq. (5)(5) $${\mathcal{B}}_{\mathrm{I},2}(x,y,t)=-\frac{\sqrt{\mathcal{S}} \left(1+e^{i\mathrm{ }\sqrt{\delta }\mathrm{ }(c\mathrm{ }t+x+i\mathrm{ }y)}\right)}{-1+e^{i\mathrm{ }\sqrt{\delta }(c\mathrm{ }t+x+i\mathrm{ }y)}}.$$(5) in three different types (3 D, 2 D, contour) for its real, imaginary and absolute values.

Figure 3. Distinct graphs of Eq. (6)(6) $${\mathcal{B}}_{\text{II},1}(x,y,t)=i \sqrt{\mathcal{S}} \hspace{0.17em}\text{tan}\hspace{0.17em}\left(\sqrt{\delta } (c t+x+i y)\right),$$(6) in three different types (3 D, 2 D, contour) for its real, imaginary and absolute values.

Figure 4. Distinct graphs of Eq. (7)(7) $${\mathcal{B}}_{\text{II},2}(x,y,t)=-i \sqrt{\mathcal{S}} \text{cot}\left(\sqrt{\delta } (c t+x+i y)\right).$$(7) in three different types (3 D, 2 D, contour) for its real, imaginary and absolute values.

Figure 5. Matching between computational and approximate solutions based on the calculated values in Table 1.

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.