Peakon solutions of a b-Novikov equation

Figures & data

Figure 1. The curves of a versus ε for the 4th-order approximation and for $\hslash =-1,-2,-2.5$.

Table 1. $W_M^{\prime }\left(0^{+}\right),a_M,I_M$ and corresponding errors for $b=3,ϵ=1/10$ and the optimum ℏ.

Order M	ℏ optimum	Error $E_M$	$W_M^{\prime }\left(0^{+}\right)$	${\alpha }_M$	$I_M$
8	−1.617353369	8.449544572$\times {10}^{-7}$	−1.008493714	0.973736	1.999996348
9	−1.444432581	4.530287629$\times {10}^{-8}$	−1.009953643	0.973721	2.000013154
10	−1.305368461	4.378280872$\times {10}^{-11}$	−1.011095919	0.97371	1.999995500
15	−0.8827970199	7.718036623$\times {10}^{-9}$	−1.014312999	0.973679	2.000057168
17	−2.178369930	1.052302111$\times {10}^{-7}$	−1.015023210	0.973832	2.000134411
20	−0.6676018923	1.083197693$\times {10}^{-9}$	−1.015800585	0.973671	1.999779293
24	−1.668253179	6.404635995$\times {10}^{-16}$	−0.9730587006	0.973798	2.012913
25	−1.614707284	7.123265317$\times {10}^{-17}$	−0.9687500000	0.973798	2.031331769

Figure 2. ℏ-curve for a for 25 terms in the case $b=3,ϵ=1/10$.

Figure 3. Comparison of the HAM solution in the case b = 3 for M = 25 with the exact solution for c = 1.

Table 2. $W_M^{\prime }\left(0^{+}\right),a_M,I_M$ and corresponding errors for $b=4,ϵ=1/10$ and the optimum ℏ.

Order M	ℏ optimum	Error $E_M$	$W_M^{\prime }\left(0^{+}\right)$	${\alpha }_M$	$I_M$
8	0.2224414534	0.001025598693	−0.8843128928	1.22975	2.009428999
9	−2.093167808	5.053988212$\times {10}^{-7}$	−1.053522706	0.975303	2.002245310
12	0.1513458008	0.001140536645	−1.054005292	1.234840	2.002222841
15	−1.262094776	3.351849415$\times {10}^{-9}$	−1.054279739	0.975295	2.002209671
18	−1.054498366	3.775279136$\times {10}^{-10}$	−1.054456710	0.975288	2.002201011
20	−2.143279832	1.419991572$\times {10}^{-10}$	−1.029730319	0.975293	2.005545567
22	−1.971675913	4.030587317$\times {10}^{-13}$	−1.030694337	0.975293	2.033309664
25	−0.7626484933	3.856846270$\times {10}^{-11}$	−0.9999145242	0.975278	2.000000

Table 3. $W_M^{\prime }\left(0^{+}\right),a_M,I_M$ and corresponding errors for $b=5,ϵ=1/10$ and the optimum ℏ.

Order M	ℏ optimum	Error $E_M$	$W_M^{\prime }\left(0^{+}\right)$	${\alpha }_M$	$I_M$
8	0.2643960987	0.001094055946	−1.044673189	1.13628	2.001854527
10	0.6058643104	0.001379775954	−1.045096267	1.41501	2.001837876
12	0.1810971670	0.001223653884	−0.9419726231	1.14019	2.003916570
15	0.4097508998	0.0008456267111	−0.9450279818	1.43047	2.003661605
17	0.1300017376	0.001306487514	−0.946835635	1.14271	2.003498190
22	−1.024664469	2.074067836$\times {10}^{-8}$	−1.045894174	0.969545	2.001801420
25	−0.9036690081	2.258940949$\times {10}^{-8}$	−0.9520986085	0.969552	2.003014430

Table 4. $W_M^{\prime }\left(0^{+}\right),a_M,I_M$ and corresponding errors for $b=7,ϵ=1/10$ and the optimum ℏ.

Order M	ℏ optimum	Error $E_M$	$W_M^{\prime }\left(0^{+}\right)$	${\alpha }_M$	$I_M$
8	0.363427489	0.001228857332	−1.035050491	0.982103	2.001473782
10	0.2971040509	0.001328117791	−1.035411545	0.983408	2.001456406
12	0.6007517097	0.004309264124	−1.035612117	0.971101	2.001444318
15	0.2043253458	0.001472465928	−1.035780877	1.001360	2.001431777
17	0.1816863830	0.001508201948	−1.035849219	0.993373	2.001425705
20	0.3871955685	0.0009707760879	−0.9809711265	0.985117	2.001547175
22	0.3510259272	0.0007746105176	−0.9816103837	0.981237	2.001483956
25	0.3072731873	0.0006458985424	−0.9824790221	0.976949	2.00001