TY - JOUR
T1 - Corrigendum to “Regular and singular pulse and front solutions and possible isochronous behavior in the short-pulse equation
T2 - Phase-plane, multi-infinite series and variational approaches” [Commun Nonlinear Sci Numer Simul 20 (2015) 375–388, (Communications in Nonlinear Science and Numerical Simulation (2015) 20(375-388) (S1007570414002743), (10.1016/j.cnsns.2014.06.011))
AU - Choudhury, S. R.
AU - Ghose Choudhury, A.
AU - Gambino, G.
AU - Guha, P.
AU - Tanriver, U.
N1 - Publisher Copyright:
© 2014 Elsevier B.V.
PY - 2022/11
Y1 - 2022/11
N2 - Section 7 of the original paper contained several errors which are corrected here. Equations [Formula presented] and [Formula presented] are incorrect. In the following, the corrected versions of these equations are given and the subsequent results of Section 7 are also revised. The corrected variational Euler–Lagrange equations, which substitute the old equations [Formula presented] and [Formula presented], are the following system of algebraic equations: [Formula presented] [Formula presented] As is the case for most variational solitons, explicit solutions of Eqs. (1)–(2) for the optimized amplitude and width, and which replace the incorrect [Formula presented] and [Formula presented] in the original paper, are now somewhat lengthy. They satisfy the cubic equation [Formula presented] with: [Formula presented] Rather than write out the lengthy roots for (3), the optimized regular soliton is thus considered for some typical cases. These replace the incorrect Figure 7 in the original paper. For [Formula presented], we obtain [Formula presented] and [Formula presented]. The corresponding optimized regular solitons, given by the equation [Formula presented] in the original paper, is shown in Fig. 1(a). Analogously, once we choose [Formula presented], the solution of system (1) is [Formula presented] and [Formula presented] and the corresponding regular soliton is shown in Fig. 1(b). Note that, unlike the high-accuracy results in earlier sections of the original paper, variational solitons are directly obtained, but do not have a high degree of numerical accuracy. For this reason, considering the discriminant of the variational cubic (3) to distinguish parameter regimes of [Formula presented] where the number of roots go from one to three (via a saddle–node bifurcation) is not considered here. The original cubic equation itself does not have the required accuracy to really conduct such further analysis reliably. We may also substitute our variational soliton given by [Formula presented] in the original paper for the above cases directly into the left hand side of the governing model [Formula presented] in the original paper to gauge the residual error at various points of our spatial [Formula presented] domain. For [Formula presented], this residual error is shown in Fig. 2(a). Analogously, for [Formula presented], the corresponding error is shown in Fig. 2(b). These replace the incorrect Figure 8 in the original paper. The second half of Section 7, i.e. Section 7.2 of the original paper, is correct as it stands, except that Equation [Formula presented] should be replaced by: [Formula presented] Note that the extension of the variational method to embedded solitary waves in this sub-section is still not that widely known outside the embedded solitary waves community.
AB - Section 7 of the original paper contained several errors which are corrected here. Equations [Formula presented] and [Formula presented] are incorrect. In the following, the corrected versions of these equations are given and the subsequent results of Section 7 are also revised. The corrected variational Euler–Lagrange equations, which substitute the old equations [Formula presented] and [Formula presented], are the following system of algebraic equations: [Formula presented] [Formula presented] As is the case for most variational solitons, explicit solutions of Eqs. (1)–(2) for the optimized amplitude and width, and which replace the incorrect [Formula presented] and [Formula presented] in the original paper, are now somewhat lengthy. They satisfy the cubic equation [Formula presented] with: [Formula presented] Rather than write out the lengthy roots for (3), the optimized regular soliton is thus considered for some typical cases. These replace the incorrect Figure 7 in the original paper. For [Formula presented], we obtain [Formula presented] and [Formula presented]. The corresponding optimized regular solitons, given by the equation [Formula presented] in the original paper, is shown in Fig. 1(a). Analogously, once we choose [Formula presented], the solution of system (1) is [Formula presented] and [Formula presented] and the corresponding regular soliton is shown in Fig. 1(b). Note that, unlike the high-accuracy results in earlier sections of the original paper, variational solitons are directly obtained, but do not have a high degree of numerical accuracy. For this reason, considering the discriminant of the variational cubic (3) to distinguish parameter regimes of [Formula presented] where the number of roots go from one to three (via a saddle–node bifurcation) is not considered here. The original cubic equation itself does not have the required accuracy to really conduct such further analysis reliably. We may also substitute our variational soliton given by [Formula presented] in the original paper for the above cases directly into the left hand side of the governing model [Formula presented] in the original paper to gauge the residual error at various points of our spatial [Formula presented] domain. For [Formula presented], this residual error is shown in Fig. 2(a). Analogously, for [Formula presented], the corresponding error is shown in Fig. 2(b). These replace the incorrect Figure 8 in the original paper. The second half of Section 7, i.e. Section 7.2 of the original paper, is correct as it stands, except that Equation [Formula presented] should be replaced by: [Formula presented] Note that the extension of the variational method to embedded solitary waves in this sub-section is still not that widely known outside the embedded solitary waves community.
UR - http://www.scopus.com/inward/record.url?scp=85131552525&partnerID=8YFLogxK
U2 - 10.1016/j.cnsns.2022.106592
DO - 10.1016/j.cnsns.2022.106592
M3 - Comment/debate
AN - SCOPUS:85131552525
SN - 1007-5704
VL - 114
JO - Communications in Nonlinear Science and Numerical Simulation
JF - Communications in Nonlinear Science and Numerical Simulation
M1 - 106592
ER -