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Differential invariants of nonlinear equations vtt=f(x,vx)vxx+g(x,vx)
Communications in Nonlinear Science and Numerical Simulation 9 (2004) 81 www.elsevier.com/locate/cnsns
Addendum to the paper
Differential invariants of nonlinear equations vtt ¼ f ðx; vx Þvxx þ gðx; vxÞ q N.H. Ibragimov a, M. Torrisi b, A. Valenti a
b,*
Research Centre ALGA: Advances in Lie Group Analysis, Department of Health, Science and Mathematics, Blekinge Institute of Technology, 371 79 Karlskrona, Sweden b Department of Mathematics and Informatics, University of Catania, A. Doria, 6, 95125 Catania, Italy
We assumed in Theorem 2 that l1 6¼ 0, where l1 is the expression defined by Eq. (77). If l1 ¼ 0, the corresponding Eq. (8) should be considered separately. Furthermore, the cases l2 ¼ 0 and l3 ¼ 0 (see Eqs. (78) and (79)) are also exceptional because of the following lemma: Lemma. The equations l1 ¼ 0, l2 ¼ 0 and l3 ¼ 0 are invariant with respect to the equivalence group E. Proof. The invariance test Y ðli Þjli ¼0 ¼ 0;
i ¼ 1; 2; 3;
is manifestly satisfied for the twice prolonged operators Y5 , Y6 , Y7 , and YF . One can verify that the invariance test is satisfied for the remaining operator Yuð2Þ as well, namely: Yuð2Þ ðl1 Þjl1 ¼0 ¼ 2l1 jl1 ¼0 ¼ 0; Yuð2Þ ðl2 Þjl2 ¼0 ¼ 2l2 jl2 ¼0 ¼ 0;
Yuð2Þ ðl3 Þjl3 ¼0 ¼ 4l3 jl3 ¼0 ¼ 0:
Acknowledgement The authors would like to thank Professor C. Sophocleous for drawing their attention to the special cases discussed above.
q
doi of original article 10.1016/S1007-5704(03)00016-9. Corresponding author. Fax: +39-95-7337039. E-mail address: [email protected] (A. Valenti).
*
1007-5704/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2003.09.001
Addendum to the paper
Differential invariants of nonlinear equations vtt ¼ f ðx; vx Þvxx þ gðx; vxÞ q N.H. Ibragimov a, M. Torrisi b, A. Valenti a
b,*
Research Centre ALGA: Advances in Lie Group Analysis, Department of Health, Science and Mathematics, Blekinge Institute of Technology, 371 79 Karlskrona, Sweden b Department of Mathematics and Informatics, University of Catania, A. Doria, 6, 95125 Catania, Italy
We assumed in Theorem 2 that l1 6¼ 0, where l1 is the expression defined by Eq. (77). If l1 ¼ 0, the corresponding Eq. (8) should be considered separately. Furthermore, the cases l2 ¼ 0 and l3 ¼ 0 (see Eqs. (78) and (79)) are also exceptional because of the following lemma: Lemma. The equations l1 ¼ 0, l2 ¼ 0 and l3 ¼ 0 are invariant with respect to the equivalence group E. Proof. The invariance test Y ðli Þjli ¼0 ¼ 0;
i ¼ 1; 2; 3;
is manifestly satisfied for the twice prolonged operators Y5 , Y6 , Y7 , and YF . One can verify that the invariance test is satisfied for the remaining operator Yuð2Þ as well, namely: Yuð2Þ ðl1 Þjl1 ¼0 ¼ 2l1 jl1 ¼0 ¼ 0; Yuð2Þ ðl2 Þjl2 ¼0 ¼ 2l2 jl2 ¼0 ¼ 0;
Yuð2Þ ðl3 Þjl3 ¼0 ¼ 4l3 jl3 ¼0 ¼ 0:
Acknowledgement The authors would like to thank Professor C. Sophocleous for drawing their attention to the special cases discussed above.
q
doi of original article 10.1016/S1007-5704(03)00016-9. Corresponding author. Fax: +39-95-7337039. E-mail address: [email protected] (A. Valenti).
*
1007-5704/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2003.09.001