A Riemann-Hilbert Approach for the Novikov Equation
We develop the inverse scattering transform method for the Novikov equation ut−utxx+4u²ux=3uuxuxx+u²uxxx considered on the line x∈(−∞,∞) in the case of non-zero constant background. The approach is based on the analysis of an associated Riemann-Hilbert (RH) problem, which in this case is a 3×3 matri...
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| Дата: | 2016 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2016
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/147860 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | A Riemann-Hilbert Approach for the Novikov Equation / A. Boutet de Monvel, D. Shepelsky, L. Zielinski // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 44 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1478602019-02-17T01:23:13Z A Riemann-Hilbert Approach for the Novikov Equation Boutet de Monvel, A. Shepelsky, D. Zielinski, L. We develop the inverse scattering transform method for the Novikov equation ut−utxx+4u²ux=3uuxuxx+u²uxxx considered on the line x∈(−∞,∞) in the case of non-zero constant background. The approach is based on the analysis of an associated Riemann-Hilbert (RH) problem, which in this case is a 3×3 matrix problem. The structure of this RH problem shares many common features with the case of the Degasperis-Procesi (DP) equation having quadratic nonlinear terms (see [Boutet de Monvel A., Shepelsky D., Nonlinearity 26 (2013), 2081-2107, arXiv:1107.5995]) and thus the Novikov equation can be viewed as a ''modified DP equation'', in analogy with the relationship between the Korteweg-de Vries (KdV) equation and the modified Korteweg-de Vries (mKdV) equation. We present parametric formulas giving the solution of the Cauchy problem for the Novikov equation in terms of the solution of the RH problem and discuss the possibilities to use the developed formalism for further studying of the Novikov equation. 2016 Article A Riemann-Hilbert Approach for the Novikov Equation / A. Boutet de Monvel, D. Shepelsky, L. Zielinski // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 44 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35Q53; 37K15; 35Q15; 35B40; 35Q51; 37K40 DOI:10.3842/SIGMA.2016.095 http://dspace.nbuv.gov.ua/handle/123456789/147860 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We develop the inverse scattering transform method for the Novikov equation ut−utxx+4u²ux=3uuxuxx+u²uxxx considered on the line x∈(−∞,∞) in the case of non-zero constant background. The approach is based on the analysis of an associated Riemann-Hilbert (RH) problem, which in this case is a 3×3 matrix problem. The structure of this RH problem shares many common features with the case of the Degasperis-Procesi (DP) equation having quadratic nonlinear terms (see [Boutet de Monvel A., Shepelsky D., Nonlinearity 26 (2013), 2081-2107, arXiv:1107.5995]) and thus the Novikov equation can be viewed as a ''modified DP equation'', in analogy with the relationship between the Korteweg-de Vries (KdV) equation and the modified Korteweg-de Vries (mKdV) equation. We present parametric formulas giving the solution of the Cauchy problem for the Novikov equation in terms of the solution of the RH problem and discuss the possibilities to use the developed formalism for further studying of the Novikov equation. |
| format |
Article |
| author |
Boutet de Monvel, A. Shepelsky, D. Zielinski, L. |
| spellingShingle |
Boutet de Monvel, A. Shepelsky, D. Zielinski, L. A Riemann-Hilbert Approach for the Novikov Equation Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Boutet de Monvel, A. Shepelsky, D. Zielinski, L. |
| author_sort |
Boutet de Monvel, A. |
| title |
A Riemann-Hilbert Approach for the Novikov Equation |
| title_short |
A Riemann-Hilbert Approach for the Novikov Equation |
| title_full |
A Riemann-Hilbert Approach for the Novikov Equation |
| title_fullStr |
A Riemann-Hilbert Approach for the Novikov Equation |
| title_full_unstemmed |
A Riemann-Hilbert Approach for the Novikov Equation |
| title_sort |
riemann-hilbert approach for the novikov equation |
| publisher |
Інститут математики НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/147860 |
| citation_txt |
A Riemann-Hilbert Approach for the Novikov Equation / A. Boutet de Monvel, D. Shepelsky, L. Zielinski // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 44 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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