Periodic Waves in the Fractional Modified Korteweg–de Vries Equation
| Autor: | Uyen Le, Fábio Natali, Dmitry E. Pelinovsky |
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| Rok vydání: | 2021 |
| Předmět: |
Partial differential equation
010102 general mathematics Mathematical analysis Saddle-node bifurcation 01 natural sciences 010101 applied mathematics Quadratic equation Norm (mathematics) Ordinary differential equation 0101 mathematics Korteweg–de Vries equation Analysis Bifurcation Mathematics Energy functional |
| Zdroj: | Journal of Dynamics and Differential Equations. 34:1601-1640 |
| ISSN: | 1572-9222 1040-7294 |
| DOI: | 10.1007/s10884-021-10000-w |
| Popis: | Periodic waves in the modified Korteweg–de Vries (mKdV) equation are revisited in the setting of the fractional Laplacian. Two families of solutions in the local case are given by the sign-definite dnoidal and sign-indefinite cnoidal solutions. Both solutions can be characterized in the general fractional case as global minimizers of the quadratic part of the energy functional subject to the fixed $$L^4$$ norm: the sign-definite (sign-indefinite) solutions are obtained in the subspace of even (odd) functions. Morse index is computed for both solutions and the spectral stability criterion is derived. We show numerically that the family of sign-definite solutions has a generic fold bifurcation for the fractional Laplacian of lower regularity and the family of sign-indefinite solutions has a generic symmetry-breaking bifurcation both in the fractional and local cases. |
| Databáze: | OpenAIRE |
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