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Construction of Partial Differential Equations with Conditional Symmetries

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Integrability, Supersymmetry and Coherent States

Part of the book series: CRM Series in Mathematical Physics ((CRM))

Abstract

Nonlinear PDEs having given conditional symmetries are constructed. They are obtained starting from the invariants of the conditional symmetry generator and imposing the extra condition given by the characteristic of the symmetry. Series of examples of new equations, constructed starting from the conditional symmetries of Boussinesq, are presented and discussed thoroughly to show and clarify the methodology introduced.

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Acknowledgements

DL has been supported by INFN IS-CSN4 Mathematical Methods of Nonlinear Physics. DL thanks ZT and the SUNY Polytechnic Institute for their warm hospitality at Utica when this work was started. DL thanks the Departamento de Física Téorica of the Complutense University in Madrid for its hospitality. MAR was supported by the Spanish MINECO under project FIS 2015-63966-P. D. Nedza, summer student of ZT, contributed to the verification of some of the computations.

Author information

Authors and Affiliations

  1. INFN, Sezione Roma Tre, Roma, Italy

    Decio Levi

  2. Dept. de Física Teórica, Universidad Complutense de Madrid, Madrid, Spain

    Miguel A. Rodríguez

  3. SUNY Polytechnic Institute, Utica, NY, USA

    Zora Thomova

Authors
  1. Decio Levi
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  2. Miguel A. Rodríguez
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  3. Zora Thomova
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Corresponding author

Correspondence to Decio Levi .

Editor information

Editors and Affiliations

  1. Department of Physics, Ankara University, Ankara, Turkey

    Şengül Kuru

  2. Department of Theoretical Physics, Atomic Physics and Optics, University of Valladolid, Valladolid, Spain

    Javier Negro

  3. Department of Theoretical Physics, Atomic Physics and Optics, University of Valladolid, Valladolid, Spain

    Luis M. Nieto

Appendix: Determining Equations for \(\hat X_2\) and \(\hat X_6\)

Appendix: Determining Equations for \(\hat X_2\) and \(\hat X_6\)

For completeness we present here the determining equation of the conditional symmetries when ξ = 1 and η = 0 for which we have been able just to present some particular solutions

$$\displaystyle \begin{aligned} &g_{xx}+3 f g_x+(5f'+4f^{2})g+4f^{4}y+7 f'f^{2}y-2f'f =0, {} \end{aligned} $$
(45)
$$\displaystyle \begin{aligned} &g_{yy}+ \left(5 yf^2 -5 y f'+3 g\right)g_x+2\left(8y f^2 +10 y f'+g\right) f g \\ & +2 y^2 f \left(13 f^4-20 f^{\prime 2}+16 f^2 f'\right) =0. {} \end{aligned} $$
(46)
$$\displaystyle \begin{aligned} &\phi_{uuuu}\phi^4+ (4 \phi_{uu}^2+6 \phi_{u} \phi_{uuu} +4 \phi_{xuuu} ) \phi^3+ (7 \phi_{uu} \phi_{u}^2+2 (6 \phi_{xuu} +1 ) \phi_{u} \\ & +\phi_{uu} (12 \phi_{xu} +u )+6 (\phi_{uuu} \phi_{x} +\phi_{xxuu} ) ) \phi^2+ (8 \phi_{xu}^2+2 (2 \phi_{u}^2+u ) \phi_{xu} \\ & +\phi_{x} (10 \phi_{u} \phi_{uu} +12 \phi_{xuu} +3 )+4 \phi_{uu} \phi_{xx} +6 \phi_{u} \phi_{xxu} +4 \phi_{xxxu} ) \phi \\& +3 \phi_{uu} \phi_{x}^2 +4 \phi_{u} \phi_{x} \phi_{xu} +u \phi_{xx} +4 \phi_{xu} \phi_{xx} +6 \phi_{x} \phi_{xxu} +\phi_{xxxx} +\phi_{y}=0. \end{aligned} $$
(47)

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Levi, D., Rodríguez, M.A., Thomova, Z. (2019). Construction of Partial Differential Equations with Conditional Symmetries. In: Kuru, Ş., Negro, J., Nieto, L. (eds) Integrability, Supersymmetry and Coherent States. CRM Series in Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-20087-9_17

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