On Existence of Solitons for the Second Harmonic Equations of a Laser Beam

V. Cao Longa, P.P. Goldsteinb and S. Vu Ngoc aInstitute of Physics, University of Zielona Góra, Podgórna 50, 65-246 Zielona Góra, Poland bThe Andrzej Soltan Institute for Nuclear Studies, Hoża 69, 00-681 Warsaw, Poland cFaculty of High Technology, Vinh University, Nghe An, Vietnam

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Received: 17 08 2004;

We look for conditions of existence of soliton solutions for equations governing propagation of a monochromatic laser beam coupled to its second harmonic in a nonlinear medium. The system proves to be non-integrable in the sense of Painlevé. However it is partially integrable for some values of its parameters. We further check the possibility of solving the equations by the Hirota bilinear method. The system is found to be solvable this way provided that amplitudes of both modes are equal while the complex phase of the second harmonic is equal to the double phase of the fundamental mode (modulo π). The Hirota scheme is found to work merely for exact resonance, i.e. for the ratio of the dispersion coefficients equal to the ratio of frequencies. Finally, all these conditions may only be satisfied by single envelope travelling waves, in which the envelope has locally the shape of the Weierstrass function.

DOI: 10.12693/APhysPolA.106.843 PACS numbers: 42.81.Dp