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We present a wide class of novel solitary and periodic waves in anon-centrosymmetric waveguide exhibiting second- and third-ordernonlinearities. We show the existence of bright, gray, and W-shaped solitarywaves as well as periodic waves for extended nonlinear Schr\"{o}dinger equationwith quadratic and cubic nonlinearities. We also obtained the exact analyticalalgebraic-type solitary waves of the governing equation, including bright andW-shaped waves. The results illustrate the propagation of potentially rich setof nonlinear structures through the optical waveguiding media. Such privilegedwaveforms characteristically exist due to a balance among diffraction,quadratic and cubic nonlinearities.

Novel solitary and periodic waves in quadratic-cubic non-centrosymmetric waveguides