Discontinuous Galerkin Methods for the Ostrovskyxe2x80x93Vakhnenko Equation In this paper, we develop discontinuous Galerkin methods for the Ostrovskyxe2x80x93Vakhnenko (OV) equation, which yields the shock solutions and singular soliton solutions, such as peakon, cuspon and loop solitons. The OV equation has also been shown to have a biHamiltonian structure. We directly develop the energy stable or Hamiltonian conservative discontinuous Galerkin schemes for the OV equation. Error estimates for the two energy stable schemes are also proved. For some singular solutions, including cuspon and loop soliton solutions, the hodograph transformation is adopted to transform the OV equation or the generalized OV system to the coupled dispersionless (CD) system. Subsequently, two discontinuous Galerkin schemes are constructed for the transformed CD system. Numerical experiments are provided to demonstrate the accuracy and capability of the proposed schemes, including shock solution and, peakon, cuspon and loop soliton solutions.

BrightDark and Multi Solitons Solutions of (3  1)Dimensional CubicQuintic Complex Ginzburgxe2x80x93Landau Dynamical Equation with Applications and Stability In this paper, brightdark, multi solitons, and other solutions of a (3  1)dimensional cubicquintic complex Ginzburgxe2x80x93Landau (CQCGL) dynamical equation are constructed via employing three proposed mathematical techniques. The propagation of ultrashort optical solitons in optical fiber is modeled by this equation. The complex Ginzburgxe2x80x93Landau equation with broken phase symmetry has strict positive spacexe2x80x93time entropy for an open set of parameter values. The exact wave results in the forms of darkbright solitons, breathertype solitons, multi solitons interaction, kink and antikink waves, solitary waves, periodic and trigonometric function solutions are achieved. These exact solutions have key applications in engineering and applied physics. The wave solutions that are constructed from existing techniques and novel structures of solitons can be obtained by giving the special values to parameters involved in these methods. The stability of this model is examined by employing the modulation instability analysis which confirms that the model is stable. The movements of some results are depicted graphically, which are constructive to researchers for understanding the complex phenomena of this model.