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The nonlocal symmetry is derived from the known Darboux transformation (DT) of the Hirota-Satsuma coupled Korteweg-de Vries (HS-cKdV) system, and infinitely many nonlocal symmetries are given by introducing the internal parameters. By extending the HS-cKdV system to an auxiliary system with five dependent variables, the prolongation is found to localize the so-called seed nonlocal symmetry related to the DT. By applying the general Lie point symmetry method to this enlarged system, we obtain two main results: a new type of finite symmetry transformation is derived, which is different from the initial DT and can generate new solutions from old ones; some novel exact interaction solutions among solitons and other complicated waves including periodic cnoidal waves and Painlevé waves are computed through similarity reductions. In addition, two kinds of new integrable models are proposed from the obtained nonlocal symmetry: the negative HS-cKdV hierarchy by introducing the internal parameters; the integrable models both in lower and higher dimensions by restricting the symmetry constraints.

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A method is proposed to seek the nonlocal symmetries of nonlinear evolution equations. The validity and advantages of the proposed method are illustrated by the applications to the Boussinesq equation, the coupled Korteweg-de Vries system, the Kadomtsev-Petviashvili equation, the Ablowitz-Kaup-Newell-Segur equation and the potential Korteweg-de Vries equation. The facts show that this method can obtain not only the nonlocal symmetries but also the general Lie point symmetries of the given equations.

SUMMARYIt has been observed in plate impact experiments that some brittle solids may undergo elastic deformation at the shock wave front, and fail catastrophically at a later time when they are shocked near but below the apparent Hugoniot elastic limit. Because this phenomenon appears to have features di erent from those of usual inelastic waves, it has been interpreted as the failure wave. To design an e ective numerical procedure for simulating impact failure responses, a three-dimensional computational damage model is developed in this paper. The propagation of the failure wave behind the elastic shock wave is described by a non-linear di usion equation. Macroscopic shear-induced dilatancy is assumed and treated as a one-to-one measure of the mean intensity of microcracking. The damage evolution in time is determined based on the assumption that the deviatoric strain energy in the elastically compressed material (undamaged) is converted, through the damaging process, into the volumetric potential energy in the comminuted and dilated material. For the ease in large-scale simulations, the coupled damage di usion equation and the stress wave equation are solved via a staggered manner in a single computational domain. Numerical solutions by using both the ÿnite element method and the material point method, i.e. with and without a rigid mesh connectivity, are presented and compared with the experimental data available. It is shown that the model simulations capture the essential features of the failure wave phenomenon observed in shock glasses, and that the numerical solutions for localized failure are not mesh-dependent.

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The nonlocal symmetry of the mKdV equation is obtained from the known Lax pair; it is successfully localized to Lie point symmetries in the enlarged space by introducing suitable auxiliary dependent variables. For the closed prolongation of the nonlocal symmetry, the details of the construction for a one-dimensional optimal system are presented. Furthermore, using the associated vector fields of the obtained symmetry, we give the reductions by the one-dimensional sub-algebras and the explicit analytic interaction solutions between cnoidal waves and kink solitary waves, which provide a way to study the interactions among these types of ocean waves. For some of the interesting solutions, the figures are given to show their properties.

Using the lax pair, nonlocal symmetry of the coupled Hirota equation is obtained. By introducing an appropriate auxiliary dependent variable the nonlocal symmetry is successfully localized to a Lie point symmetry. For the closed prolongation, one-dimensional optimal systems and nonlocal conservation laws of the coupled Hirota equation are studied.