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In this paper, we develop a numerical model based on a multiphasic theory to simulate the deformation response of a hydrogel strip immersed into an acidic solution under an external electric field. The deformation response consists of complicated mechano-electrochemical behaviours including mechanical effects (pressure and diffusive drag), chemical effects (concentration, chemical potential, osmotic pressure) and electric effects (electric field intensity, electric potential). The complexly coupled nonlinear governing equations are numerically solved using a recently developed meshless radial basis function method. We analyse the major factors which influence the swelling/shrinking behaviours of the hydrogel strip. The numerical results show good agreement with the experimental

Based on a multidimensional Riemann theta function, the Hirota bilinear method is extended to explicitly construct multiperiodic (quasiperiodic) wave solutions for the (2+1) -dimensional Bogoyavlenskii breaking soliton equation. Among these periodic waves, the one-periodic waves are well-known cnoidal waves, their surface pattern is one-dimensional, and often they are used as one-dimensional models of periodic waves in shallow water. The two-periodic (biperiodic) waves are a direct generalization of one-periodic waves, their surface pattern is two dimensional, that is, they have two independent spatial periods in two independent horizontal directions. The two-periodic waves may be considered to represent periodic waves in shallow water without the assumption of one dimensionality. A limiting procedure is presented to analyze asymptotic behavior of the one- and two-periodic waves in details. The exact relations between the periodic wave solutions and the well-known soliton solutions are established. It is rigorously shown that the periodic wave solutions tend to the soliton solutions under a "small amplitude" limit.

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