Discretizing the continuous nonlinear SchrMinger equation with arbitrary power nonlinearity influences the time evolution of its ground state solitary solution. In the subcritical case, for grid resolutions above a certain transition value, depending on the degree of nonlinearity, the solution will oscillate smoothly with a frequency and amplitude that depend on both the resolution and the degree of nonlinearity. Thus in this region the discrete system will give a good reproduction of the dynamics in the continuum one. However, when the discretization gets too coaxe the solution will become localized in finite time, although the degree of nonlinearity is subcritical. Thus in this region the discrete system cannot reproduce the stable solitary solution found in the continuum system. Numerical studies of the phenomenon are performed and a variational approach is used to give a qualitative explanation of the dynamics in the discrete system.

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In this paper, a simple, two-dimensional model of the deoxyribonucleic acid (DNA) is presented.In the model the two polynucleotide strands are linked together through the hydrogen bonds. The phosphodiester bridges in the backbone are described by the anharmonic potential of Toda kind, while the hydrogen bonds are described by the Lennard-Jones potential. Longitudinal wave propagation on ring-shaped DNA molecules is investigated. The model predicts a significant increase in the lifetime of the open states of the hydrogen bonds at physiological temperatures. Thus anharmonicity may play a role in DNA denaturation.

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