Abstract
The soliton theory in mathematical physics has been generally established for spatially one-dimensional(1D), continuous systems except for a few exceptional cases such as the Toda lattice.1 Underlying concepts are the inverse-scattering-theory formalism,2 the Hirota bilinear form using the D-operators,3 the Sato τ-function theory4 and so on. An open question and issue are whether or not the concepts developed in mathematical physics and mathematics are equally applicable to realistic problems in physics, such as higher dimensional continuous systems, discrete lattices, solids, small molecules, macromolecules, etc. and biological systems. In field theory, finite-energy, stable localized classical solutions of nonlinear differential equations have often been identified as solitons,5, while nonlinear differential difference equations in solid state physics, biophysics and biochemistry have been reduced to differential equations by using the continuum approximation to get solitons.
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References
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Takeno, S. (1993). Model Hamiltonian, Coherent State and Anharmonic Localized Modes as Dynamical Self-Trapping in Nonlinear Systems and Biological Macromolecules. In: Christiansen, P.L., Eilbeck, J.C., Parmentier, R.D. (eds) Future Directions of Nonlinear Dynamics in Physical and Biological Systems. NATO ASI Series, vol 312. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1609-9_34
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DOI: https://doi.org/10.1007/978-1-4899-1609-9_34
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