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Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations Quantum chains with a Catalan tree pattern of conserved charges: The Δ=−1 XXZ model and the isotropic octonionic chain

We study nonlinear dynamics of inhomogeneous DNA double helical chain under dynamic plane-base rotator model by considering angular rotation of bases in a plane normal to the helical axis. The DNA dynamics in this case is found to be governed by a perturbed sine-Gordon equation while taking into account the interstrand hydrogen bonding energy between bases and the intrastrand inhomogeneous stacking energy and by making an analogy with the Heisenberg model of the Hamiltonian of an inhomogeneous anisotropic spin ladder with ferromagnetic legs and antiferromagnetic rung coupling. In the homogeneous limit the dynamics is governed by the kink-antikink soliton of the sine-Gordon equation which represents the formation of open state configuration in DNA double helix. The effect of inhomogeneity in stacking energy in the form of localized and periodic variations on the formation of open states in DNA is studied under perturbation. The perturbed soliton is obtained using a multiple scale soliton perturbation theory by solving the associated linear eigen value problem and by constructing the complete set of eigen functions. The inhomogeneity in stacking energy is found to modulate the width and speed of the soliton depending on the nature of inhomogeneity. Also it introduces fluctuations in the form of train of pulses or periodic oscillations in the open state configuration.

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Articles you may be interested inErratum: On the integrability of the inhomogeneous spherically symmetric Heisenberg ferromagnet in arbitrary dimensions [The dynamics of an inhomogeneous spherically symmetric continuum Heisenberg ferromagnet in arbitrary (n-) dimensions is considered. By a known geometrical procedure the spin evolution equation equivalently is rewritten as a generalized nonlinear Schrodinger equation. A Painleve singularity structure analysis of the solutions of the equation shows that the system is integrable in arbitrary (n-) dimensions only when the inhomogeneity is of inverse power in the radial coordinate in the formf(r)=~,~~2~n~'~+~2~~~~~2~. This is confirmed by obtaining the associated Lax pair, Backlund transformation, and the solitonlike solution of the evolution equation. Further, calculations show that the one-dimensional linearly inhomogeneous ferromagnet acts as a universal model to which all the integrable higher-dimensional inhomogeneous spherically symmetric spin models can be formally mapped.

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