Manning-type potential induced by kink scatterings with phonons in molecular chains with hyperbolic double-well substrates

TL;DR

The paper develops a continuum nonlinear Klein-Gordon framework for a 1D molecular chain with a deformable hyperbolic double-well substrate $V_{\mu}(x)$, where the single parameter $\mu$ tunes minima positions and barrier height to model isotope effects. It derives an analytic kink soliton solution with velocity-dependent width, and computes the kink energy and rest mass, showing distinct $\mu$-dependent trends across three physical regimes. By linearizing around the kink, it obtains a Schrödinger-type eigenproblem for kink-phonon scattering whose potential $U_{\mu}$ reduces to a Manning-type form; the lowest bound state and its ground-state wavefunction are obtained exactly, indicating an integrable scattering problem in the continuum limit. The results highlight a direct link between kink stability, isotope-driven substrate deformability, and Manning-like scattering, while pointing to future work on discrete spectra and broader physical contexts in molecular chains and biomolecules.

Abstract

A rescaled Manning potential is obtained in the analysis of scatterings of small- amplitude excitations with a kink defect. The generic model is a nonlinear Klein- Gordon Hamiltonian describing a one-dimensional chain of identical molecules, sub- jected to an hyperbolic single-particle substrate potential. To account for isotope effects that are likely to affect characteristic equilibrium parameters of the molec- ular chain, including the lattice spacing (i.e. the characteristic intermolecuar dis- tance) and/or the barrier height, the hyperbolic substrate potential is endowed with a real parameter whose variation makes it suitable for the description of molecu- lar excitations in a broad range of systems with inversion symmetry. These include hydrogen-bonded molecular crystals, α-helix proteins, long polymer chains and two- state quantum-tunneling systems in general. Double-well models with deformable profiles are relevant in physical contexts where the equilibrium configurations are sensitive to atomic or molecular substitutions, dilution, solvation and so on.

Figures (5)

• Figure 1: Plot of the hyperbolic double-well potential $V{\mu}(x)$ for $\mu=0.1$ (solid line), $\mu=0.5$ (dash line), $\mu=1.0$ (dash-dotted line) and $\mu=2.0$ (dot line). From left to right graphs: tunable minima but fixed barrier, tunable barrier height but fixed minima, tunable minima and tunable barrier height.
• Figure 2: Plots of $y(z)$ versus $z$ for $\mu=0.1$ (solid line), $\mu=0.5$ (dash line), $\mu=1.0$ (dash-dotted line) and $\mu=2.0$ (dot line). From left to right graphs: tunable minima but fixed barrier height, tunable barrier height but fixed minima, tunable minima and tunable barrier height.
• Figure 3: Kink rest mass $M_{sol}(\mu)$ (in units of the $\phi^4$ kink rest mass $M_{sol}(0)$), plotted versus $\mu$. Left graph: double-well model with fixed barrier height but tunable minima, middle graph: double-well model with fixed minima but tunable barrier height, right graph: double-well model with tunable barrier height and tunable minima.
• Figure 4: Energy shift $U_{\mu}$ versus $\mu$. Left graph: double-well model with fixed barrier height but tunable minima, middle graph: double-well model with fixed minima but tunable barrier height, right graph: double-well model with tunable barrier height and tunable minima.
• Figure 5: Groundstate wavefunction $g_0$ given by (\ref{['solmod']}) versus $z$ for $\mu=0.001$ (solid line), $\mu=0.5$ (dash line), $\mu=1.0$ (dash-dotted line) and $\mu=2.0$ (dot line). From left to right graphs: variable minima but fixed barrier, variable barrier but fixed minima, variable minima and barrier.