Mobility of discrete multibreathers in the exciton dynamics of the Davydov model with saturable nonlinearities

We show that the state of amide-I excitations in proteins is modeled by the discrete nonlinear Schrödinger equation with saturable nonlinearities. This is done by extending the Davydov model to take into account the competition between local compression and local dilatation of the lattice, thus leading to the interplay between self-focusing and defocusing saturable nonlinearities. Site-centered (sc) mode and/or bond-centered mode like discrete multihump soliton (DMHS) solutions are found numerically and their stability is analyzed. As a result, we obtained the existence and stability diagrams for all observed types of sc DMHS solutions. We also note that the stability of sc DMHS solutions depends not only on the value of the interpeak separation but also on the number of peaks, while their counterpart having at least one intersite soliton is instable. A study of mobility is achieved and it appears that, depending on the higher-order saturable nonlinearity, DMHS-like mechanism for vibrational energy transport along the protein chain is possible.

Figure 1

Examples of profiles of single-hump [(a) ($\tilde{\eta };\Omega )\approx$ (1.1;96.706)], two-hump [(b) ($\tilde{\eta };\Omega )\approx$ (1.1;100.4)], three-hump [(c) ($\tilde{\eta };\Omega )\approx$ (1.1;101.75)], and four-hump [(d) ($\tilde{\eta };\Omega )\approx$ (1.1;102)] solitons.

Figure 2

Examples of profiles of two-hump [(a) ($\tilde{\eta };\Omega )\approx$ (1.1;100.9)], two-hump [(b) ($\tilde{\eta };\Omega )\approx$ (1.1;100.9)], and three-hump [(c) ($\tilde{\eta };\Omega )\approx$ (1.1;101.75)] solitons for bc-sc mode, bc-bc mode, and sc-bc-sc mode, respectively.

Figure 3

Stability diagram of the fixed points of Eq. (19) (top left panel) in the $\Omega -\tilde{\eta }$ parameter plane. The areas marked in blue, red, and yellow are those for which $p_0$, $p_1$, and $p_2$ are, respectively, unstable saddle nodes. In the top right and bottom left panels, we have intersections of stable and unstable manifolds for ($\Omega ,\tilde{\eta }$) = ($4.4,0.06$) and ($5,0.06$), respectively. The red dots being the fixed points. In the bottom right panel, we have the map orbits around the fixed point $p_0$ according to $\Omega$, $\tilde{\eta }=0.06.$

Figure 4

Domain of existence of the intersection of manifolds. Condition of obtaining bright soliton solution.

Figure 5

Profiles of $\text{two}-\text{hump}$ solitons (left column) and their correponding eigenvalue spectrum (right column) with increasing (from top to bottom) values of ${\ell }_2$: ${\ell }_2=4$; 8; 12 and 32. $\Omega \approx$ 3.6, $\tilde{\eta }\approx$ 0.0464.

Figure 6

Profiles of $\text{four}-\text{hump}$ solitons (left column) and their correponding eigenvalue spectrum (right column) with increasing (from top to bottom) values of ${\ell }_4$: ${\ell }_4=16$; 26 and 50. $\Omega \approx$ 4.9, $\tilde{\eta }\approx$ 0.056.

Figure 7

Development of instability of two-hump solitons having at least one intersite soliton for [($\Omega ,\tilde{\eta }$) = ($100.9,1.1$)]. The left and right panels correspond to bc-sc mode and bc-bc mode, respectively.

Figure 8

Existence and stability diagrams for all observed types of sc DMHS solutions. In the top left panel, we have the existence diagram for one-hump solitons (red), two-hump solitons (blue), and three-hump solitons (black). Inset (d) shows the case where $\tilde{\eta }>1$. Other insets, (a), (b), and (c), are related to observed types of sc DMHS. The stability diagram of one-hump and two-hump (${\ell }_2=$ 32) solitons is shows in the top right and bottom left panels, respectively. The bottom right figure illustrates the existence (dot) and stability (circle) diagram for four-hump (${\ell }_4=50$) solitons. Insets show the case where $\tilde{\eta }>1$.

Figure 9

Density plot $|{\varphi }_n{|}^2$ for not kicked (a) two-hump, (b) three-hump, (c) four-hump solitons and their kicked counterparts, (d), (e), and (f), respectively. We have (d) $\varpi =0.55$, $\Omega \approx$ 3.6, $\tilde{\eta }\approx$ 0.0464; (e) $\varpi =0.3$, $\Omega \approx$ 5.635, $\tilde{\eta }\approx$ 0.065; (f) $\varpi =0.3$, $\Omega \approx$ 4.9, $\tilde{\eta }\approx$ 0.056; respectively. Mobility is achieved for $\Omega \simeq \eta \left(\frac{1}{1+{\alpha }^2}\right)$, (d) $\alpha =0.4777$, (e) $\alpha =0.2765$, (f) $\alpha =0.2549$.