(Top panel) Soliton solution for the coupling , which arise out of the background . The form of the soliton was calculated numerically with the help of Eq. (28) and the velocity is . Here and in the following figures, bold dots show the soliton on the lattice and the dashed line the solution of QCA (5) which has an additional offset for better visibility. (Bottom panel) The soliton in the logarithmic scale.
(Top panel) The shape of the compacton for the coupling . The background is and the velocity was set to . Markers show the compacton on the discrete lattice and the dashed line is the corresponding solution in the quasicontinuum approximation. (Bottom panel) The same plot in the logarithmic scale. The super exponentially decaying tails are clearly visible.
The values of uniform background states that can be connected by kinks, in the QCA for the particular coupling and various values of .
(Top panel) The shape of the kink for , the velocity is . (Center panel) The kink in the logarithmic scale. (Bottom panel) The kink shown from its top in the logarithmic scale. Markers show the kink on the lattice and the dashed line is the QCA with an additional offset for better visibility.
(Top panel) The wave form of the compact kink solution between and for the coupling . Markers show the wave form on the discrete lattice and the dashed line represents the quasicontinuum approximation of the kink. (Bottom panel) The sine of the kink in the logarithmic scale.
(Top panel) The shape of a kink with one exponential and one compact tail. The coupling for this specific wave is with . The position of the kink point and the wave velocity can be obtained from Eqs. (31) and (23) , their values are and . (Middle panel) The kink in the logarithmic scale. The compact tail of the kink is near clearly visible. (Bottom panel) The kink shown from its top in logarithmic scale, the exponential tail becomes visible.
The shape of a periodic wave on the lattice. The offset is , the wavelength is and the velocity .
Phase space of Eq. (10) for and (a) and (b) . The critical velocity is . It is clearly visible that below the critical velocity the fixed point is a center, hence periodic waves exist in Eq. (4) . Above one can observe solitonic solutions: the background is a saddle point and a homoclinic orbit exists.
(a) Wave speed in dependence of the imaginary eigenvalue , see Eq. (26) . The dots mark possible bifurcation points to real eigenvalues. (b) Wave speed vs the real and imaginary part of the eigenvalues, with . Only the positive parts of the real and imaginary axes are shown.
Solitary wave with periodic and exponentially decaying tails. and . The plot was generated with the help of the traveling wave scheme, Eq. (28) for the lattice equation; it has no QCA counterpart.
Evolution of different initial pulses for the coupling and . The initial conditions were set according to Eq. (32) . (a) and , (b) and , and (c) and . The dashed line shows the initial condition and the solid line the lattice at the time .
Evolution of different initial pulses for the coupling and the background . The initial conditions were set according to Eq. (32) . (a) and , (b) and , (c) and , (d) and , and (e) and . The dashed line shows the initial condition and the solid line the lattice at the time . Furthermore the position of the kink point at is shown.
Transition to chaos. The background is and the lattice contains sites with periodic boundary conditions. The field is shown in a gray scale vs time (horizontal axis) and space (vertical axis). (Upper plot) A kink and several solitons emerge out of a pulse. The time interval is . (Lower plot) Emergence of chaos after a collision of two solitons which creates an soliton–antisoliton pair. The time interval is .
Transient times to chaos for different backgrounds . The length of the lattice is and the initial -pulse is for and else. In order to obtain an average of the transient times we also varied the width of the initial pulse from to and calculated the transient time as the average over the transient times for different initial pulses. The line is an exponential fit and the transient times scales with .
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