Consider the following map (complex plane to the complex plane itself) which conserves its absolute value.

or

Here are complex numbers and functions of. denotes the complex conjugate of X. As the numerator and the denominator of Equation (4) are complex conjugate with each other, the absolute value of Equation (4) is 1. Therefore Equation (4) represents a phase rotation map. The equivalence of Equations (4) and (5) is proved easily as follows. From Equation (5)

Then

It is easily shown that LFMT phase rotations are closed with respect to inversion and composition. (see Equations (44) and (55) in Appendix A)

Generally A, B can be any function of r. However we restrict our discussion to the following 3 cases for simplicity. We discuss type-0 mainly in this study.

(1) type-0

(2) type-1

(3) type-2

Here, are constant complex numbers, and we assume is constant for all cases. There are several formulas on LFMT phase rotations, and we will summarize them in appendix A. We also summarize the extension of this discrete phase rotation to the continuous one in appendix B. Additionally, we use the following notation for simplicity.

[Definition]

denotes the function.

denotes (namely the phase rotation of the double angle of the argument).

For example, by using the above definitions, is simplified

as or.

The function which multiplies the constant k is written as [k]. Note that $ is not included in [] in this case. Complex conjugate operator C, inversion operator V are defined as

respectively. We will omit the function composition symbol when [..] are uses.

In LFMT phase rotation (type-0)

corresponds to small or large amplitude region respectively.

In small amplitude region this map becomes a linear map and in large amplitude region, this becomes a linear map with complex conjugation. In Figure 2, we illustrate LFMT phase rotation in the case of A = B = 1. We

can see that this map closes to a linear map at the limits of.

Approaches from fixed point analysis are useful when we investigate the characteristics of complex-LFMT NLQCA. In general a map from a circle to the circle itself is called a circle map. LFMT phase rotation of Equation (4) is a circle map for any fixed r. Now we find fixed points of this circle map. The equation for the fixed points is

Apparently the Equation (10) is satisfied when, Namely

Therefore, Equation (11) has a real solution if and only if. In order to investigate the stability of these fixed points, we calculate the gain of the linearized map around the fixed point. Let x be the small angle deviation from, then we obtain the linearized map as follows.

This means the phase gain at the fixed point is

The 2nd equation in the parentheses of Equation (13) is obtained by using Equation (11). If is a fixed point then is another fixed point and it can be proved that the product of the gains for these two fixed points is 1 as shown in Equation (14).

because the product of the two denominators is

Therefore one fixed point is stable and the other fixed point is unstable or both fixed points are critical (marginally stable/unstable). From Equation (13), the critical situation means, namely or . Moreover, from (11) we can see that in the case and in the case. is merely the lower bound of the condition for a critical point to exist

From this section we investigate real-LFMT NLQCA. Firstly we explain the parameters A, B we use in the case of type-0 described by Equation (17).

As, we have only to investigate 32 out of 64 (= 8 × 8) parameter sets, and moreover about a half of those 32 is sufficient if we employ the space reflection symmetry (P) which we explain next.

We consider the following CPTA symmetry (Equations (18)-(21)). We find that the NLQCAs with a parameter which belongs to the same parameter groups behave qualitatively similarly.

・ C-inversion (see Equations (45) and (46) in Appendix A)

・ P-inversion (see Equations (45) and (46) in Appendix A)

・ T-inversion (see Equation (44) in Appendix A)

・ A-inversion (see Equation (45) in Appendix A)

These C, P, T, A have the following properties.

(2) A commutes with P, T, C

(3) P commutes with T (PT = TP)

(4) C “anti”-commute with T, P (CP = PCA, CT = TCA)

We define the 6 groups C, D, E, F, G, H so that ID1 and ID2 belongs to the same group if ID1 and ID2 can be transformed each other by C, P, T, A. IDs that belongs to the same group have qualitatively similar behaviors.

[Remark]

These symmetry formulas are same for the type-2, because A, B can be regarded as function of |$| and we can replace A with in Equations (18)-(21).

The CPTA mapping for ID = (n, m) of Equation (17) is shown in Table 1.

As illustrated in Figure 6, C-inversion is simply a redefinition of the sign of the amplitudes. A-inversion is also redefinition of the sign (in this case sign inversion of the all amplitudes).

In this section we show simulated waveform of type-G/E/C. For these cases, evolution of the waveform can be explained almost by comparing to the corresponding continuous time counterpart equation we discuss later. In Figure 7 and Figure 8 the simulated waveforms of type-G are shown. In Figure 9 the simulated waveform of type-E is shown. In Figure 10 the simulated waveform of type-C is shown. In all cases initial (t =

0) waveform is set to the Gaussian form and

simulated both in forward time (blue) and backward time (red) with a periodic boundary condition. Backward time evolution is performed using T-inversion formula (20).

Two-point-averaged are plotted and the spatial axis indicates.

Type-G (Figure 7) corresponds to the in viscid Burgers equation, where T-inversion simply corresponds to the inversion of moving direction. After the steep slope collapses, the NLQCA waveform goes out of the applicable range of in viscid Burgers equation approximation and KdV’s soliton-like wave packets spawn (see Figure 8).

In type-E (Figure 9) we can observe very slow diffusive behavior. This can be approximated by the certain kind of non-linear diffusion equation which we discuss later. In backward time simulation the sign of the amplitude is inverted at the beginning then behaves as a non-linear diffusion equation. This initial amplitude inversion behavior can be interpreted as the transition process from the non-positive diffusion constant to the positive one.

Type-C (Figure 10) lies between type-G and type-E and behaves like a Burgers equation with viscosity, namely the steep slope does not collapse and keeps on moving with lowering its height. In backward time simulation the sign of the amplitude is inverted at the beginning as in type-E then behaves like a viscid Burgers equation.

In this section we show that type-F and type-H can be understood from the view point of fixed point. In the case of real-LMFT NLQCA the meaning of a fixed point is slightly different from that in the case of complex-LMFT NLQCA we discussed before. In this case, a fixed point waveform does not keep still but propagates to ±45 deg. direction in the spacetime. Now we consider the (pseudo) fixed point equation for a fixed waveform moving to the left or right at the speed of one. Namely

or equivalently using,

Here means a true fixed point or a pseudo fixed point respectively, and “pseudo” means accepting temporal sign alternation.

As

(from Equations (45) (46) (48) in Appendix A), Equation (23) can be rewritten as

Obviously, in order for z to be a true fixed point or a pseudo fixed point, must be

where R, I denotes the set of real numbers and the set of pure imaginary numbers respectively. Sufficient (and presumably necessary) conditions for Equation (25) are

Namely, using Equation (24)

From Equation (17) and Table 1, we conclude that type-F has true fixed points, and type-H has pseudo fixed points.

In Figure 11, Figure 12, Figure 13, simulated waveforms of type-H, type-F, type-D are shown respectively. In all cases initial (t = 0) waveform is set to the Gaussian form

and simulated both in forward time (blue)

and backward time (red) with a periodic boundary condition. Backward time evolution is performed using T-inversion formula (20). Two-point-averaged are plotted

and the spatial axis indicates x/2. In all cases

waveforms around the start time, after a short time (t = 9900 to 10100) and after a long time (t = 499900 to 500100) are shown. Type-H (Figure 11) is half stable and Type-F (Figure 12) is super-stable and Type-D (Figure 13) is unstable. These results correspond to the above fixed point analysis. In these simulations we used

the NLQCA parameter in all cases therefore in the small amplitude limit the phase rotation becomes. This corresponds to the

advection type linear QCA with advection speed where the time evolution of the waveform is described as the superposition of the left-moving and right- moving wave packets (see [22] ). In this case,

component moves right with temporal sign alternation, and component moves left with a constant sign. (corresponds to and corresponds to in Equation (27) respectively.)

In type-H (Figure 11) two mountain-shaped wave packets move to opposite directions. The right-moving wave packet (which corresponds to the pseudo fixed point) is stable forever whereas the left-moving wave packet becomes unstable in the long run. Here only the waveforms for even t are plotted, therefore the temporal sign alternation of the right-moving wave packet cannot be seen. In type-F (Figure 12) the right-mov- ing wave packet disappears soon whereas the left-moving wave packet (which corresponds to the true fixed point) is super-stable. In type-D (Figure 13) the waveform is similar to type-F, namely the right-moving wave packet disappears soon whereas the left-moving wave packet keeps moving. However this left-moving wave packet becomes unstable in the long run.

It is known that the continuum limit of the simplest QCA becomes linear advection equation or TDSE (see for example [22] ). We find that in the case of type-0 real-LFMT NLQCA the continuum limit becomes a nonlinear advection-diffusion equation. Concretely, in type-0 real-LFMT NLQCA, the continuum limit exists if B is real. Consider the case where the velocity depends on the amplitudes (a, b, c, d) in the advection type QCA (see for example [22] ).

As is the average of the amplitude and is the difference

between the right side average and the left side average, p and (q/2) means the coeffi-

cients of ψ and in. Let be respectively then we have

By inserting Equation (28) to Equation (29) we have

Therefore

As Equation (31) implies that, we thus obtain the type-0 real- LFMT NLQCA

As NLQCA is unitary, its continuum limit must be unitary time evolution equation. The best candidate is

Note that the operators are anti-Hermite.

Equation (33) can also be rewritten in the following forms.

or if we set

This implies is a conserved quantity. We can regard this equation as nonlinear advection diffusion equation of which advection coefficient and diffusion coefficient are proportional to. When p = 0 this type of nonlinear diffusion equation is called as a porous-medium equation [21] [23] . (In this case for Equation (35) the degree of the

porosity). In the article [21] a certain kind of NLQW was proposed and it was

shown numerically that its continuum limit obeys a porous-medium equation with the degree of the porosity approximately 1.5.

In Figure 14, we demonstrate using numerical simulation that the continuum limit of NLQCA of Equation (32) is indeed well described by Equation (33). The PDE (33) is solved using Finite Difference Method (FDM) (Runge-Kutta (4th order)). Real PDE parameters are taken from A using Equation (32).

Type-2 real-LFMT NLQCA is related to type-0 real-LFMT NLQCA with a space inversion (see Equation (48) in Appendix A) and the basic 2 × 2 matrix of type-2 real LFMT NLACA becomes the form of Equation (36) in small amplitude limit.

The linear QCA governed by Equation (36) is related to the TDSE-type linear QCA of which basic 2 × 2 matrix is given by Equation (37).

Both linear QCAs (Equations (36) and (37)) have essentially the same dispersion relation Equation (38) and basically behave as TDSE [22] .

Note that according to the argument in [22] , the 2 × 2 unitary matrices in the Z- transformation representation for Equations (36) and (37) are

And the eigenvalues of and have essentially the same value except the constant phase factor, which leads to Equation (38). However in the limit,

eigenvectors of are, whereas eigenvectors of are. Therefore the linear QCA governed by Equation (36) (which does not have eigenvector) is thought to be described by the superposition of two (something like “forward going” and “backward going” corresponding to) TDSEs in the wavenumber

limit just like the advection type linear QCA (see [22] ). Now we assume the form of Equation (40) as in the case of type-0 NLQCA (Equations (28) (29)) expecting to obtain type-2 NLQCA.

However this time is expressed as Equation (41), not in a rational form.

It is not straightforward to represent type-2 real-LFMT NLQCA using continuum limit approach as in the case of type-0 NLQCA.

In this section, we try to understand the large amplitude behavior of type-2 NLQCA by relating it to the small amplitude behavior of type 0. Using both-sides inversion and

conjugation formula ((54) in Appendix A), we can state that if z evolves by type-0 NLQCA then evolves by type-2 NLQCA. So if obeying type-0 NLQCA varies slowly in space, and its continuum limit is some PDE for, then the continuum limit of type-2 NLQCA is expected to become approximately the PDE rewritten for.

(Note that it is important to consider the pair not of but of. Using means using

for the adjacent grid points pair which causes sign alternating behavior of.)

We numerically examine the validity of the approximation in the case of inviscid Burges equation. Inviscid Burgers case (type-G) is the most promising for the above

continuum limit argument to be applicable because the PDE for is unitary time

evolution too. (This can be easily verified by the fact that the flux in Equation (35) contains only.)

In Figure 15, we show the simulated NLQCA waveforms of both for

and for. Two waveforms match

well, although a certain adjusting parameter need to be introduced.

Note that we do not plot the solution of the inversed Burgers equation itself but plot its inverted values. Initial waveform for Burgers equation is

and its inverse is used for the inversed Burgers equation.