Dynamics of solitons for nonlinear quantum walks

We present some numerical results for nonlinear quantum walks (NLQWs) studied by the authors analytically \cite{MSSSS18DCDS, MSSSS18QIP}. It was shown that if the nonlinearity is weak, then the long time behavior of NLQWs are approximated by linear quantum walks. In this paper, we observe the linear decay of NLQWs for range of nonlinearity wider than studied in \cite{MSSSS18DCDS}. In addition, we treat the strong nonlinear regime and show that the solitonic behavior of solutions appears. There are several kinds of soliton solutions and the dynamics becomes complicated. However, we see that there are some special cases so that we can calculate explicit form of solutions. In order to understand the nonlinear dynamics, we systematically study the collision between soliton solutions. We can find a relationship between our model and a nonlinear differential equation.


Introduction
Quantum walks (QWs), which are quantum analog of classical random walks [1,7,10,18] are now attracting many interest because of its connection to various regime in mathematics, physics and applications such as quantum algorithms [2,5,20] and topological insulators [3,4,6,9,12,13,14]. Nonlinear quantum walks (NLQWs), which are nonlinear version of QWs, have been recently proposed by several authors [8,11,19] and in particular related to some nonlinear differential equations such as nonlinear Dirac equations [11]. In [16], we have initiated analytical study of NLQWs using the methods developed for the study of nonlinear dispersive equations. More precisely, it was shown in [16] that the scattering phenomena for NLQWs in the weak nonlinear regime, that is, the behavior of solutions of NLQWs can be approximated by that of corresponding to linear quantum walks. We further proved a weak limit theorem, which is one of the main theme of the study of QWs since the celebrated result by Konno [15], for NLQWs in [17]. In [16,17] we have concentrated in the case when the nonlinearity is weak and proved results showing the similarity of QWs and NLQWs.
In this paper, we present some theoretical and numerical results of NLQWs to understand the effect of nonlinearity when the nonlinearity is strong. Solitonic behavior is one of the characteristics of nonlinear systems. We focus on such behavior to understand the dynamics of NLQWs in a strong nonlinear regime, in which the dynamics generally becomes complicated. However, there are some special cases where we can obtain explicit forms of solutions.
We now introduce the NLQWs we study in this paper. Let C : R × R → U (2), where U (2) is the set of 2 × 2 unitary matrices. For a matrix A ∈ U (2) and a vector u ∈ C 2 , A T and u T denote the transpose of A and the transpose of u, respectively. We define the (nonlinear) quantum coin C : l 2 (Z; C 2 ) → l 2 (Z; C 2 ) by (Ĉu)(x) = C(|u 1 (x)| 2 , |u 2 (x)| 2 )u(x), (1.1) where u(·) = (u 1 (·), u 2 (·)) T ∈ l 2 (Z; C 2 ). For (T ± u)(x) = u(x ∓ 1), we set that is, Here, e j (j = 1, 2) denote e 1 = (1, 0) T and e 2 = (0, 1) T . Let the one-step nonlinear time evolution operator U depending on a given state u ∈ l 2 (Z, C 2 ) be which is a map from l 2 (Z; C 2 ) to itself. Then, the state u t of the walker at time t is defined by the recursion relation with some initial state u 0 ∈ l 2 (Z; C 2 ) with u 0 l 2 = 1. Notice that l 2 norm will be conserved, i.e. u t l 2 = 1 for all t ∈ N. We define the nonlinear evolution operator U (t) as Throughout this paper, we shall consider the following concrete quantum coin: where g ∈ R, p ≥ 1 and R(θ) is the θ-rotation matrix, that is, Thus for given u,Ĉ acts as The value of |g| is a strength of the non-linearity. Thus in the case where |g| is not small, it is expected that a nonlinear effect arise. Indeed, in [16], it has been shown that u t scatters for p ≥ 2 and |g| ≪ 1. To this end, let us prepare the following remark. In order to make the dependence on g explicit, we write U g (t) for U (t) defined by (1.4) with the coin (1.5). We observe that, for v 0 = |g| p/2 u 0 with u 0 l 2 = 1, . Hence, we can fix |g| = 1 and vary the norm value of u 0 l 2 . Here, a small u 0 l 2 corresponds to the case where |g| is small which means a small nonlinear strength and vice versa. Therefore, we study the case g = ±1 for (1.5) and see the dependence on the initial state.
The organization of this paper is the following. In Section 2, we give some explicit example of soliton solutions of NLQWs and discuss its stability/instability. In Section 3, we numerically study the behavior of l ∞ norm ( u l ∞ = sup x∈Z u(x) C 2 ) and observe three types of behavior. That is, linear decay, soliton and oscillation. The oscillation solution is a new type of solution and we will give some theoretical explanation in Subsection 3.1. In Section 4, we systematically study the collision of solitons by numerical simulation. In Section 5, we summarize our results. 2 Decision of the solitonic behavior in a dynamical system In this section, we discuss the existence and stability of soliton solutions of NLQWs. First, we give an example of explicit non-scattering solutions. More precisely, we let j = 1, 2 and define δ j,x ∈ l 2 (Z; C 2 ) as δ j,x (y) = e j if y = x and δ j,x (y) = 0 if y = x. Then, setting where a is a root of π 4 + ga 2p = 0, ϕ t becomes a solution of NLQWs. Observe that this solution is localized, travels by constant speed and does not decay. Thus, we call this solution a soliton solution. We remark that in Section 4, we will observe a typical phenomena for soliton solutions, which behave like KdV solitons. That is, when we collide two solitons, then they pass though each other without changing their shapes but only changing its position and phase.
It can be shown that for the existence of such non-scattering solutions, it suffices to have π 4 + ga 2p ∈ πn (n ∈ N) so even for g > 0. Then, there exist solutions with solitonic behavior. Moreover we can show that ϕ t defined by (2.1) is unstable. Indeed, for 0 < ε < 1, we let ϕ ε (t) be a solution of NLQW with its initial state ϕ ε (0) := a(1 − ε)δ 1,0 . Then, it is clear that ϕ ε (0) → ϕ 0 as ε ↓ 0. We see that ϕ t ε (−t) C 2 → 0 as t → ∞ and obtain ϕ t − ϕ t ε l 2 → ϕ t l 2 + ϕ t ε l 2 ∼ 1 as then we see that ψ t ε (−t) → (a, 0) T as t → ∞. Thus, there is a large set of initial data which persists soliton-like behavior with speed 1 in the left edge. We will explain this picture for the soliton using a dynamical system as follows.
Our interest is now the left most amplitude at each time step, that is, u t (−t). Then we regard the initial state as u 0 (x) = δ 0 (x)e 1 . We put r t := ||u t (−t)|| 2 l 2 and θ t := π/4 + gr p t . Here g ∈ {±1}. Since u t (−t) = cos θ t−1 · u t−1 (−t + 1), we have Then the problem is simply reduced to the dependence on the initial data r 0 with respect to the convergence destination in the above dynamical system (2.2). Now to obtain the fixed point of this discrete-time dynamical system, let us consider the following function.
We pick up the important properties of f (x) as follows.
(ii) f (x) takes a local maximum if and only if cot[π/4 + gx p ] = 2pgx p ; (iii) f (x) takes a local minimum 1 if and only if gx p ∈ (4Z + 1)π/4; Remark that the points which accomplish (i), (ii), (iii), respectively, appear periodically as follows: which is independent of g and p. We set , which is the set of all the points satisfying (i) except 0. Let x (m) be the m-th smallest element of P . Let y (m) > x (m) be the smallest value of the solution of n+1 . See Fig. 1. Then from a simple observation to this iterative dynamical system (see Figs.2), we can completely determine the set in R + for the initial value r 0 providing a solitonic behavior whose fixed point is Remark 2.2. We do not exclude the possibility that ϕ ε converges (in some sense) to another unknown traveling wave type solution.
3 Behavior of l ∞ norm In this section, we study the behavior of l ∞ norm of the solutions. We set C ± the coin (1.5) with g = ±1 and p = 1.
First, we discuss the behavior of u t l ∞ with small initial state, which implies that the nonlinear effect is week. From Theorem 2.1 of [16], the behavior of u t l ∞ of the linear model (the case g = 0 in (1.5)) is approximated by t −1/3 as t → +∞. It can be verified that u t l ∞ to the nonlinear model (1.5) with small initial state behaves like a linear. Indeed, Figure 4 shows that the behavior with u 0 = 0.2δ 1,0 is approximated by a linear model. We remark that the case p = 1 was not proved to scatter in [16].
On the other hand, if u 0 = δ 1,0 for both coins C ± , then u t l ∞ does not decay and becomes almost constant (see Figure 3). Thus, it is expected that the solitonic behavior which has been discussed in Section 2 appears. For this case, we can find the explicit value of a = u t l ∞ by solving the equation π/4 + a 2 = π/2 for C + and π/4 − a 2 = 0 for C − . In Table 1, we put the value of (a)  u t l ∞ obtained by the numerical calculations starting from u 0 = δ 1,0 , which can be compared with It is clear that a solution corresponding to the equation is soliton solution which has been discussed in Section 2. Hence, it is expressed by u t = aδ 1,−t when u 0 = aδ 1,0 , which keeps its shape and moves to left on the x-axis with its speed 1 (see Figure 5). As it has been already mentioned in Remark 2.1, solving the equation we get a periodic solution, which behaves aδ 1,y → aδ 2,y+1 → −aδ 1,y → −aδ 2,y+1 → aδ 1,y → · · · .

Oscillations
A striking feature of the coin C − is an oscillating behavior of u t l ∞ . In Figure 3 (b), we can see oscillations when u 0 = 0.8δ 1,0 and u 0 = 0.6δ 1,0 . These are different from solitons. We would like to consider the reason why such behavior appears. Let α > 0 and u 0 = αδ 1,0 . Then, it is clear that we have that  The left and right end parts of u t are on the function Since h − (0) = 0 and a graph of y = h − (x) touches a graph of y = x at x = √ π/2 which is the value in Table 1, a sequence x n cos converges to 0 if the initial condition is smaller than √ π/2 ≈ 0.886227. Thus, oscillation behavior comes from an inner part of u t . Actually, a speed of the main part of u t is less than 1 (see Figure 6 (c)).  The reason why such oscillating solution appears only in the case C − may be related to the graphs of (3.3) and (3.4). Let the initial state u 0 = 0.6δ 1,0 . In the case of C + , it is easily seen that the left and right end parts of u t are on the function of which graph is in Figure 6 (b). From Figure 6 (b), the decay of u t l ∞ is quick and it can be less than 0.3 after the first step. This decay is also observed in Figure 3 (a). On the other hand, we see from Figure 6 (a) that the left and the right end parts of u t decay slowly. This implies that the inner part of u t , which denotes the area between the left and the right end parts, can have strong influence coming from the end parts for long time step. Then, the dynamics becomes complicated.  If we take u 0 l 2 > 1, which implies a nonlinear effect becomes stronger, then several kinds of non-scattering solutions appear. Among such solutions, we focus on solitons. Considering a nonlinear system, we are interested in not only the existence of solitons but also an interaction between solitons. We would like to consider a collision between two solitons and observe the behavior of solitons after collision.

Collision between two solitons
We see that soliton solutions from Section 2 are divided into two types. Indeed, for the case with coin C + , solve a equation π 4 + a 2 = (2n − 1)π (n = 1, 2, . . .) (4.1) and let u 0 = aδ 1,0 . Then, we obtain that a soliton u t = (−1) t aδ 1,−t for t = 0, 1, 2, . . ., which moves to the left on the x-axis with speed 1. It is easily seen from the definition of the map S that the soliton moves to the right on the x-axis if u 0 = aδ 2,0 . We call such soliton rotating soliton. On the other hand, a solution with u 0 = aδ 1,0 where a is determined by solving π 4 + a 2 = 2nπ (n = 1, 2, . . .), becomes u t = aδ 1,−t , which keeps its shape and moves to the left on the x-axis with speed 1. It moves to the right on the x-axis if u 0 = δ 2,0 . We call this soliton traveling soliton. Furthermore, we regard the periodic solution corresponding to the equation (3.2) as a soliton since it is localized and its sup-norm does not decay for t > 0. We call such solution periodic soliton. For the case with coin C − , we have obtained the traveling soliton from (3.1). Other types of solitons are obtained by solving the following equations:  Table 2 summarizes the values of a obtained by (4.1) and (4.2) with n = 1 and by (4.3) and (4.4) with n = 0 together with the cases (3.2) and (3.1). Figure 7 shows the behavior of rotating and traveling solitons for the case C + . Treating these kinds of solitons, we would like to consider the following four types of collision: rotating soliton traveling soliton periodic soliton C + a ≈ 1.534990 a ≈ 2.344736 a ≈ 0.886227 C − a ≈ 1.981664 a ≈ 0.886227 a ≈ 1.534990 Table 2: The values of u t l ∞ of solitons, which are used in numerical studies in Section 4.
(I) collision between the same solitons; (II) collision between the rotating soliton and the traveling soliton; (III) collision between the periodic soliton and the rotating soliton; (IV) collision between the periodic soliton and the traveling soliton.
It will turn out that Case I exhibits a simple dynamics which we can calculate explicitly, while the behavior of solutions after collision in Cases II-IV becomes complicated.  where u t = (u 1 , u 2 ) T . Figure (a) shows a collision between rotating solitons. One moving to the right on the x-axis has its initial state u 0 = 1.534990δ 2,450 and another one moving to the left has its initial state u 0 = 1.534990δ 1,750 . Figure (b) shows a collision between traveling solitons. One moving to the right on the x-axis has u 0 = 2.344736δ 2,450 and another one moving to the left has u 0 = 2.344736δ 1,750 .

Collision between rotating solitons Collision between traveling solitons
We consider two cases, one is the collision between rotating soliton and itself and another is the one between traveling soliton and itself. Since the case with C − exhibits exactly the same behavior, we discuss the case C + only. Figure 8 shows the location of solitons in the case C + , where (a) is a collision between two rotating solitons and (b) is the one between two traveling solitons. It follows from these figures that location and speed of solitons do not change after collision. However, there is certainly interaction, which can be observed in Figure 9. Actually, we can explicitly calculate the behavior of u t at the collision time.
Let α be the positive root of the equation (4.2) with n = 1 and consider the collision between two traveling solitons. We demonstrate what happens in the behavior of u t with u 0 = αδ 1,750 + αδ 2,450 .
Since the soliton with u 0 = αδ 1,750 moves to the left and the one with u 0 = αδ 2,450 moves to the right on the x-axis with their speeds 1, they have no interaction for 0 ≤ t ≤ 149, and we obtain u 150 = αδ 1,600 + αδ 2,600 .
Therefore, for t = 151, we see that Noting that π/4 + α 2 = 2π and α 2 = 2π − π/4, we have We can continue the similar calculations for t = 152 to obtain that It is easily seen that there is no interaction for t > 152. Figure 9 shows the behavior of u t which has been just proved. We note that the above behavior is very similar to the phenomena of collision for KdV solitons. That is, if the two solitons did not interact with each other, then we should havẽ u 152 = αδ 1,598 +αδ 2,602 . So, comparing with u 152 , we see that only the position and phase (α became −α) have changed. If we consider two traveling solitons but different values, namely, u t with u 0 = βδ 1,750 + γδ 2,450 , where π/4 + β 2 = 2nπ and π/4 + γ 2 = 2mπ (n = m). Then, u t has the same behavior as the case n = m = 1 for 0 ≤ t ≤ 150 and it becomes u 150 = βδ 1,600 + γδ 2,600 .
However, for t = 151, we see that Since we cannot obtain the explicit values for t > 151, the behavior of u t becomes complicated. Consequently, using exactly the same solitons is an important to obtain the simple dynamics of collision. Similar behavior is observed if we consider the collision between rotating solitons for C + , and the same cases for C − . Comparing with the collision in Case I, the dynamics after collision in Cases II-IV is complicated.  Figure 10 shows a collision between a rotating soliton and a traveling soliton for the case C + in (a) and for C − in (b). Although Figure 10 (b) looks similar to Case I discussed in the previous subsection, it is not so simple and there should be interaction between two solitons. Indeed, we notice that the speed of soliton moving to the right on x-axis slightly changes after collision. We make an approximate calculation to understand the dynamics which observed in Figure 10 (b).

Collision: case II
Let consider the behavior of u t for C − with its initial state u 0 = βδ 1,750 + γδ 2,450 , where β and γ are positive roots of the equations (4.4) and (3.1), respectively. Here, the part with βδ 1,750 corresponds to a rotating soliton and the one with γδ 2,450 corresponds to a traveling soliton. Since the rotating and traveling solitons move with speed 1, they collide at t = 150 and have u 150 = βδ 1,600 + γδ 2,600 .
Then, for t = 151, we see that Noting that β 2 = π/4 + π and γ 2 = π/4, we obtain that It follows that each peak at x = 600 moves one distance to the left or to the right, respectively. We should notice that each soliton loses its property since the value of the peak has been changed. For t ≥ 152, the dynamics can be complicated because of the value of peaks. We obtain, for the peak on x = 599 at t = 151, that For the peak on x = 601 at t = 151, similar calculations leads to Since β = √ 5π/2 and γ = √ π/2, we have βγ = √ 5π/4, where Noting that we see that the main part of (4.5) is the first term on the right-hand side and the one of (4.6) is the second term on the right-hand side. Therefore, it follows from sin θ ≈ 0.184347, cos θ ≈ 0.982861 that u 152 looks like We note that Hence, the left end part of u t on the x-axis behaves like a rotating soliton and converges to the one. On the other hand, traveling soliton was changed to an oscillating solution which was considered in Section 2.3. Then, the right end part of u t tends to 0 and the part which is larger than 0.3 emerges from interior part of u t (compare with Figure 6 (c)). This shows the reason why the speed of the main part moving to the right has been changed in Figure 10 (b).  Finally, we discuss the Cases III and IV. They exhibit similar dynamics after collision. Figure  11 shows the complicated behavior of u t . For the case C + , a collision tends to lead a coalescence of soliton. On the other hand, several solitons appear after collision for the case C − .

Conclusion
In this paper, we have studied the dynamics of solutions of NLQWs in a strong nonlinear regime. If u t is localized, travels by a constant speed and u t l ∞ does not decay for all t > 0, then we call such a solution a soliton solution. The existence of soliton solutions moving to the right or to the left with speed 1 has been shown in Section 2, which consists of traveling and rotating solitons. In addition, there is a periodic soliton which evolves in a finite region with period 4. We have systematically investigated a collision between two solitons choosing from a traveling soliton, a rotating soliton and a periodic soliton. Although it is expected that the dynamics becomes too complicated to analyze it, we could calculate the explicit process of collision between the same solitons (Subsection 4.1). This behavior of solitons is very similar to the phenomena of collision for KdV solitons.
It would be interesting to study a large time behavior of oscillating solutions which has been obtained in Section 3. We would like to know whether the oscillating solution decays for large t or converges to some periodic solution. This is our future work.