Existence and non-existence of asymmetrically rotating solutions to a mathematical model of self-propelled motion

Mathematical models for self-propelled motions are often utilized for understanding the mechanism of collective motions observed in biological systems. Indeed, several patterns of collective motions of camphor disks have been reported in experimental systems. In this paper, we show the existence of asymmetrically rotating solutions of a two-camphor model and give necessary conditions for their existence and non-existence. The main theorem insists that the function describing the surface tension should have a concave part so that asymmetric motions of two camphor disks appear. Our result provides a clue for the dependence between the surfactant concentration and the surface tension in the mathematical model, which is difficult to be measured in experiments.


Introduction
Large numbers of independent individuals sometimes cooperate as a collection. Examples where the collection develops a function can be observed in biological systems such as flocks of birds [1], schools of fish swim [49], insect swarms, bacterial colonies [50] and cell group motions [35]. Vicsek [48] has shown the appearance of collective motions by using a particle model and, subsequently to his pioneering work, theoretical research has been conducted in terms of nonlinear physics for understanding the mechanism of collective motions performed by living organisms [36,48,49].
On the other hand, to study collective motions from the viewpoint of experimental systems, many researchers have utilized self-propelled materials as nonbiological systems. Examples of a simple experimental system for self-propelled materials include surfactants on particles or droplets driven by the difference of the surrounding surface tension [2, 4, 6, 10, 19, 24, 27-29, 32, 33, 41, 44, 47]. In addition, experiments to control the motions of self-propelled materials by a chemical reaction have also been reported [30,31,45]. For the theoretical understanding of these experimental results of self-propelled materials, a mathematical model for the experimental system was introduced and its mathematical analysis was carried out. In particular, one-dimensional motions of self-propelled materials, such as motions on the water surface of elongated channels, are described by a particle reactiondiffusion systems. For instance, let x c (t) and u(x, t) be the center of the surfactant material disk and the surface concentration of the surfactant, respectively. Then, a mathematical model for self-propelled motions is described as follows: where , , d u and k denote an area density of the surfactant disk, a viscosity coefficient, a diffusion coefficient and a combined rate of sublimation and dissolution of the surfactant, respectively. The function F(x) represents the supply of the surfactant molecules and a simple example of it is the supply from solid surfactant, which is described by where k u and u 0 are the supply rate and the density of the solid surfactant, respectively, and (x) is the Dirac delta function. The function G(u) is a driving force to the self-propelled particle, which is given by, for example, or (1) , (u(t, x c (t) + r)) − (u(t, x c (t) − r)) ,

3
Asymmetrically rotating solutions for a self-propelled model Here, the function (u) , which is a strictly decreasing function of the surface concentration of the surfactant, represents the surface tension of the water surface and, for instance, it is given by where a > 0 , m ∈ ℕ , and 0 , 1 > 0 represent the surface tension of pure water and that of the critical micelle concentration of the surfactant, respectively. Although it is difficult to determine the surface tension function (u) from experimental measurements, mathematical models phenomenologically assume that (u) is strictly deceasing. Indeed, this assumption has been employed in preceding studies and the following functions have been proposed as candidates for (u) [20,42]: in which u 0 is a positive constant, and To understand the mechanism of self-propelled motions theoretically, the mathematical model (1) with a strictly decreasing function has been studied by use of computer-aided analysis [3,10,19,20,23,25,38]. Moreover, a mathematical model for two-dimensional problems, such as motions on water surfaces, have been constructed [5,16,18,33] and the model has been studied theoretically [14,17,21,22]. In addition, the experiments to control motions of self-propelled materials by a chemical reaction have been reported [30,31,45], and their theoretical studies have been conducted by using the mathematical model (1) coupled with a chemical reaction model [13,26,37]. From these studies, the particle reaction-diffusion system (1) is considered as a physically relevant model for describing self-propelled motions in nonlinear physics and physical chemistry. One of the remarkable phenomena observed in self-propelled motions is the appearance of collective motions. Indeed, several patterns of collective motions have been reported in the preceding studies [11,39,40,42,43,46]. For example, Suematsu et al. have observed that camphor boats cause a phenomena like traffic jams [42], and oscillatory motions of camphor disks appear depending on the number of disks and their surface areas [43]. Nakata et al. have also reported collective motions of camphor disks such as billiard motions and traffic jam phenomena in an annular water channel [15]. Our concern is whether these collective motions appear in the mathematical model (1) as well. Since camphor disks used in the experiments are very light, Nishi et al. have analyzed motions of two camphor disks by using the following dimensionless mathematical model without the inertia term [34]: (2) (u) = a m a m + u m ( 0 − 1 ) + 1 , where > 0 and L > 4r > 0 . The function (u) > 0 is strictly decreasing for u > 0 and the function F(x) is given by Meanwhile, L denotes the map from ℝ to [0, L), that is, for any x ∈ ℝ , there exists n ∈ ℤ such that x = L (x) + nL . The periodic boundary condition is imposed by The preceding study [34] has clarified the mechanism of the emergence of billiard and traffic jam motions in the model (5) by computer aided analysis. In addition, they have also reported notable motions: a symmetrically and an asymmetrically oscillating motions, a symmetrically and an asymmetrically rotating motions (see Fig. 1), an oscillating and rotating motion. Furthermore, both the experimental measurement and the numerical computation have shown that the asymmetrically rotating motion of two camphor disks is stable as shown in Fig. 1b.
On the other hand, the density of camphor molecules just behind camphor disks is higher than that in other area in general, since camphor molecules remain behind for a while after the passing of a camphor disk. Then, the driving force for the disk behind, which is generated by the difference in the surface tensions, Asymmetrically rotating solutions for a self-propelled model seems to be weaker than that for the previous disk, and consequently the disk behind moves slower than the previous one. Hence, it seems reasonable that the distance between the two camphor disks increases gradually and their motions become symmetric in the end. In fact, reduced equations for camphor motions are derived based on the weak interaction theory, suggesting that repulsive forces act on the camphor particles through the reaction diffusion field [7][8][9]. From the above considerations, the appearance of asymmetrically rotating motions suggested numerically in [34] seems to be curious and implies that an attractive force between camphor disks exists. Moreover, the computer aided analysis in [34] has shown that the asymmetrically rotating solution appears via a pitch-fork bifurcation of the symmetrically rotating solution. Our motivation in this study is, through the analysis of the model (5), making it clear what kind of effect is essential for the existence of asymmetrically rotating motions for a strictly decreasing function with mathematical rigor. For convenience in the mathematical analysis, we introduce the uniform rotation velocity, c ≥ 0 , and consider the steady state problem of (5) via the moving coordinate z = x − ct . Then, rotating solutions of (5) on the moving coordinate are defined as follows: On the basis of the above definition, we state the following theorem about the existence of symmetrically and asymmetrically rotating solutions.
Here, we impose the assumption |z 1 c − z 2 c | L > 2r since the effect of contact between two disks are not considered in the model (6). According to Theorem 1, the existence of asymmetrically rotating solutions depends on the shape of the function (u) , which corresponds to the surface tension. We remark that it can be confirmed that there exists a function (u) in (2) satisfying the condition (7), and (u) in (3) satisfies the condition (7) with some suitable parameters. For the linear function (u) in (4), Theorem 1 claims that there is no asymmetrically rotating solution. For motions of a single camphor, we can obtain a similar result to Theorem 1, which is discussed in Appendix A. In the next section, we give the proof of Theorem 1.

Reformulation
Since solutions of (5) are recovered from those of (6), it is sufficient to show the existence and non-existence of stationary solutions of (6). That is to say, we consider the following equations: . To see the translational symmetry of (8) and (9), we introduce the following periodically extended equations: for i = 1, 2 , n ∈ ℤ , and (7)
For the sake of simplicity, we define the following functions: Throughout this paper, we will omit the discussion on c when it is fixed. We first show that the system (12) has a unique solution.
For any given c ∈ ℝ , there exists a unique solution of (12). Moreover,

expressed by
Proof Let us fix c ∈ ℝ . By a classical theory, we can represent a solution of (12) by .
Thus, it is sufficient to show the unique existence of constants a 1 ± , a 2 ± , b 1 ± and b 2 ± . The periodic boundary condition, U(L) = U(0) and U � (L) = U � (0) , requires that which is equivalent to These relations are rewritten by where Similarly, since U is continuously differentiable at l 1 , l 2 and l 3 , we have Note that it follows from det = − − + = −2 < 0 that is a regular matrix. We find that (15) and (16) are equivalent to Substituting these equalities in order, we obtain

3
Asymmetrically rotating solutions for a self-propelled model which indicates that constants a 1 ± are uniquely determined. Using (17), we obtain the other constants a 2 ± , b 1 ± and b 2 ± that are uniquely expressed by Thus, we conclude that (12) has a unique solution U ∈ C 1 [0, L) satisfying the periodic boundary condition.
On the other hand, considering we find Substituting a 1 ± into the first line in (14), we obtain for z ∈ (0, l 1 ) . Since the constants a 1 , a 2 , b 1 and b 2 are determined so that It follows from + − − = 2 and + − = −1 that Thus, we obtain The definitions of U + and U − yield Similarly, it is confirmed that ◻ For later use, we introduce the following notations: . .
We will omit the discussion on c and d when they are fixed. Before stating the proof of Theorem 1, we show some properties of U 1f , U 1r , U 2f and U 2r .

Lemma 2
Let c > 0 be a fixed constant. Then, we have Proof We consider U i , U if , and U ir as functions of d, that is, We have Note that Since it follows from 0 < 4r < L that 2 (x;r, L∕4) is strictly decreasing for x > 0 (see Appendix A.2), we find that 2 ( + ;r, L∕4) < 2 ( − ;r, L∕4) , that is, Since we have U ± E ± (d)(1 − E ± (2r)) > 0 and 1 − E ± (z) has the same sign as that of z, we obtain the desired result. (c) Note that 4r − 2d)).
Since we have and, similarly, (18) is rewritten by For the case of L − 4r − 2d < 0 , we have and thus Finally, we easily confirm that L − 4r − 2d = 0 is equivalent to
Next, we consider the case of c = 0 . Note that (8)

Proof of Theorem 1(c)
We prove Theorem 1(c) by contradiction. Let c > 0 be a fixed constant. If there exists a rotating solution, then (20) is satisfied with U 1 < 0 and U 2 < 0 . Thus, it is sufficient to prove that (20) is not satisfied provided that U 1 < 0 and U 2 < 0 . For asymmetric solutions, that is, d ≠ L∕2 − 2r , we should consider the following three cases.
For the case 3, it follows from L∕2 − 2r − d > 0 and Lemma 2(c) that U 2f < U 1f . Hence, we have U 2f < U 1f ≤ U 2r . In the same manner as that for the case 2, we assume that . Hence, we find which contradicts (20). 2r (c, d)) . We show that there exists a constant d 0 ∈ (0, L∕2 − 2r) such that (c, d 0 ) = 0 . Our strategy is to investigate the asymptotic behavior of (c, d) as d → +0 and d → L∕2 − 2r − 0 for sufficiently large c. We first consider the limit of (c, d) as d → +0 . It follows that and in the same limit. Note that

Asymmetrically rotating solutions for a self-propelled model
Similarly, it follows from U * * < U * that there exists U * * * satisfying U * * < U * * * < U * such that Using (29) and (31), we find Here, we introduce the function 0 (c) defined by Then, owing to (U * − U * * )(1 − E + (2r)) > 0 , the sign of (c, 0) coincides with that of 0 (c) . Since it follows from (30) that defining the function ̃ 0 (c) by we find 0 < � 0 , owing to ′ < 0 . We remark that if the limit of ̃ 0 as c → ∞ is negative, then � 0 (c) < 0 holds for sufficiently large c by the continuity of ̃ 0 , which implies that 0 (c) and (c, 0) are also negative. Thus, it is enough to investigate the limit of ̃ 0 (c) as c → ∞ to determine the condition on which (c, 0) is negative for sufficiently large c. For simplicity of notation, we set ≡ 4r∕L . Then, we have Indeed, it follows from (30) and U * * < U * * * < U * that and Lemma 3 in Appendix C gives As for the limit of ̃ 1 (c) as c → ∞ , since we have Lemma 3 in Appendix C yields ̃ 1 (c) → 1 2 � ( ) as c → ∞ . In the same manner as that for ̃ 1 , we confirm that ̃ 2 (c) → Note that u f (c) < 0 . Indeed, it follows that which concludes u f (c) < 0 . We now investigate the limit of ̃ d (c) as c → ∞ . Since Lemma 3 in Appendix C gives U f (c) , U r (c) → so that U * (c) → as c → ∞ , we have � (U r (c)) → � ( ) and �� (U * (c)) → �� ( ) . Then, It follows from Lemma 3 in Appendix C that .

3
Asymmetrically rotating solutions for a self-propelled model To estimate the limit of (u , it remains to see the limits of (1 − E + (c, 2r))(U r (c) − U f (c)) −1 and E − (c, L∕2)(U r (c) − U f (c)) −1 as c → ∞ . It follows that where Thus, we find from Lemma 3 in Appendix C that which leads to As a consequence, we obtain and conclude that ∕ d(c, L∕2 − 2r) < 0 holds for sufficiently large c provided that

Concluding remarks
In this paper, we have shown the existence of symmetrically rotating solutions and have given necessary conditions for the existence and non-existence of asymmetrically rotating solutions for the two camphor model (5). In particular, it has been shown that a concave part of the function , which describes the surface tension, is necessary for the existence of asymmetrically rotating solutions. Our result clarifies an essential condition for the existence of solutions and provides a clue for the dependence between the surfactant concentration and the surface tension in the mathematical model, which is difficult to measure in experiments. Moreover, it is suggested that the characteristic motion varies depending on the form of the surface tension function and, thus, the change of the qualitative motion is caused by in other mathematical models. Although this study treats the mathematical model without the inertial force, we note that our proof of the existence theorem for symmetrically or asymmetrically rotating solutions is still valid for the following model including the inertial term d 2 x i c ∕dt 2 : where is the area density of a camphor disk. Our analysis shows that there exists an asymmetrically rotating solution for N = 2 . For the model (35) with N ≥ 2 , the preceding study [12] has reported the appearance of collective motions of camphor disks. As the future work of the present study, it is worthwhile to investigate the stability of asymmetrically or symmetrically rotating solutions mathematically. Indeed, it has been shown numerically in [34] that an asymmetrically rotating solution is stable as far as it exists and moreover, an oscillating and rotating motion appears via a Hopf bifurcation from a symmetrically rotating solution branch.   so that U f (c) < U r (c) for any c > 0 . Thus, since (u) is strictly decreasing for u > 0 , we conclude that, for any c > 0 , there exists a unique constant > 0 such that (41) holds.