Symmetry Breaking of a Time-2D Space Fractional Wave Equation in a Complex Domain

: (1) Background: symmetry breaking (self-organized transformation of symmetric stats) is a global phenomenon that arises in an extensive diversity of essentially symmetric physical structures. We investigate the symmetry breaking of time-2D space fractional wave equation in a complex domain; (2) Methods: a fractional differential operator is used together with a symmetric operator to deﬁne a new fractional symmetric operator. Then by applying the new operator, we formulate a generalized time-2D space fractional wave equation. We shall utilize the two concepts: subordination and majorization to present our results; (3) Results: we obtain different formulas of analytic solutions using the geometric analysis. The solution suggests univalent (1-1) in the open unit disk. Moreover, under certain conditions, it was starlike and dominated by a chaotic function type sine. In addition, the authors formulated a fractional time wave equation by using the Atangana–Baleanu fractional operators in terms of the Riemann–Liouville and Caputo derivatives.


Introduction
Symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations performing on a system possessing a critical point (fixed point, the roots of the transform operator) adopts the system's outcome, by defining which division of a bifurcation is occupied. This procedure is known symmetry breaking, because such changes typically transform the system from a symmetric but the disorganized state into one or more certain conditions. Symmetry breaking is studied theoretically in the nonlinear optics, lasers, liquid crystals and other areas in physics (see [1][2][3]).
Recently, Sa et al. [4] presented a review study on a complex-plane generalization of the successive distribution utilized to distinguish regular from chaotic quantum spectra. The approach structures the spreading of complex valued ratios between nearest-and next-to-nearest-neighbor spacing. Some results are discussed in the open unit disk.
In this study, we propose a time-2D space fractional wave equation of a complex variable using the modified Atangana-Baleanu fractional differential operator without singular kernel, which is catting in a special class of normalized analytic functions in the open unit disk. Some of its properties are discussed geometrically. The fractional differential operator is used together with a symmetric operator to define a new fractional symmetric operator. Then the new operator is employed to formulate a generalized time-2D space fractional wave equation. Our method is based on the subordination and majorization theory in the open unit disk [5,6]. For two analytic functions f and g in the open unit disk U = {z ∈ C : |z| < 1}, we say that f is majorized by g( f g) if there is an analytic function in the open unit disk such that f (z) = (z)g(z). Moreover, f is subordinated to g ( f ≺ g) if f (z) = g( (z)) (see [5]). As a result, we obtain altered formulas of analytic geometric solutions based on the geometric function theory.

Materials and Methods
Our methods are divided into two subsections as follows:

Complex Fractional Differential Operator
Fernandez [7] formulated Atangana-Baleanu complex fractional differential operator in terms of the Caputo derivative and the Riemann-Liouville formula respectively, as follows: where β(ν) is normalized function by β(0) = β(1) = 1 and Ξ ν (w) is the Mittag-Leffler function Moreover, Fernandez [7] introduced the following fractional differential operator where C ∆ ν h(z) is the complex Atangana and Baleanu differential operator in Caputo formula and R ∆ ν h(z) is the complex Atangana and Baleanu differential operator in the Riemann-Liouville formula. The Atangana-Baleanu fractional operators are used the generalized Mittag-Leffler function as non-local and non-singular kernel. Therefore, they have ability for practising in physics and computational studies. They are recommended in filtering and information theory. To modify the above operators, we present a class of analytic functions by This class is denoted by Λ and knowing as the class of univalent functions which is normalized by Then the modified operators of (1) and (2) are formulated by the integrals respectively where υ indicates the power of z.
Proof. For f ∈ Λ, a computation brings Similarly, we have R ∆ ν z f (z) ∈ Λ. This completes the first part. For the second part, it is adequate to show that (see [11]) The last part immediately recognizes by [11]-Corollaries 1 and 2 respectively.
Similarly, we can use C ∆ ν z f (z) to obtain the following SFDO Obviously, R (S ν ð ) k f (z) and C (S ν ð ) k f (z) are in the normalized class Λ with functional connected coefficients. Next, we shall introduce a generalized time-2D space wave equation using the SFDO. In this place, we note that there are different applications of the class of complex differential operators (see [12][13][14][15][16]).

Corollary 2. Consider the wave Equations
Proof. According to Proposition 2, we obtain In view of [11]-Theorem 1, where sin(ω) is of the second kind of locally univalent function, we get the require assertions.

Time-Fractional Wave Equation
Wave equation can be generalized to time-fractional wave equation by using the Riemann-Liouville derivative: x, y)] yy = sin(t, x, y), where z = x + iy and is a positive coefficient known as the diffusion of the medium. By using the complex transformation ω = c 1 t 2ν Γ(1 + 2ν) + c 2 x + c 3 y,
Since R (S ν ð )(ω) ∈ Λ then we have 1 = 0 and 2 = 1. Similar conversation can be considered for the operator C (S ν ð )(ω) ∈ Λ. We conclude the above construction in the following result Proposition 3. Consider the operator R (S ν ð )(t, x, y), z = x + iy. Then the wave Equation (11) has an analytic chaotic solution of the form where K is defined in (13) (see Figure 2).