Wave Packets of Mandelbrot Type in Boundary Problems of Quantum Mechanics

In last years, in physics studied the nonlinear interaction of light which can mimic the physics at so called an event horizon. As shown in a study [1], this analogue arises when a weak probe wave is unable to pass through an intense soliton, despite propagating at a different velocity”. These dynamics arises as a soliton-induced refractive index barrier. In all paper this barrier characterize the volume optic properties of a fibre with linear boundary conditions. In this paper, we consider the opposite problem when the optical medium is ideal or linear, but boundaries of the medium have the nonlinear optic properties, and describes, for example, the all-optical transistor [2]. It may be also a case when a bright soliton is passing through the soliton. In this case, the intensity of light depends on the fibre refractive index (that is take place the Kerr effect). Thus, the soliton creates a moving refractive index perturbations which passage through the another soliton [36]. This interaction between such surface solitons plays the mane role of distributions of the light in the linear medium with nonlinear surface interaction. Thus for the ideal medium the main role plays the fibre surface nonlinear refractive index. In a study [2], it has been mimicked two spectral modes of solitons when the mode-locked laser diode generate picosecond solitons. This generation will be described as a functional boundary conditions generated these solitons. The simulation is described by the hyperbolic type equations which models an evolution of amplitude of electric field A(x, t) [7]. These equations depends on the dispersion coefficients and on the nonlinear interaction coefficient of the fibre, depending on A(x, t), where x is a coordinate of a beam and t is the time. Of course, these equations must be follows from the Schrodinger type of equations. Thus phases and amplitudes of lights describes by systems of linear quantum equations. Boundary condition describes surface beams in an optical resonator with feedback. Thus the equations are linear without interaction potential, but the feedback is modeled by the nonlinear boundary conditions.


Introduction
In last years, in physics studied the nonlinear interaction of light which can mimic the physics at so called an event horizon. As shown in a study [1], this analogue arises when a weak probe wave is unable to pass through an intense soliton, despite propagating at a different velocity". These dynamics arises as a soliton-induced refractive index barrier. In all paper this barrier characterize the volume optic properties of a fibre with linear boundary conditions. In this paper, we consider the opposite problem when the optical medium is ideal or linear, but boundaries of the medium have the nonlinear optic properties, and describes, for example, the all-optical transistor [2]. It may be also a case when a bright soliton is passing through the soliton. In this case, the intensity of light depends on the fibre refractive index (that is take place the Kerr effect). Thus, the soliton creates a moving refractive index perturbations which passage through the another soliton [3][4][5][6]. This interaction between such surface solitons plays the mane role of distributions of the light in the linear medium with nonlinear surface interaction. Thus for the ideal medium the main role plays the fibre surface nonlinear refractive index. In a study [2], it has been mimicked two spectral modes of solitons when the mode-locked laser diode generate picosecond solitons. This generation will be described as a functional boundary conditions generated these solitons. The simulation is described by the hyperbolic type equations which models an evolution of amplitude of electric field A(x, t) [7]. These equations depends on the dispersion coefficients and on the nonlinear interaction coefficient of the fibre, depending on A(x, t), where x is a coordinate of a beam and t is the time. Of course, these equations must be follows from the Schrodinger type of equations. Thus phases and amplitudes of lights describes by systems of linear quantum equations. Boundary condition describes surface beams in an optical resonator with feedback. Thus the equations are linear without interaction potential, but the feedback is modeled by the nonlinear boundary conditions. The problem of the coherent interaction of impulses in a nonlinear medium is well-known. As noted in a study [8]," an interaction may be utilized for the transmission of information, for frequency conversion, and for the description of processes which proceed in more intensive fields and at times shorter than to relaxation time" [9].
where ψ:=ψ (x; t): R 2 → C, where C is a complex space, is an unknown function,  is the Planck constant. Let us divide the two parts of equation on a value mv 2 , where m is the mass of particles, v is their velocity and p=mv is an impulse. Further, we introduce t t τ = , τ is a relaxation time of a wave function to some equilibrium, and we consider a dimensionless constant h λ ντ = . As a result, we obtain the dimensionless equation: We can consider the functional two-point boundary conditions with the real or complex parameter θ. For these conditions there is a theorem of existent and uniqueness of solutions the problem [10][11][12][13]. But our aim is instead of the linear boundary conditions to consider nonlinear conditions where Φ: C → C is nonlinear structural stable map, with additional initial conditions where h>0 is a small parameter. We assume that conditions and the similar conditions for second derivatives at points (0, 0) and (0, 0) are satisfied. This ensures the existence of solutions of C 2 [(0, l)×[0, t 0 )] -class. Of course, real and imaginary parts of the map Φ are from C 2 -class. This ensures that the map Φ: C → C is structural stable. It means that a corresponding map G: R 2 → R 2 is structural stable. It means that ( )  (18) And, for the transport equation We have the boundary condition Here maps Φ 1 , Φ 2 ∈ C 1 (I → I) are structural stable, where I is an open closed interval. The structural stable maps form an open dense subset [5].
In order to solve equations, we use the method of characteristics. To do this, we consider the Hamilton system of ordinary difference equations with hamiltonian ( ) With the initial condition, For the given constant p the function x:=x(p,t) is the solution of equation Where p can be considered as additional coordinate in (x,p,t)space.

Interpretation of the equation follow from the relation
Here, L= Φ 0 along a curve x:=ϑ 0 (t, x, p) which is determined from a solution of equation (R31). Then L ≥ Φ 0 along each other curve and the curve ϑ 0 minimizes the functional . In other words, the difference L -Φ 0 as a function of x  has a minimal value 0 as x  =p. It follows that , Let us define x y L =  and H L yx − = −  . These values are called by spatial and temporal impulses in a space t, x, x  . The the differential form y dx -H dt is differential dS(x, t) if x:=p(x, t) [14]. The same result follows from the Euler equation  From the variational problem for a lagrangian L(t, x, x  ) we can obtain a local hamiltonian L(t, x, y) and the corresponding local Hamilton-Jacobi equation. Then on characteristics dx(p, t)=p we have the equation that spectre σ(TG r ) ∩ {z: |z|=1}=ф, where ф is empty set.
Further we assume that h>0 is a small parameter, and we consider the problem with accuracy O(h 2 ), where O(h 2 ) → 0) as t → +∞. We find solutions in the form: where S(x, t) is a real phase, and φ(x, t, h) is real amplitude. (Below, where it will not cause misunderstandings, the parameter h will be omitted).
We assume that C 0 ([0; l] × [0+∞) is the space of bounded continuous functions, and C 2 is the space of twice differentiable functions with the norm: l] × [0,+∞. The function ψ ∈ C 2 belongs to if its real and imaginary parts belong C 2 ([0, l] × [0,+∞. Then in C 2 -norm there is the following convergence: and A ± is a set of saddle points of codimension one, and ϙ(ϛ)=(S 0 (ϛ), φ 0 (ϛ)) is an initial curve in R 2 , which is determined by initial data of the boundary problem, and N is least common multiple of the map G: (S,φ) → (Φ 1 (S,φ ), Φ 2 (S,φ)).

Method of Reduction of Problem to System of Integro-Difference Equations
Now we return to the problem. Indeed, substituting (7) in eqn. (2), we obtain that We find solution with accuracy O(h 2 ) so that Then we obtain the Hamilton-Jacobi equation The boundary condition can be written in form: Where ℜ and ℑ real and imaginary parts of a complex number. Let us denote F1:= ℜΦ and F1: = ℑ Φ . Then from eqn. (13) it follows that: We define the function : arctan F F Φ = , then the boundary condition (4) can be written as ( ) For simplicity, we consider, initially, the case, when : ϕ Φ = Φ By integration along characteristics dx∕dt=p and with help of boundary conditions the problem can be reduced to the system of integro-difference equations: where we used the boundary conditions for the phase. Integrodifference eqn. (33) follows from the transport eqn. (19) which can be written as Hence, S t x p S l t l p S l t l p x p S l t l p S l l p p p Here, a function S(l, t can be found from difference eqn. (32). It is known [5] that, in typical cases, symptotic solutions of difference equations tends to elements of attractor of corresponding dynamic system which are piecewise constant periodic functions p 1 (t) such that p 1 (t) ∈ A + 1 , where A + 1 is a set of attractive fixed points of the map Integro-difference equation can be written as Since k<0, all trajectories are attracted by a plane S,φ. But on this plane we obtain two independent differential equations for the phase and amplitude. asymptotic solutions of the boundary problem are determined by asymptotic solutions of the system of difference equations.
Let as define P a set of fixed points of the map Hence, for k<0 all trajectories in R 3 are attracted by a plane (S,φ) ⊂ R 2 . It means that asymptotic behaviour of trajectories is determined of codimension 1 as shown on Figure 1. Then from [15][16][17] it follows that that S(l, t) ⇒ p 1 (t) and φ(l, t) ) p 2 (t) as t → ∞, where p 1 (t) ∈ A + 1 and p 2 (t) ∈ A+ 2 . Here, A + 1 , A + 2 are sets of attractive fixed points of maps (φ 1 ,φ 2 ), correspondingly. The functions p 1 (t), p 2 (t) are periodic with periods 2 N is least common multiple of periods of attractive circles of the map G.

Example 1
For example, we consider a quadratic map Φ: z → z 2 +μ, where μ, z ∈ C and C is a complex space. Then from the boundary conditions follows the relations: We assume that μ 2 =0. then Similarly, since integral term tend to zero as e kt , where k<0, as shown above, we have If μ 2 1 -μ 1 <3∕4, a point β 1 is repelling, but instead appears attractive circle of period 2 which contains the points

Asymptotic of System of Difference Equations
Thus asymptotic behaviour of difference equations is known for the so-called hyperbolic maps [18]. Indeed, if the map G: R 2 → R 2 , which is produced by these equations, has a finite number of fixed points A, then functions S(l, t) and φ(l, t) tends to asymptotic 2 N l∕p periodic piecewise constant functions p 1 (t) ∈ A and p 2 (t) ∈ A for almost all points t ∈ (-l∕p,1), excluding finite or infinite points of discontinuities ( Figure 2). It is possible if the map G is hyperbolic. It means that the spectrum of the differential T(G) has no real points with values equal in modulus 1. If also stable manifolds W s (A + ) intersect unstable manifolds W u (A -) and W u (A + ) transversally, and an initial curve (S 0 (-l∕p, 0), φ(-l∕p; 0)) interact an unstable manifolds W u (A ± ), then the map G is structural stable and hyperbolic.
Here, A+ is a set of attractive points the map G, Ais a set of repelling points, and Ais a set of saddle type points.
It must be noted that Thus asymptotic solution has the form Thus it has been proved that the initial boundary value problem can be reduced to a system of integro-difference equations. These system can be, in its turn, reduced to a system of non-autonomic difference equations in R 3 , where non-autonomic perturbations are produced by the phase S(x, t) on characteristics dx∕dt=p. These perturbations tends to zero as t → +∞. As a result, the difference equations in R 3 are reduced to a difference equations in R 2 at (S,φ) -plane. Solutions of these equations (S,φ) are piecewise constant asymptotic periodic functions (p 1 , p 2 ) with finite 'points' of discontinuities on a period. It should be noted that for difference equations we get asymptotic distributions from some class of initial data from C 2 (I, I), but for integro-difference equations it has been proved only that if solutions of integro-difference equations exists, then these solutions tends to functions (p 1 , p 2 ) for a special class of initial data, which belongs to a small neighbourhood of limit functions (p 1 , p 2 ). It should be noted also that if asymptotic invariant distributions of difference equations are asymptotic invariant solutions of integro-difference equations, becomes functions (p 1 , p 2 ) are piecewise constant asymptotic periodic functions. It should be noted also that if a limit function p2(t-x∕p) is piecewise constant, the a function

Physical sense of the boundary conditions
The boundary conditions describes the changing of phases between input S(0, t) and output phase in the electronic device the multiplier of the phase N and the amplifier J of a frequency of the input signal at a point x=l. Additionally, there is the resistor R. The device is designed so that we can change independently the phase and amplitude. The map Φ -1 1 describes actions of the amplifier J and the multiplier on the input phases independently on the values of the amplitude of input signal [14,15].

Physical Applications for Mandelbrot and Julia Sets
In this section, we use the paper [19] where it the dynamics of a kicked particle which moves in a double-well potential. In this paper, it is shown that the stroboscopic dynamics of the particle can be reduced to the research of dynamics of orbits of the known complex logistic map. The logistic map is homeomorphic to the quadratic map Q c (z):=z 2 +c where C ∈ C is a complex parameter. But this map has well-known Julia and Mandelbrot sets. Indeed, the map Q c (z): C → C is connected or totally disconnected [20,21]. The Mandelbrot set represent is a 'delimiter' between c -values with connected Julia and totally disconnected Julia sets [22]. The Mandelbrot set is connected [23]. They also discussed also the non-solved hypotheses that the Mandelbrot set is locally connected [20]. Next, by Brunner and Hubbard it has been studied the Mandelbrot set for cubic. There are two critical orbits for cubics [21]. Roshon generalize the study of the case n=3,4. He proved that in a case n=4 the Mandelbrot set is connected. In a study [20] the cases n=2,3,4,5 has been considered. Let f:X → X any function, where X is a topological space. Let us define the Picard orbit by the recurrent formula: u n =f(u n -1), n=1,2….
Let M be the Mandelbrot set for the map Q c (z):=z 2 + c where c ∈ C such that the orbit of the point 0 is bounded so that M:={c∈C:Q k c (0), k=0,1…, is bounded} (69) where Q k c (0) are iterations of the function with a condition Q c (0)=0 which produce a critical orbit. Then there is known for the quadratic map that if f(c)>2 and z>c, then f n (z) → ∞ as n → ∞. For example, the point c is not in the set M. As shown in a study [20], if c ∈ [-2,1∕4), then f n (z)<const. In this case, the Julia set is connected.

Conclusion
Thus an initial boundary problem for the linear Shrodinger equation with functional nonlinear two-points boundary conditions is considered. A structure of attractors of the problem has been constructed. It is shown that the attractor contains piecewise constant periodic functions with finite 'points' of discontinuities on a period. Thus phases and amplitudes of wave functions of the Shrdinger equation are piecewise constant periodic functions. The method of reduction of the problem to a system of integro-difference equations has been applied. It is shown that these equations have invariant piecewise constant periodic solution. It is proved that perturbations of such solutions are asymptotically stable.