Space-time breather solution for nonlinear Klein-Gordon equations
SSpace-time breather solutionfor nonlinear Klein-Gordon equations
Yasuhiro Takei , Yoritaka Iwata , ∗ Mizuho Information & Research Institute, Tokyo, Japan Kansai University, Osaka, JapanE-mail: ∗ iwata [email protected] Abstract.
Klein-Gordon equations describe the dynamics of waves/particles in sub-atomicscales. For nonlinear Klein-Gordon equations, their breather solutions are usually knownas time periodic solutions with the vanishing spatial-boundary condition. The existence ofbreather solution is known for the Sine-Gordon equations, while the Sine-Gordon equations arealso known as the soliton equation. The breather solutions is a certain kind of time periodicsolutions that are not only play an essential role in the bridging path to the chaotic dynamics,but provide multi-dimensional closed loops inside phase space. In this paper, based on the high-precision numerical scheme, the appearance of breather mode is studied for nonlinear Klein-Gordon equations with periodic boundary condition. The spatial periodic boundary conditionis imposed, so that the breathing-type solution in our scope is periodic with respect both to timeand space. In conclusion, the existence condition of space-time periodic solution is presented,and the compact manifolds inside the infinite-dimensional dynamical system is shown. Thespace-time breather solutions of Klein-Gordon equations can be a fundamental building blockfor the sub-atomic nonlinear dynamics.
1. Introduction
Let us consider one-dimensional wave equations. The existence of breather solution [1, 2, 3, 4] hasbeen known for some nonlinear Klein-Gordon equations; e.g., for Sine-Gordon equations. Thebreather mode is regarded as a kind oscillation. Indeed, for one-dimensional cases, it behavesasymptotically damping for | x | → ∞ , and periodic for t . Such a periodic property leads to theoscillation. On the other hand, the breather mode is not necessarily stable in most of nonlinearKlein-Gordon equations (for a textbook, see [5]).In this paper, utilizing the high-precision numerical code [Iwata-Takei], for hyperbolicevolution equations, the breather solution is explored in the double-well type nonlinear Klein-Gordon equations. By assuming the periodic boundary condition for the spatial direction x ,here we are seeking a periodic solution for both time and space. In this sense it is likely to becalled the space-time breather solution. On the other hand, since the model equation exactlycorrespond to the φ -theory in the quatum field theory, the obtained solution is expected tobring about a new insight on the existence of nonzero mass states. a r X i v : . [ n li n . PS ] S e p t=0 u x -1.5-1-0.500.511.5 0 1 2 3 4 5 6 7 8 t=0 v x Figure 1. (Color online) Initial functions f ( x ) = A sin( πx/
4) and g ( x ) = 0 with A = 1, wherethe spatial range is fixed [0 ,
2. Mathematical model
Let x ∈ [0 , L ] be a finite domain of space. The positive evolution problem is considered ( t ≥ ∂/∂t and ∂/∂x be denoted by ∂ t and ∂ x respectively. We consider the nonlinear Klein-Gordon equation with the double-well type interaction, which is also known as φ -theory. ∂ t u + α∂ x u + ( βu − µ ) u = 0 ,u ( x,
0) = f ( x ) , u (0 , t ) = u ( L, t ) ,∂ t u ( x,
0) = g ( x ) , ∂ t u (0 , t ) = ∂ t u ( L, t ) . (KG)where α , β , and µ are real constants. Since Eq. (KG) is solved by the Fourier transform,initial functions f ( x ) and g ( x ) are given as L -functions, and the periodic boundary conditionis imposed for x -direction. In the numerical calculations of this paper, the initial functions arefixed to f ( x ) = A sin( πx/
4) and g ( x ) = 0 with A > v = ∂ t u , the firstequation of (KG) is written by ∂ t v + α∂ x u + ( βu − µ ) u = 0 . (1)If β = 0 is satisfied, it is simply a linear Klein-Gordon equation in which µ means the squareroot of the mass. Otherwise if µ = 0 is satisfied (massless case), we see that Eq. (KG) is ageneralization of Klein-Gordon equation with cubic nonlinearity (cf. φ -theory in the contextof quantum field theory (for a textbook, see [6])). This equation holds the symmetry breaking,which is known as the Higgs mechanism. The free-particle solutions are useful to identify the condition for the appearance of breathersolution. Let β be a real constants satisfying β >
0. By taking ( βu − µ ) u = 0, the constant V ( u ) Figure 2.
Higgs potential (4) in case of ( µ, β ) = (0 . , u = ±√ . u = 0 shows a maximum of the potential.distributions (corresponding to three vacuums in the context of Higgs mechanism) follow: u = 0 , ± (cid:113) µ/β (2)which trivially satisfy the initial and boundary value problem (KG). These solutions correspondto constant stationary solutions of (KG) without any interaction. That is, u = 0 , ± (cid:112) µ/β are regarded as free particle solutions, and Eq. (KG) always holds the free-particle solutions.Needless to say, three solutions are degenerated to massless cases if µ ≤ µ = β = 0) include other solutionsthan the constant solutions. For example, let A be a real number, u = Ae ± i ( kx − ωt ) (3)is a massless free-particle solution with the equality k = − αω , where initial functions shouldbe f ( x ) = Ae ± ikx and g ( x ) = Aωe ∓ i ( kx + π/ in this case. These solutions correspond to typicalsolutions of (KG) in a limited setting. which are also useful to the mode analysis. In this papermuch attention is paid to the dynamics of nonlinear solutions. We will see that the dynamicsof nonlinear solutions (interacting solutions) are highly affected by the free-particle solutions(non-interacting solutions). In the theory of dynamical systems, there is a technical concept“absorbing set’. The constant solutions, which are also regarded as the stationary solutions,may or may not play a role of absorbing set (for the definition, see [7]).
3. Theoretical estimates for the appearance of breather mode
Following the usage in the quantum field theory, let us call V ( u ) = 14 βu − µu (4)the Higgs potential in case of the present inhomogeneous term ( βu − µ ) u . t=0 u x -0.4-0.200.20.4 0 1 2 3 4 5 6 7 8 t=12 u x -0.4-0.200.20.4 0 1 2 3 4 5 6 7 8 t=24 u x -0.4-0.200.20.4 0 1 2 3 4 5 6 7 8 t=36 u x -0.4-0.200.20.4 0 1 2 3 4 5 6 7 8 t=48 u x -0.4-0.200.20.4 0 1 2 3 4 5 6 7 8 t=60 u x Figure 3. (Color online) (
A, α ) = (0 . , − . u ( x, t )with x ≤ u ( x, t ) with x > u = 0. The breather solution is the periodic solution for both t and x , so that it is a kind of oscillation.The breather solution can be distinguished from a simple oscillation (Fig. 3; for short, we calloscillation in the following) by the appearance of certain kinds of collectivity, where activatedmodes result in the resonance. Here the breather solution (Fig. 4) includes the resonatinglarge amplitude oscillation, which localized only in the positive or negative side of u = 0.This localization property is not satisfied by the simply-oscillating solution. In this sense theterminology the breather mode makes sense in which many modes achieves the resonance. Simplyspeaking, the breather solution is realized by the instability of constant solution u = 0 and thestability of constant solutions u = ± (cid:112) µ/β . Here we obtain a guiding criterion in advance to a systematic calculation. Let a function G ( u )be defined by G ( u ) = − α∂ x u − ( βu − µ ) u, where α < β >
0. If we confine ourselves to a free-particle solution u = ( − Ai/ e i ( kx − ωt ) − e − i ( kx − ωt ) ) (see also Eq. 3; in the following we call the formal solution) in the fully interactingcases, let G ( u ) be replaced with˜ G ( u ) = αk u − ( βu − µ ) u = βu (cid:32) αk + µβ − u (cid:33) . This setting corresponds to the mode analysis for the stability. Since f ( x ) has the same formas A sin( kx ) = − Ai ( e ikx − e − ikx ) /
2, ˜ G ( u ) is true at t = 0 at the least. By considering thestationary condition G ( u ) = 0, three roots are represented by u = 0 , ± (cid:115) αk + µβ , where three real roots exist if µ > − αk , three real roots are exactly the same if µ = − αk ,and one real root with two imaginary root exist if µ < − αk . These solutions correspond to t=0 u x -0.4-0.200.20.4 0 1 2 3 4 5 6 7 8 t=12 u x -0.4-0.200.20.4 0 1 2 3 4 5 6 7 8 t=24 u x -0.4-0.200.20.4 0 1 2 3 4 5 6 7 8 t=36 u x -0.4-0.200.20.4 0 1 2 3 4 5 6 7 8 t=48 u x -0.4-0.200.20.4 0 1 2 3 4 5 6 7 8 t=60 u x Figure 4. (Color online) (
A, α ) = (0 . , − . u ( x, t ) with x ≤ u ( t, x ) >
0) and a part u ( x, t ) with x > u ( t, x ) < µ ,and more definitely on the activated spatial frequency k .Under the condition µ > − αk , let us limit ourselves to the constant distributions. In caseof ¯ u = (cid:113) αk + µβ , the absorbing set is calculated by (0 , ¯ u max ] satisfying (cid:90) ¯ u max ˜ G ( u ) du = (cid:90) ¯ u max βu (cid:32) αk + µβ − u (cid:33) du = (cid:20) (cid:16) αk + µ (cid:17) u − β u (cid:21) ¯ u max = 0 , and consequently, the absorbing set is calculated to be(0 , ¯ u max ] = , (cid:115) αk + µ ) β , (5)where ( αk + µ ) /β < αk + µ ) /β is always satisfied, and ¯ u ∈ (0 , ¯ u max ] is true.On the other hand, in terms of the curvature of spatial distribution, let us focus on thedifferential operator without approximation. A part of spatial distribution is picked out bysetting an interval [¯ x , ¯ ξ ]. The differential operator satisfies (cid:90) ¯ x blc ¯ x ∂ x u dx = (cid:90) ¯ ξ ¯ x blc ∂ x u dx, if the the 2nd derivative balances at x = ¯ x blc . Here the starting point ¯ x is assumed tosatisfy ∂ x u (¯ x ) = A (i.e. A · ξ is assumed tosatisfy ∂ x u ( ¯ ξ ) = 0 (i.e. ikA · x , ¯ ξ ]. The condition followsas ∂ x u (¯ x blc ) = 12 ∂ x u ( x ) = 12 A. ● ● ● ● ● ● A − α Figure 5. (Color online) The border of breather and oscillation solutions, where parametersare fixed to ( µ, β, L ) = (0 . , , − α = k (cid:16) µ − βA (cid:17) . Note that we have confirmed by taking several time intervals that nobreather solution can exist in the right-upper region of the graph.If the formal solution u = ( − Ai/ e i ( kx − ωt ) − e − i ( kx − ωt ) ) is applied, u (¯ x blc ) = √ A (6)is obtained, where the condition is checked for the discrete time ωt = π, π, · · · with respect tothe mode analysis. The resulting wave amplitude makes sense. Indeed, √ A/ − α ≤ k (cid:18) µ − βA (cid:19) (7)is satisfied, the positive constant solution ¯ u = (cid:113) αk + µβ behaves as a local attractor with aabsorbing set (5). The similar analysis is valid for the negative constant solution ¯ u = − (cid:113) αk + µβ ,while ¯ u = 0 is expected to repulse neighbor solutions at least for µ > − αk . As a result, astatement to confirm in this paper is • the breather solutions appear and survive at least for a while, if the initial function f ( x ) = A sin( πx/
4) and g ( x ) = 0 is given to satisfy the condition (7)where the further details are obtained as a life time formula.
4. Numerical experiments
The numerical calculations are carried out based on the high-precision numerical code using theFourier spectral method [8, 9]. In the present version, the implicit third order Runge-Kutta ● ● ● ● ● ● A − α Figure 6. (Color online) Life time estimates for breather solutions, where parameters arefixed to ( µ, β, L ) = (0 . , , , T ] , ( T =128 , , − α > T = 128 , , T = 128 , , − α = k (cid:16) µ − βA (cid:17) . method with two intermediate steps is utilized for the time direction, and spectral treatmentis implemented for the space direction. The spatial discretization is carried out based on thespectral method. The solution is assumed to be expanded by the Fourier series, and terminatedat the 2 th term, which corresponds to the resolution for the spatial direction. The timediscretization used in the implicit calculation is fixed to ∆ t = 2 − . The size L of the space isfixed to [0 , L ] = [0 , T is flexible in order to identify the lifetime of breather solution. T = 64 , , The coefficients are fixed to µ = 0 . β = 1. As the initialization of this research, webegin with searching for breather solution with a low-frequency mode, and k is taken as k = 2 π/L = π/
4. In terms of checking the validity of Eq. (7), the amplitude A and the squaredspeed of wave α are taken as free parameters. We have carried out systematics: 4 T × A × α and the other random choices, which is up to ∼
500 calculations.The transient appearance and disappearance of breather solutions are distinguished bywhether the mixture of plural numbers of mode are activated or not, and by whether thevalues of u ( t, x ) for a given spatial interval keep the positivity or negativity. For giving thecriterion of choosing the values of α , it is necessary to take sufficient numbers of α to identifythe border between the appearance and disappearance of breather solutions. Based on thebisection method, the border points are plotted if the relative error of the interval width is lessthan 0.10%. Here is a reason why we perform 10 α times calculations for one combination of T and A .Even starting from exactly the same initial functions (Fig. 1), some waves result in the simpleoscillation (Fig. 3), and the breathing oscillation (Fig. 4) is achieved in the other cases. Thosedifference is only in the difference of α value -0.220 and -0.075. In Fig. 5 the result for shorttime interval T = 64 is shown. The theoretical prediction (red curve) agrees quite well with ll − a T lll lll lll A=0.03A=0.06A=0.12A=0.25
Figure 7. (Color online) Parameter dependence of lifetime T of breather solution. Parametersare fixed to ( µ, β, L ) = (0 . , , A, ( A =0 . , . , . , .
25) of initial function is shown in order to find the relationship between T and α . The calculated values are shown by circles. Red, blue, green, purple curves are obtainedby the polynomial regression.the numerical systematics (black points). Here we confirm the validity of the existence limit ofbreather solution for a given initial setting: | α | ≤ k (cid:18) µ − βA (cid:19) , (8)which is obtained by the polynomial regression of obtained border points.Let us move on to the lifetime estimates of breather solution. At points x = 0 , ,
8, thesufficiently small wave amplitude condition: | u ( x, t ) | << x = 0 , , u ( t, x ) without satisfying | u ( x, t ) | << | u ( x ) | < − for thesufficiently small wave amplitude condition. This criterion is exploited when we consider largertime intervals. Note that all the calculations performed in this paper satisfies this smallnesscondition. That is, we focus on the competitive existence between the breather solution and thesimple oscillation.Blue, green and purple curves in Fig. 6 are depicted in terms of a maximum value of parameter − α > T = 128 , , A and α , and the ordinary oscillations appear instead. In this sense, the value T is regardedas the lifetime of breather solution.Here, based on the numerical results, the relationship between the lifetime of the breathersolution and the squared speed of wave α is shown in Fig.7. As for the overall trend, it canbe seen that the larger the squared speed of wave α , the shorter the lifetime of the breathersolution. On the other hand, for a fixed the squared speed of wave α , the larger the amplitude A of the initial function results in the longer lifetime of the breather solution. In conclusion thelifetime of the breather solution are determined only by the squared speed of wave α and theamplitude A of the initial function. . Conclusion In this paper, the space-time periodic breather solutions are numerically searched. Since theperiodic boundary condition is imposed, the obtained breather solution forms a closed curve inthe phase space with respect both to time and space. In this sense, what is meant by breathersolution in this paper is closed compact manifold in the phase space. From a different pointof view, the breather solution is the localized oscillations around the constant distributions u = ± (cid:112) µ/β .The appearance condition of short- and long-lived breather solution is obtained in a purelytheoretic mode analysis. The validity of existence-limit formula (7) is supported by systematicnumerical experiments with full nonlinearity. A mutual relation between coefficients is provided;the amplitude of wave must be smaller for the waves with higher speeds. In the present settings,we see that the breather solution seems to decay asymptotically. Consequently what we haveobtained in this paper is the breather solutions with a finite lifetime..The appearance condition of short- and long-lived breather solution is obtained in a purelytheoretic mode analysis. The validity of existence-limit formula (7) is supported by systematicnumerical experiments with full nonlinearity. A mutual relation between coefficients is provided;the amplitude of wave must be smaller for the waves with higher speeds. In the present settings,we see that the breather solution seems to decay asymptotically. Consequently what we haveobtained in this paper is the breather solutions with a finite lifetime.