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Fourth Conference AMITaNS June 11 - 16, 2012, St. St. Konstantine and Helena, Bulgaria . A class of localized solutions of the linear and nonlinear wave equations. D. A. Georgieva, L. M. Kovachev. AMITaNS, 2012. - PowerPoint PPT Presentation
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A class of localized solutions of the linear and A class of localized solutions of the linear and nonlinear wave equationsnonlinear wave equations
D. A. Georgieva, L. M. Kovachev
Fourth Conference AMITaNS June 11 - 16, 2012, St. St. Konstantine and Helena, Bulgaria
2
Our aim is to investigate narrow-band and broad-band
laser pulses in femto- and attosecond regions in linear and
nonlinear regime.
AMITaNS, 2012AMITaNS, 2012
3
AMITaNS, 2012AMITaNS, 2012
With the progress of laser innovations it is very important to study the localized waves, especially pulses which admit few cycles under envelope only and pulses in half-cycle regime.
One important experimental result is that even in femtosecond region the waist of no modulated initially laser pulse continue to satisfy the Fresnel’s low of diffraction. The parabolic diffraction equation governing Fresnel’s evolution of a monochromatic wave in continuous regime (CW regime) is suggested for firs time from Leontovich and Fock
.04 02
22
2
z
wik
y
w
x
w
The equation admits solutions of the kind of fundamental Gaussian mode, higher order modes such as Laplace-Gauss, Helmholtz-Gauss and Bessel-Gauss beams.From other hand the optics of laser pulses, especially in the femtosecond (fs) region operates with strongly polychromatic waves – narrow-band and broad-band pulses. Additional possibility appear to perform a fs pulse to admits approximately equal size in x, y, z directions – Light Bullet – or relatively large transverse and small longitudinal size – Light Disk.
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AMITaNS, 2012AMITaNS, 2012
Our work is devoted to obtaining and investigating of analytical solutions of the wave equation governing the evolution of ultra-short laser pulses in air and nonlinear vacuum.
5
AMITaNS, 2012AMITaNS, 2012
The linear Diffraction - Dispersion Equation (DDE) governing the propagation of laser pulses in an approximation up to second order of dispersion for the amplitude function V = V(x,y,z,t ) of the electrical field is
where = k''k0v2 is a number counting the influence of the second order of dispersion, k'' is the group velocity dispersion, k0 is main wave number and v is the group velocity.In dispersionless media it is obtained also the following Diffraction Equation (DE) for the amplitude function A = A(x,y,z,t)
BASIC EQUATIONS
In air 2.110-5 0, and hence DDE is equal to DE, and at hundred diffraction lengths appear only diffraction problems. This means that we can use an approximation 0 and investigate DE only on these distances.
.11
22
2
20 t
A
vA
t
A
vz
Aik
,11
22
2
20 t
V
vV
t
V
vz
Vik
6
AMITaNS, 2012AMITaNS, 2012
generates the same amplitude equation DE
We note that a simple Courant-Hilbert ansatz
01
2
2
2
t
E
vE
)(exp,,,),,,( 0 vtziktzyxAtzyxE
applied to the wave equation
.11
22
2
20 t
A
vA
t
A
vz
Aik
• The initial problem for a parabolic type equation is correctly posed, while the initial problem for a hyperbolic type equation is problematical ones.
Hence, a solution of the wave equation can be obtained solving the amplitude equation DE and multiplying it with the main phase.
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AMITaNS, 2012AMITaNS, 2012
FOURIER TRANSFORM
When When 0 0 the fundamental solutions are equal. Therefore we will investigate only the second the fundamental solutions are equal. Therefore we will investigate only the second one.one.
The equations DDE and DE are solved by applying the spatial Fourier transform to the components of the amplitude functions A and V. The fundamental solutions of the Fourier images and  in (kx, ky, kz, t) space are correspondinglyV
tkkkkkkkv
ikkkVtkkkV zzyxzyxzyx 02222
00 211
exp)0,,,(ˆ),,,(ˆ
.exp)0,,,(ˆ),,,(ˆ 20
220
tkkkkkivkkkAtkkkA zyxzyxzyx
and
8
AMITaNS, 2012AMITaNS, 2012
The exact solution of DE can be obtained by applying the backward Fourier transform
tkkkkkivkkkAFA zyxzyx
20
220
1 exp)0,,,(ˆ
Substituting in the latter integral the backward Fourier transform takes the formzz kkk ˆ0
or in details
.exp
exp)0,,,(ˆ2
1 20
2203
zyxzyx
yxzyx
dkdkdkzkykxki
tkkkkkivkkkAA z
.ˆˆexp
ˆexp)0,ˆ,,(ˆexp2
1 222003
zyxzyx
zyxzyx
kddkdkkzykxki
tkkkivtkkkkAvtzikA
9
AMITaNS, 2012AMITaNS, 2012
EXACT SOLUTIONS
An analytical solution of DE is obtained for first time for initial Gaussian light bullet
In this case the 3D backward Fourier transform becomes
I. Gaussian light bullet
.2/exp)0,,,( 20
222 rzyxzyxA
,ˆˆexpˆexp
2
ˆexp
2exp
2
1
02
0222
20
222
0
20
20
3
zyxzyxzyx
zyx
kddkdkkkirzykxkikkkivt
rkkkvtzik
rkA
10
AMITaNS, 2012AMITaNS, 2012
which in spherical coordinates can be presented in the following way
,ˆˆˆsinˆexp2
ˆexpˆ
ˆ1
2exp
2
1 20
2
0
0
20
20
2 rrrr
r kdkrkivtrk
kr
vtzikrk
A
.ˆ2
ˆ2
1expˆ
ˆ2
ˆ2
1expˆ
2exp
ˆ2),,,(
0
2
20
0
2
20
0
20
20
rvtr
ierfcrvt
rrvti
rvtr
ierfcrvt
rrvti
vtzikrk
r
itzyxA
where and . . 2
02
022ˆ kirzyxr 222 ˆˆ
zyxr kkkk
The corresponding exact solution is
11
-0.04-0.02
00.02
0.04z cm
-0.04-0.02
00.02
0.04y
0
0.2
0.4
0.6
0.8
1
Ax 0,y,z,t-0.04
-0.0200.02
0.04z cm
-0.04-0.02
00.02
0.04y
7.827.84
7.867.88
7.9z cm
-0.04-0.02
00.02
0.04y
0
0.2
0.4
0.6
0.8
1
Ax 0,y,z,t7.82
7.847.86
7.887.9
z cm
-0.04-0.02
00.02
0.04y
23.52523.55
23.57523.6
z cm
-0.04-0.02
00.02
0.04y
0
0.2
0.4
0.6
0.8
1
Ax 0,y,z,t23.525
23.5523.575
23.6z cm
-0.04-0.02
00.02
0.04y
15.67515.7
15.72515.75
z cm
-0.04-0.02
00.02
0.04y
0
0.2
0.4
0.6
0.8
1
Ax 0,y,z,t15.675
15.715.725
15.75z cm
-0.04-0.02
00.02
0.04y
Narrow-band pulses – Fresnel’s diffraction
12
-0.04-0.02
00.02
0.04zcm
-0.04-0.02
00.02
0.04y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-0.04
-0.0200.02
0.04zcm
-0.04-0.02
00.02
0.04y
7.827.84
7.867.88
7.9zcm
-0.04-0.02
00.02
0.04y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t7.82
7.847.86
7.887.9
zcm
-0.04-0.02
00.02
0.04y
15.67515.7
15.72515.75
zcm
-0.04-0.02
00.02
0.04y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t15.675
15.715.725
15.75zcm
-0.04-0.02
00.02
0.04y
23.52523.55
23.57523.6
zcm
-0.04-0.02
00.02
0.04y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t23.525
23.5523.575
23.6zcm
-0.04-0.02
00.02
0.04y
31.37531.4
31.42531.45zcm
-0.04-0.02
00.02
0.04y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t31.375
31.431.425
31.45zcm
-0.04-0.02
00.02
0.04y
13
0.000050.0001
0.000150.0002
zcm
-0.0001
-0.0000500.00005
0.0001y
0
0.1
0.2
0.3
0.4
Ax0,y,z,t0.00005
0.00010.00015
0.0002zcm
-0.0001
-0.0000500.00005
0.0001y
00.00005
0.00010.00015zcm
-0.0001-0.00005
00.00005
0.0001y
0
0.1
0.2
0.3
0.4
Ax0,y,z,t0
0.000050.0001
0.00015zcm
-0.0001-0.00005
00.00005
0.0001y
-0.0001-0.00005
00.00005
0.0001zcm
-0.0001
-0.0000500.00005
0.0001y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-0.0001
-0.0000500.00005
0.0001zcm
-0.0001
-0.0000500.00005
0.0001y
Broad-band pulse – semi-spherical diffraction
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AMITaNS, 2012AMITaNS, 2012
Multiplying A(x,y,z,t) with the main phase we find an analytical solution of the wave equation
.ˆ2
ˆ2
1expˆ
ˆ2
ˆ2
1expˆ
2exp
ˆ2),,,(
0
2
20
0
2
20
20
20
rvtr
ierfcrvt
rrvti
rvtr
ierfcrvt
rrvti
rk
r
itzyxE
20
2220 2/exp0,,, rzyxzikzyxE •
• We can observe a translation of the solution in z direction – due to the term exp(ik0z)
• Not stable with time – the amplitude function decreases and the energy distributes over whole space
15
-0.001
0
0.001z
-0.001
0
0.001y
-1
-0.5
0
0.5
1
Ex0,y,z,t-0.001
0
0.001z
-0.001
0
0.001y
-0.001
0
0.001z
-0.001
0
0.001y
-1
-0.5
0
0.5
1
Ex0,y,z,t-0.001
0
0.001z
-0.001
0
0.001y
-0.001
0
0.001z
-0.001
0
0.001y
-1
-0.5
0
0.5
1
Ex0,y,z,t-0.001
0
0.001z
-0.001
0
0.001y
-0.001
0
0.001z
-0.001
0
0.001y
-1
-0.5
0
0.5
1
Ex0,y,z,t-0.001
0
0.001z
-0.001
0
0.001y
-0.001
0
0.001z
-0.001
0
0.001y
-1
-0.5
0
0.5
1
Ex0,y,z,t-0.001
0
0.001z
-0.001
0
0.001y
Fresnel’s type of diffraction as exact solution of the wave equation
16
-0.001
0
0.001z
-0.001
0
0.001y
-1
-0.5
0
0.5
1
Ex0,y,z,t-0.001
0
0.001z
-0.001
0
0.001y
-0.001
0
0.001z
-0.001
0
0.001y
-1
-0.5
0
0.5
1
Ex0,y,z,t-0.001
0
0.001z
-0.001
0
0.001y
-0.001
0
0.001z
-0.001
0
0.001y
-1
-0.5
0
0.5
1
Ex0,y,z,t-0.001
0
0.001z
-0.001
0
0.001y
-0.001
0
0.001z
-0.001
0
0.001y
-1
-0.5
0
0.5
1
Ex0,y,z,t-0.001
0
0.001z
-0.001
0
0.001y
-0.001
0
0.001z
-0.001
0
0.001y
-1
-0.5
0
0.5
1
Ex0,y,z,t-0.001
0
0.001z
-0.001
0
0.001y
-0.001
0
0.001z
-0.001
0
0.001y
-1
-0.5
0
0.5
1
Ex0,y,z,t-0.001
0
0.001z
-0.001
0
0.001y
-0.001
0
0.001z
-0.001
0
0.001y
-1
-0.5
0
0.5
1
Ex0,y,z,t-0.001
0
0.001z
-0.001
0
0.001y
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AMITaNS, 2012AMITaNS, 2012
Our main purpose is to find exact solutions of the DE and wave equations. Let us consider again the solution of the amplitude function A presented as integral in 3D Fourier space
.ˆˆexp
ˆexp)0,ˆ,,(ˆexp2
1 222003
zyxzyx
zyxzyx
kddkdkkzykxki
tkkkivtkkkkAvtzikA
II. Spherically symmetric finite energy solutions of the wave equation
All terms except the Fourier image of the initial conditions depend on the translated wave number . We can use the Transition theorem to present as a function of only. Let the initial conditions are in form
0,ˆ,,ˆ0kkkkA zyx
zk A
zk
Then, applying the Transition theorem we have
1. The Transition theorem xkgFxixgF exp
.exp0,,,0,,, 0* zikzyxAzyxA
.0,ˆ,,ˆ0,,,ˆ0,,, *0 zyxzyx kkkAkkkkAzyxAF
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AMITaNS, 2012AMITaNS, 2012
Thus, all functions in the backward Fourier transform depend on translated wave number only
.ˆˆexp
ˆexp)0,ˆ,,(ˆexp2
1 222*03
zyxzyx
zyxzyx
kddkdkkzykxki
tkkkivtkkkAvtzikA
In our study we shall consider spherically-symmetric functions A*(x,y,z,0), i.e.
.,0,0,,, 222** zyxrrAzyxA
The Fourier image of a spherically-symmetric function is also spherically-symmetric, i.e.
.,0,ˆ0,0,,, 222***zyxrr kkkkkArAFzyxAF
The backward radial Fourier integral is
.ˆˆsinˆexp0,ˆˆˆ1exp
2
1 *
0
02 rrrrr kdkrkivtkAkr
vtzikA
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AMITaNS, 2012AMITaNS, 2012
2. Spherically-symmetric exact solutions with finite energy
A.A. Localized algebraic function of kind 20
2* /1/1 rrA
20
2
0 1/exp0,,,r
rzikzyxA
*2
0
2
2
02
0
2
2
02
0
2
0
1/10,,,
.1/1,,,
1/exp,,,
Ar
rzyxE
r
ivt
r
rtzyxE
r
ivt
r
rvtziktzyxA
.ˆexpˆ2
0,ˆˆexp2
0,ˆ00
*r
r
rrr
r krk
kAkrk
kA
The Fourier image of the initial data is
Hence, the corresponding solutions of the amplitude and wave equations are
20
-4-2
02
4zcm
-4-2
024y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
024y
-4-2
02
4zcm
-4-2
024y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
024y
-4-2
02
4zcm
-4-2
024y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
024y
-4-2
02
4zcm
-4-2
024y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
024y
-4-2
02
4zcm
-4-2
024y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
024y
-4-2
02
4zcm
-4-2
024y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
024y
21
-4-2
02
4zcm
-4-2
02
4y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
02
4y
-4-2
02
4zcm
-4-2
02
4y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
02
4y
-4-2
02
4zcm
-4-2
02
4y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
02
4y
-4-2
02
4zcm
-4-2
02
4y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
02
4y
-4-2
02
4zcm
-4-2
02
4y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
02
4y
-4-2
02
4zcm
-4-2
02
4y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
02
4y
-4-2
02
4zcm
-4-2
02
4y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
02
4y
-4-2
02
4zcm
-4-2
02
4y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
02
4y
-4-2
02
4zcm
-4-2
02
4y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
02
4y
-4-2
02
4zcm
-4-2
02
4y
0
0.2
0.4
0.6
0.8
1
Ax0,y,z,t-4
-20
24
zcm
-4-2
02
4y
22
AMITaNS, 2012AMITaNS, 2012
B.B. Localized algebraic function of kind 220
2* /1/1 rrA
2
20
2
0 1/exp0,,,r
rzikzyxA
220
2
0
220
2
00
2,,,
2exp,,,
itvrr
itvrtzyxE
itvrr
itvrvtziktzyxA
.ˆexp4
0,ˆˆexp4
0,ˆ0000
*rrrr krrkAkrrkA
The Fourier image of the initial data is
Hence, the corresponding solutions of the amplitude and wave equations are
23
AMITaNS, 2012AMITaNS, 2012
C.C. Localized algebraic function of kind 42* 1/1 rA
420 1/exp0,,, rzikzyxA
42
02
22222
420
2
22222
0
13203282986,,,
1320328298
exp6,,,
itvrr
itvtvirtvitvtvirvtrtvivttzyxE
itvrr
itvtvirtvitvtvirvtrtvivt
vtziktzyxA
22* ˆˆ33ˆexp96
0,ˆˆ33exp96
0,ˆrrrrrrrr kkkkAkkkkA
The Fourier image of the initial data is
Hence, the corresponding solutions of the amplitude and wave equations are
24
AMITaNS, 2012AMITaNS, 2012
D.D. Localized algebraic function of kind 422* 1/1 rrA
4220 1/1exp0,,, rrzikzyxA
44
440
11
32
3,,,
11exp
32
3,,,
irvtirvt
itzyxE
irvtirvtvtzik
itzyxA
.ˆˆexp48
0,ˆˆexp48
0,ˆ 22*rrrrrr kkkAkkkA
The Fourier image of the initial data is
Hence, the corresponding solutions of the amplitude and wave equations are
25
AMITaNS, 2012AMITaNS, 2012
E. Localized algebraic function of kind 322* 1/3 rrrA
3220 1/3exp0,,, rrrzikzyxA
33
330
11
2
1,,,
11exp
2
1,,,
irvtirvtrtzyxE
irvtirvtvtzik
rtzyxA
.ˆˆexp48
0,ˆˆexp4
0,ˆ *rrrrrr kkkAkkkA
The Fourier image of the initial data is
Hence, the corresponding solutions of the amplitude and wave equations are
26
AMITaNS, 2012AMITaNS, 2012
,,01
220
20
32
2
2nkCC
t
C
vC
where γ = 2.
After neglecting two small perturbations terms the corresponding nonlinear amplitude equation for broad-band femtosecond pulses can be reduced to
NONLINEAR PROPAGATION OF BROAD-BAND PULSES IN AIR. LORENTZ TYPE SOLITON.
where γ is the nonlinear coefficient, C0 is the amplitude maximum and n2 is the nonlinear refractive index. The equation admits exact soliton solution propagating in forward direction only
,/~,~1
2~
~lnsec 22222
viatviazyxrrr
rhC
• The 3D+1 soliton solution has Lorentz shape with asymmetric kz spectrum
• The solution preserves its spatial and spectral shape in time
27
299990
300000
300010z
-10
0
10y
0
0.5
1
1.5
2
Ay,z299990
300000
300010z
-10
0
10y
599990
600000
600010z
-10
0
10y
0
0.5
1
1.5
2
Ay,z599990
600000
600010z
-10
0
10y
3.29999 106
3.3 106
3.30001 106z
-10
0
10y
0
0.5
1
1.5
2
Ay,z3.29999 106
3.3 106
3.30001 106z
-10
0
10y
-10
0
10z
-10
0
10y
0
0.5
1
1.5
2
Ay,z-10
0
10z
-10
0
10y
28
-10
0
10z
-10
0
10y
0
0.5
1
1.5
2
Ay,z-10
0
10z
-10
0
10y
299990
300000
300010z
-10
0
10y
0
0.5
1
1.5
2
Ay,z299990
300000
300010z
-10
0
10y
599990
600000
600010z
-10
0
10y
0
0.5
1
1.5
2
Ay,z599990
600000
600010z
-10
0
10y
29
AMITaNS, 2012AMITaNS, 2012
,7245
7 22
74
4
kiikik BBBE
cm
e
AMITaNS, 2012AMITaNS, 2012
In 1935 Euler and Kockel predict one intrinsic nonlinearity to the electromagnetic vacuum due to electron-positron nonlinear polarization. This leads to field-dependent dielectric tensor in form
PROPAGATION OF BROAD-BAND PULSES IN NONLINEAR VACUUM. VORTEX LOCALIZED SOLUTION.
where complex form of presentation of the electrical Ei and magnetic Bi components is used. The term containing BiBk vanishes when localized electromagnetic wave with only one magnetic component Bl is investigated. Thus, the dielectric response relevant to such pulse is
.45
14 22
74
4
BE
cm
eikik
30
AMITaNS, 2012AMITaNS, 2012
.90
7,0
1
01
74
420
22
2
2
2
22
2
2
2
cm
ekBBE
t
B
cB
EBEt
E
cE
AMITaNS, 2012AMITaNS, 2012
The magnetic field, rather than the electrical filed, appears in the expression for the dielectric response and the nonlinear addition to the intensity profile (effective mass density) of one electromagnetic wave in nonlinear vacuum can be expressed in electromagnetic units as Inl = (|E|2 – |B|2). When the spectral width of one pulse Δkz exceeds the values of the main wave vector Δkz k0 the system of amplitude equations in nonlinear vacuum becomes
We present the components of the electrical and magnetic fields as a vector sum of circular and linear components
.zl
yxc
z
BB
EiEE
E
31
AMITaNS, 2012AMITaNS, 2012
.0
1
01
01
222
2
2
2
222
2
2
2
222
2
2
2
llczl
l
clczc
c
zlczz
z
BBEEt
B
cB
EBEEt
E
cE
EBEEt
E
cE
AMITaNS, 2012AMITaNS, 2012
Hence we get the following scalar system of equations
Let us now parameterize the space-time by pseudo-spherical coordinates (r, τ, θ, φ)
,cossincosh
sinsincosh
coscosh
sinh
rx
ry
rz
rct
where .22222 tczyxr
32
AMITaNS, 2012AMITaNS, 2012
,cosh
1tanh2
131,2222
2
22
2
2
2
2
rrrrrrtc
AMITaNS, 2012AMITaNS, 2012
The d’Alambert operator in pseudo-spherical coordinates is
Where Δθ,φ is the angular part of the usual Laplace operator
),,()()(,,,
,,),,()()(,,,
lll
iii
YTrRrB
cziYTrRrE
which separates the variables. The nonlinear terms appear in radial part only.
,),(),(),(222222
constYTYTYT llcczz
.sin
1sin
sin
12
2
2,
We solve the system of equations using the method of separation of the variables
with an additional constrain on the angular and “spherical“ time parts
33
AMITaNS, 2012AMITaNS, 2012
,,,,03 2
22
2
lcziRRRr
A
r
R
r
R
ri
AMITaNS, 2012AMITaNS, 2012
The radial parts obey the equations
where Ai are separation constants. We look for localized solutions of the kind
.2;1 22 iA
where Ci are another separation constants connected with the angular part of the Laplace operator Yi(θ,φ).
The separation constants Ai, α and γ satisfy the relations
The corresponding τ – dependent parts of the equations are linear
,,,,0coshcoshsinh2cosh 22
22 lcziTAC
d
dT
d
Tdiii
ii
.
lnsec
r
rhR
34
AMITaNS, 2012AMITaNS, 2012
sinh;cosh;cosh lcz TTT
AMITaNS, 2012AMITaNS, 2012
The solutions of the τ – equations which satisfy the constrain condition are
with separation constants for the electrical part Az = Ac = 3; Cz = Cc = 2 and for the magnetic part Al = 3; Cl = 0 correspondingly. The magnetic part of the system of equations do not depend from the angular components, i.e. Yl(θ,φ) = 0, while for the electrical part we have the following linear system of equations
.8;2;42
Using the relation between the separation constants Ai and the real number α we have
The solutions which satisfy the above system and the constrain condition are
.,,2., cziY
Y
i
i
.expsin;cos iTY cz
35
AMITaNS, 2012AMITaNS, 2012AMITaNS, 2012AMITaNS, 2012
Finally we can write the exact localized solution of the system of nonlinear equations representing the propagation of electromagnetic wave in vacuum
The solution can be rewritten in Cartesian coordinates
.sinh
lnsec),,,(
)exp(sincoshlnsec
),,,(
coscoshlnsec
),,,(
2
2
2
r
rhrB
ir
rhrE
r
rhrE
l
c
z
.,1
2),,,(
1
)(2),,,(
1
2),,,(
222224
4
4
tczyxrr
ctrB
r
iyxrE
r
ztzyxE
l
c
z
36
AMITaNS, 2012AMITaNS, 2012AMITaNS, 2012AMITaNS, 2012
The linear part of the energy density (intensity) of the solution can be expressed as
• The solution admits own orbital momentum l = 1 for the electrical components • The solution is with finite energy and presents nonlinear spherical shock wave
.1
4
1
84),,,(
22222
2222
24
222
tczyx
tczyx
r
tcrtzyxI
37
-4 -2 2 4r
-1
-0.5
0.5
1
Ir,t
8 10 12 14r
-1
-0.5
0.5
1
Ir,t
-4 -2 2 4r
0.2
0.4
0.6
0.8
1Ir,t
8 10 12 14r
0.005
0.01
0.015
0.02
0.025
0.03Ir,t
t = 0 t = 10
38
1) New exact localized solutions of the linear amplitude and wave
equations are presented. The amplitude function decreases and the
energy distributes over whole space for finite time.
2) The soliton solution of the nonlinear wave equation admits
Lorentz’ shape and propagates preserving the initial form and
spectrum.
3) The system of nonlinear vacuum equations is solved in 3D+1
Minkowski time-space. The solution admits own angular momentum
of the electrical part. The solution is with finite energy and presents
nonlinear spherical wave.