scattering of weakly nonlinear dispersive wave on a ... · oleg kiselev [email protected] institute of...

Scattering of weakly nonlinear dispersive wave ona parametric resonance

Oleg Kiselev [email protected] of Mathematics, Ufa, Russia

collaborated with

S.Glebov, USPTU N. Tarkhanov, Potsdam University

Nonlinear Physics. Theory and Experiment. V, June, 12-20,2008

Oleg Kiselev, Institute of Mathematics, Ufa, Russia Scattering on a local parametric resonance

Outlines

1 Parametric driven nonlinear Klein-Gordon equation

2 Small amplitude solution

3 Problem of control for weak nonlinear wave

4 Simulations

5 Asymptotic approach outside of resonance

6 Primary local parametric resonance

7 Scattering problem for the primary local resonance equation

8 Matching and connection formula for solution before and afterthe resonance

9 Conclutions

Outlines

4 Simulations

9 Conclutions

Outlines

4 Simulations

9 Conclutions

Outlines

4 Simulations

9 Conclutions

Outlines

4 Simulations

9 Conclutions

Outlines

4 Simulations

9 Conclutions

Outlines

4 Simulations

9 Conclutions

Outlines

4 Simulations

9 Conclutions

Outlines

4 Simulations

9 Conclutions

Mathematical model

An object of this talk is solutions of parametric perturbednonlinear Klein-Gordon equation:

∂2t U − ∂2

(1 + εf cos

(S(ε2x , ε2t)

))U + γU3 = 0.

Here ε is small parameter, γ, f ∈ R and S(y , z) is smoothfunction.

Mathematical model

An object of this talk is solutions of parametric perturbednonlinear Klein-Gordon equation:

∂2t U − ∂2

(1 + εf cos

(S(ε2x , ε2t)

))U + γU3 = 0.

Here ε is small parameter, γ, f ∈ R and S(y , z) is smoothfunction.

Small amplitude solution

We study a small amplitude solution in the form of amodulated oscillating wave:

U(x , t, ε) ∼ εu1(x1, t1, x2, t2) exp{i(kx + ωt)}+ c .c ..

This solution depends on groups of scaled variables:

fast variables are x , t;slow variables are x1 = εx , t1 = εt;very slow variables are x2 = ε2x and t2 = ε2t.

Small amplitude solution

We study a small amplitude solution in the form of amodulated oscillating wave:

U(x , t, ε) ∼ εu1(x1, t1, x2, t2) exp{i(kx + ωt)}+ c .c ..

This solution depends on groups of scaled variables:

fast variables are x , t;slow variables are x1 = εx , t1 = εt;very slow variables are x2 = ε2x and t2 = ε2t.

Problem

Background

In a general approach the shape of the weak nonlinear wave isdefined by Nonlinear Schrodinger equation.There exist resonant curves on the plane (x2, t2) such that thisapproach is not valid near these curves and the main role playsthe perturbation.

To control of weak nonlinear dispersive waves.To find a connection formula for the solution before and afterthe resonance.

Problem

Background

In a general approach the shape of the weak nonlinear wave isdefined by Nonlinear Schrodinger equation.There exist resonant curves on the plane (x2, t2) such that thisapproach is not valid near these curves and the main role playsthe perturbation.

To control of weak nonlinear dispersive waves.To find a connection formula for the solution before and afterthe resonance.

Simulations

Annihilation of NLSE soliton

Simulations

Generation of NLSE soliton

Bounds of NLSE approach

It is well-known that the weak nonlinear waves are defined byNLSE.

Bibliography

P. L. Kelley, 15 (1965), pp. 1005-1008.

V. I. Talanov, ZhETF Letters, 1965, n2, pp. 218-222.

V. E. Zaharov, J Appl Mech. and Tech. Phys. 1968, n2, pp.86-94.

Further we show the borders of this approach for solutions of theparametric driven equation.

Theorem about external expansion

The asymptotic solution of PNGKE has the form

U = ε(u1(t1, x1, x2, t2) exp{i(kx2 + ωt2)/ε

2}+ c .c .)+

ε2(u+

2 exp(iφ+(x2, t2)/ε2) + u−2 exp(iφ−(x2, t2))/ε

c .c .)

+ . . . .

u1 = Ψexp

∫ t2[

L[φ−]+

L[φ+]

Function Ψ is determined by the NLSE:

iω∂t2Ψ− ∂2ζ Ψ + 3γ|Ψ|2Ψ = 0, ζ = ωx1 + kt1.

2}+ c .c .)+

ε2(u+

c .c .)

+ . . . .

u1 = Ψexp

∫ t2[

L[φ−]+

L[φ+]

iω∂t2Ψ− ∂2ζ Ψ + 3γ|Ψ|2Ψ = 0, ζ = ωx1 + kt1.

2}+ c .c .)+

ε2(u+

c .c .)

+ . . . .

u1 = Ψexp

∫ t2[

L[φ−]+

L[φ+]

iω∂t2Ψ− ∂2ζ Ψ + 3γ|Ψ|2Ψ = 0, ζ = ωx1 + kt1.

The correction term is defined by following formulas:

u±2 = − u1f

2L[φ±].

Here φ± is a phase function and L[ψ] is an eikonal operator

φ± = kx2 + ωt2 ± S(x2, t2), L[φ] ≡ −(∂t2φ

(∂x2φ

)2+ 1.

The expansion is valid in the domains

−ε−1

(− (∂t2φ±)2 + (∂x2φ±)2 + 1

)� 1,

u±2 = − u1f

2L[φ±].

φ± = kx2 + ωt2 ± S(x2, t2), L[φ] ≡ −(∂t2φ

(∂x2φ

)2+ 1.

−ε−1

(− (∂t2φ±)2 + (∂x2φ±)2 + 1

)� 1,

u±2 = − u1f

2L[φ±].

φ± = kx2 + ωt2 ± S(x2, t2), L[φ] ≡ −(∂t2φ

(∂x2φ

)2+ 1.

−ε−1

(− (∂t2φ±)2 + (∂x2φ±)2 + 1

)� 1,

Local parametric resonance

The condition of primary local parametric resonance is anequation:

−(∂t2(kx2+ωt2±S(x2, t2))

(∂x2(kx2+ωt2±S(x2, t2))

)2+1 = 0

A typical local resonance generates new harmonics (seeGlebov, Kiselev, Lazarev, 2006) with

k1 = k ± ∂x2S |L[χ1,±1]=0, as |k1| 6= k

The condition of primary local parametric resonance is anequation:

−(∂t2(kx2+ωt2±S(x2, t2))

(∂x2(kx2+ωt2±S(x2, t2))

)2+1 = 0

A typical local resonance generates new harmonics (seeGlebov, Kiselev, Lazarev, 2006) with

k1 = k ± ∂x2S |L[χ1,±1]=0, as |k1| 6= k

Bibliography

Firstly the local resonance was studied since of 1970’s:

J. Kevorkian, SIAM J. Appl. Math., 20 (1971), pp. 364-373.L. Rubenfeld, Stud. Appl. Math., 57 (1977), pp. 77-92.J. C. Neu, SIAM J. Appl. Math., 43 (1983), pp. 141-156.L. A. Kalyakin, Mathematical notes, 44 (1988),pp. 697-699.S. G. Glebov, Differential equations., 31 (1995),pp. 1402-1408.

The local resonance may be used for soliton generation andcontrol:

S.G. Glebov, O.M. Kiselev, V.A. Lazarev, Proceedings of theSteklov Institute of Mathematics. Suppl., 2003, 1, S84-S90.S.G.Glebov, O.M. Kiselev, V.A. Lazarev, SIAM J. Appl.Math., 2005, vol.65, n6, pp.2158-2177.

Bibliography

Firstly the local resonance was studied since of 1970’s:

J. Kevorkian, SIAM J. Appl. Math., 20 (1971), pp. 364-373.L. Rubenfeld, Stud. Appl. Math., 57 (1977), pp. 77-92.J. C. Neu, SIAM J. Appl. Math., 43 (1983), pp. 141-156.L. A. Kalyakin, Mathematical notes, 44 (1988),pp. 697-699.S. G. Glebov, Differential equations., 31 (1995),pp. 1402-1408.

The local resonance may be used for soliton generation andcontrol:

S.G. Glebov, O.M. Kiselev, V.A. Lazarev, Proceedings of theSteklov Institute of Mathematics. Suppl., 2003, 1, S84-S90.S.G.Glebov, O.M. Kiselev, V.A. Lazarev, SIAM J. Appl.Math., 2005, vol.65, n6, pp.2158-2177.

Here we consider a special case which is called by localparametric resonance

k − ∂x2S = −k.

In this case new harmonics are not generated in theleading-order term but the envelope function changes.

The local parametric resonance was studied for ordinarydifferential equations by

Bibliography

V.S. Buslaev, L.A.Dmitrieva. Theor. Math. Phys., 1987, v.73,n3, pp.430-441.S.H. Oueini, C.-M. Chin and A.H. Nayfeh J. Of Vibration andcontrol 2000, 6, pp.1115-1133.

Here we consider a special case which is called by localparametric resonance

k − ∂x2S = −k.

In this case new harmonics are not generated in theleading-order term but the envelope function changes.The local parametric resonance was studied for ordinarydifferential equations by

Bibliography

V.S. Buslaev, L.A.Dmitrieva. Theor. Math. Phys., 1987, v.73,n3, pp.430-441.S.H. Oueini, C.-M. Chin and A.H. Nayfeh J. Of Vibration andcontrol 2000, 6, pp.1115-1133.

In a neighborhood of the parametric resonant curve the formalasymptotic expansion is

U(x , t, ε) = ε(w1(x1, t1, x2, t2) exp(iS(x2, t2)/ε) + c .c .

ε2(w2,1 exp(iS(x2, t2)/ε)+w2,2 exp(2iS(x2, t2)/ε)+ c .c .

)+ . . . .

where w1(x1, t1, x2, t2) is a solution of

Local Parametric Resonance Equation

i∂x2S∂x1w1 − i∂t2S∂t1w1 + λw1 + f2w1 = 0.

and λ is a function which depends on slow and very slowvariables:

λ(x1, t1, x2, t2) = ε−1(−

(∂t2φ±

(∂x2φ±

)+ . . . .

i∂x2S∂x1w1 − i∂t2S∂t1w1 + λw1 + f2w1 = 0.

λ(x1, t1, x2, t2) = ε−1(−

(∂t2φ±

(∂x2φ±

)+ . . . .

i∂x2S∂x1w1 − i∂t2S∂t1w1 + λw1 + f2w1 = 0.

λ(x1, t1, x2, t2) = ε−1(−

(∂t2φ±

(∂x2φ±

)Oleg Kiselev, Institute of Mathematics, Ufa, Russia Scattering on a local parametric resonance

)+ . . . .

i∂x2S∂x1w1 − i∂t2S∂t1w1 + λw1 + f2w1 = 0.

λ(x1, t1, x2, t2) = ε−1(−

(∂t2φ±

(∂x2φ±

)Oleg Kiselev, Institute of Mathematics, Ufa, Russia Scattering on a local parametric resonance

Scattering problem

Our goal is to solve a scattering problem for the local parametricresonance equation. This problem is solved by four steps.

Obtain an asymptotic reduction to

the ordinary primary parametric resonance equation

i dWdσ + σW + FW = 0

Solve this equation using the parabolic cylinder functions.

Use formula for the solution to solve a scattering problem.

Estimate a domain of validity for the internal asymptoticsolution.

Scattering problem

Matching asymptotic expansions and connection formula

In the domain l > 0 the formal asymptotic solution has the sameform

U(x , t, ε) ∼ εv1(x1, t1, t2) exp{i(kx + ωt)}+ c .c ..

The amplitude v1 = Ψ+ exp

4ωG (x2, t2)

}and Ψ is determined

by the nonlinear Schrodinger equation also and initial datum onthe curve l = 0

Ψ+(t2, ζ)|l=0 = ef 2π8 Ψ− +

(1 + i)ef 2π16 e i f 2

16ln(2)f

2Γ(1− i f 2

8 )Ψ−,

where ψ = Ψ(t2, ζ)|l=−0Oleg Kiselev, Institute of Mathematics, Ufa, Russia Scattering on a local parametric resonance

Conclutions

We construct the small asymptotic solution for the parametricperturbed Klein-Gordon equation.

The shape of this solution defines by NSE.

This approximation is valid before and after the parametricresonant curve.

On this curve the solution has a jump. The solution of NSEbefore the line and after the line are connected by:

Ψ+(t2, ζ)|l=0 =

ef 2π8 Ψ−(t2, ζ)|l=0 + (1+i)e

f 2π16 e i f 2

16 ln(2)f√

2Γ(1−i f 2

Ψ−(t2, ζ)|l=0,

Conclutions

Ψ+(t2, ζ)|l=0 =

ef 2π8 Ψ−(t2, ζ)|l=0 + (1+i)e

f 2π16 e i f 2

16 ln(2)f√

2Γ(1−i f 2

Ψ−(t2, ζ)|l=0,

Conclutions

Ψ+(t2, ζ)|l=0 =

ef 2π8 Ψ−(t2, ζ)|l=0 + (1+i)e

f 2π16 e i f 2

16 ln(2)f√

2Γ(1−i f 2

Ψ−(t2, ζ)|l=0,

Conclutions

Ψ+(t2, ζ)|l=0 =

ef 2π8 Ψ−(t2, ζ)|l=0 + (1+i)e

f 2π16 e i f 2

16 ln(2)f√

2Γ(1−i f 2

Ψ−(t2, ζ)|l=0,

New results

Primary parametric resonance partial differential equation

i∂x2S∂x1w1 − i∂t2S∂t1w1 + λw1 +f

2w1 = 0.

Connection formula:

Ψ+(t2, ζ)|l=0 = ef 2π8 Ψ−(t2, ζ)|l=0 +

(1 + i)ef 2π16 e i f 2

16ln(2)f

2Γ(1− i f 2

8 )Ψ−(t2, ζ)|l=0,

New results

Primary parametric resonance partial differential equation

i∂x2S∂x1w1 − i∂t2S∂t1w1 + λw1 +f

2w1 = 0.

Connection formula:

Ψ+(t2, ζ)|l=0 = ef 2π8 Ψ−(t2, ζ)|l=0 +

(1 + i)ef 2π16 e i f 2

16ln(2)f

2Γ(1− i f 2

8 )Ψ−(t2, ζ)|l=0,

Bibliography

Preprint: arXiv:0806.3338

scattering of weakly nonlinear dispersive wave on a ... · oleg kiselev [email protected] institute of...

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