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In project we propose a new approach to the solution for the problem of generating the solitons with given parameters.
They are generated due to the passing through local resonance with the external oscillating perturbation.
Multi-scale and matching methods allow us to investigate the solutions that describe the rearrangements in soliton generation.
we intend to obtain the explicit formulas for the dependence of the external force and parameters of the solitons appearing after the resonance.
The phenomenon of solitary waves generation by the small external oscillating force
in the solutions of the nonlinear equations is known mainly in numerical analysis. See, e.g.,
L. Friedland, A.G. Shagalov, Exсitation of solitons by adiabatic
multyresonant forcing, Phys. Rev. Lett., v. 8, 20, 1998, p. 4357-4360.
The sufficient rearrangement of the solutions structure implies the difficulties
in analytic description of the process of soliton generation.
On the local resonance in linear ordinary equations see, e.g., the papers of J. Kevorkyan (1971) and L. Rubinfeld (1977).
Later the local resonance was studied for the partial differential equations, linear, see Neu (1983),
and weakly nonlinear, see L.A. Kalyakin (1988) and S.G. Glebov (1995).
Wave propagation in nonlinear media leads to the deformation of the wave envelope.
It is well known that the envelope can satisfy different nonlinear equations, in particular
nonlinear Schr\"odinger equation (L.P. Kelley (1965), V.I. Talanov (1965), V.E. Zaharov (1968)).
Soliton solutions of the nonlinear Schr\"odinger equation don't alter their properties in time.
Therefore they can be interesting as envelopes for rapidly oscillating wave packets.
The problem of soliton generation in nonlinear equations is important for applications.
The effect of appearing solitary waves because of the modulation instability is well known for some nonlinear equations.
For example, there is complete analytic description of such soliton annihilation (generation)
for the Kadomtsev-Petviashvili equation.
Results on soliton generation in the perturbed nonlinear Schr\"odinger equation because of the instability of
certain modes are given, e.g., in N.V. Alexeeva, I.V. Barashenkov,
D.E. Pelinovsky, Dynamics of the parametrically driven NLS solitons
beyond the onset of the oscillatory instability. Nonlinearity, v. 12,
1999, p. 103-140. Unstable character of the solution restricts the possibilities
of soliton generation with preassigned parameters.
Whereas for applications it is important to control the parameters of generated wave.
In this project we'll show that the solitary wave generation by the local resonance is universal phenomenon in the
theory of wave propagation. It can be used for the generation of waves with preassigned parameters.
Unlike the known numeric results our approach reveals the dependence the generated wave parameters
on the external perturbation in explicit form.
Problems
Studying processes of wave generation in the systems with strong and weak dispersion:
- Study the process of solitary waves scattering in the nonlinear Schr\"odinger equation
with a small external perturbation.
- Investigate the generation of solitary wave packets in the nonlinear Klein-Gordon equation.
- Obtain the asymptotic solution describing the genesis of the large amplitude waves for the perturbed Boussinesq equation.
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Results
We began to study the above posed problems in 2003.
Results of 2003:
1. The formal asymptotic solution for the perturbed Scr\"odinger equation was obtained that describes
the process of soliton genesis. Asymptotically this solution was studied in detail. The result is given in
S.G. Glebov ,O.M. Kiselev, and V.A. Lazarev, Solitons generation by local resonance
interaction, Proceedings of the Steklov Institute of Mathematics, Suppl.1, 2003, pp. S84-S90.
Solitons generation by local resonance interaction.
2. It was studied the process of solitary wave packets generation in solving
perturbed nonlinear Klein-Gordon equation. It was demonstrated the wave packet forming in the solution
qf Klein-Gordon equation under the passing through the local resonance.
The envelope of the wave packet satisfies the nonlinear Schr\"odinger equation.
The special form of the perturbation allows to form the wave packets with the soliton envelope that
is not prone to the dispersion smearing of the packet in time. The results are announced in
S.G.Glebov, V.A.Lazarev, O.M. Kiselev. Generation of solitary packets
of waves by resonance, Proceedings of International seminar
"Day on Diffraction-2003", SPb. Russia. pp.46-51.
Generation of solitary packets.
Profile of solitary packets.
3. The first stage in studying the wave generation in the Boussinesq equation was made.
It was shown that the small periodic perturbation with a slowly varying frequency leads to
the appearing finite amplitude waves moving in opposite directions. The principal member satisfies the Hopf equation.
The results are given in
N.K. Gorbatova, Wave generation in the perturbed Boussinesq equation,
Degree work, Ufa, USATU, 2003.
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Publications
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S.G. Glebov ,O.M. Kiselev, and V.A. Lazarev, Solitons generation by local resonance
interaction, Proceedings of the Steklov Institute of Mathematics, Suppl.1, 2003, pp. S84-S90.
- S.G.Glebov, V.A.Lazarev, O.M. Kiselev, Generation of solitary packets
of waves by resonance, Proceedings of International seminar
"Day on Diffraction-2003", SPb. Russia. pp.46-51.
- N.K. Gorbatova, Wave generation in the perturbed Boussinesq equation,
Degree work, Ufa, USATU, 2003..
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