Spectral Approach to Numerical Integration of the GKPClass Equations in the Problems of Nonlinear Wave Dynamics Simulation

Elena S. Belashova*, Vasily Yu. Belashov* and Sergey V. Vladimirov**
*Kazan State Power Engineering University, Kazan, Russia
**School of Physics, University of Sydney, Sydney, Australia
Abstract. The spectral approach to numerical integration of the generalized Kadomtsev-Petviashvili-class equations in
the problems of nonlinear wave dynamics simulation is presented.
Keywords: Soliton, Nonlinear waves, Numerical simulation, Spectral method, KP equation, Plasma
PACS: 5235, 9420, 9430, 0270, 0340K
INTRODUCTION
The 2D and 3D nonlinear waves propagating in dispersive media (for example, ion-acoustic (IA) and
magnetosonic (MS) waves in space plasma) are described by the equations’ class [1]

where u = u(t, x, r⊥ ) is a function defining the wave field, and ℜ = ℜ[u] is a some linear functional of u. The form
of right-hand side of eq. (1) depends on the wave properties of medium and the dispersion sign, and the value of L is
defined by the dispersion character. For example, in the cases of IA waves propagating in isotropic plasma and MS
waves propagating in plasma near transverse direction to magnetic field when the dispersion law has form

where signs ‘+’ and ‘−’ correspond to first and second cases, respectively, c0 is a phase velocity of oscillations at
| k |→0 , and δl are the dispersion “scales”, functional ℜ has the form ℜ = κ∇⊥w , ∂ xw = ∇⊥u . In this case for
L=1, 2 eq. (1) is the Kadomtsev-Petviashvili (KP) or the generalized KP equation, respectively:

and it can have the 1D, 2D and 3D wave solutions localized in space dependently on L value and signs of
coefficients βl and κ . In case of strong anisotropic media functional ℜ = κΔ⊥∂ xu , and eq. (1) known as
Zakharov-Kuznetsov equation has the soliton-like solutions too.
The 2D KP equation (L=1, ∂ z = 0 ) can be integrated analytically using the IST method, but this method allows
to obtain the exact solution only under certain initial conditions [2]. The technique of IST method for the integration
of the 3D KP equation and the 2D and 3D generalized eq. (2) with L=2 is not developed now. Therefore,
development of the numerical technique for integration of the eq. (1) class is of indubitable interest for the nonlinear
plasma physics.
The hard original method for numerical integration of the KP equation (eq. (2) with L=1) was proposed in [3].
However, it doesn’t enable to study the solution’s evolution at initial stage and to consider the soliton interaction’s
dynamics. There are also some other methods for the numerical integration of the different equations of class (1)
(see, for example, review in [2]), but they are rather cumbersome.
In this paper we consider spectral approach for the numerical integration of the equations of class (1) describing the
dynamics of IA and MS waves in a plasma on the basis of eq. (2) with L=2 which is rather simple and very effective
and doesn’t require big time and memory expenditures. This approach was first used for study of some problems of
nonlinear evolution.

In case (a) the dispersion in eq. (2) is negative and the
solution in 2D space has form of 1D IA soliton (fig. 1)
propagating in isotropic plasma with v = const and integrals
ℑ1, ℑ2 , ℑ3 = const . That corresponds the analytical
results for the KP equation with negative dispersion
obtained in [6] by use of the IST technique.
The case (b) is more complicate because the dispersion
sign is defined by correlation of signs of dispersive
coefficients β1 and β2 ≠ 0 . Therefore, the form of the FMS
wave essentially depends on angle θ between vector k and
magnetic field B.
The results obtained were discussed in [2, 4], and we
show here only the most interest case when in 3D case the nonlinear
stabilization of the FMS wave beam propagating in plasma near the
cone π / 2 − θ ≤ (m/ M)1/ 2 takes place after the stages of its initial
sub-focusing and nonlinear saturation (fig. 2) [2, 4]. We studied the
problem for eq. (2) with L=2 rewritten, using transition to variables
x →−st, y →−sκ1/ 2 y, z →−sκ1/ 2 z, t →sx, h→−(6 / α)h,
1/ 4
s = γ 2 , κ= vA / 2 , as ∂t (∂ x h + h∂t h − ε∂t h − λ∂t h)=Δ⊥h 6 3 5 (we
assumed that Δ⊥ = ∂ρ2 + (1/ ρ)∂ρ ) and describing the propagation of
the FMS wave beam along the x-axis from boundary x=0 with initial
condition ( ,0, ) cos( )exp( 2 )
h0 = h t ρ = mt −ρ .
The results presented show that spectral approach considered
above can be successively used for study of the problems of 2D and 3D nonlinear wave dynamics in space plasma.
REFERENCES
1. V.Yu. Belashov, Proc. ISSS-5, Kyoto, Japan, 118 (1997)
2. V.Yu. Belashov, S.V. Vladimirov, Solitary Waves in Dispersive Complex Media, Springer-Verlag, 2005.
3. V.I. Petviashvili, Sov. Plasma Phys., 2, 469 (1976).
4. V.Yu. Belashov, Plasma Phys. and Contr. Fusion, 36, 1661 (1994)
5. Yu.A. Berezin, Nonlinear wave processes simulation. Novosibirsk, Nauka, 1982.
7. V.E. Zakharov, S.V. Manakov, S.P. Novikov, L.P. Pitaevsky, Theory of solitons. Moscow, Nauka, 1980.

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