EXACT LOCALIZED SOLUTIONS OF THE (2+1) DIMENSIONAL COMPLEX GINZBURG-LANDAU EQUATION

EMMANUEL YOMBA^{1,2} AND TIMOLÉON CRÉPIN KOFANÉ^{3}

Abstract. The (2+1) dimensional (D) complex Ginzburg-Landau (CGL) equation is analytically studied using a method derived from the Painlevé test for integrability and Hirota’s multilinear formalism. We show that 2D propagating pulse, 2D stationary pulse, 2D stationary dark and 2D propagating hole solutions are the structures that organize the dynamic of this extended systems. The degeneracies of this (2+1) dimensional CGL equation are examined as well as their solutions. This equation reveals, to exhibit rich repertoire of behaviors and, to be a generalization of an integrable system (nonlinear Schroedinger equation) and a variational equation (real Ginzburg-Landau equation).

KEYWORDS

(2+1) dimensional complex Ginzburg-Landau equation, Painlevé test for integrability, Hirota method, propagating pulses, stationary pulses, propagating hole solutions and stationary dark solutions.

1 Max-Planck- Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany.

.

2 Department of Physics, Faculty of Sciences  University of Ngaoundéré, Po. Box 454

Ngaoundéré, Cameroon. .

3 Department of Physics, Faculty of Sciences  University of Yaoundé I, Po. Box. 812,

Yaoundé, Cameroon..

2 EMMANUEL YOMBA^{1,2} AND TIMOLÉON CRÉPIN KOFANÉ^{3}

1. Introduction

Many spatially extended systems display the formation of patterns when driven sufficiently far from equilibrium. Since near threshold the form of the amplitude or envelope equation depends mainly on the symmetries and on the nature of the primary bifurcation, the amplitude description has become an important organizing principle of the theory of non-equilibrium pattern formation. This envelope equation plays a crucially important role in understanding of the power of nonlinearities in pattern selection and pattern dynamics. When subject to large external constraints, these systems may decay into spatio-temporal disorder, a state which is often reffered to weak turbulence. The best model which allows such phenomena is the complex Ginzburg-Landau (CGL) equation. Typical systems modelled by this equation include chemical reactions, liquid crystals, hydrodynamic convection, nonlinear optics, directional solidification or parametrically driven systems [1-6]. The (2+1) dimensional CGL equation model reads:

U_{t} + pΔU = aU − δ|U|^{2}U,  (1)

which can be derived universally in the vicinity of a homogeneous Hopf bifurcation in extended systems [7-10] with the complex variable U(x,y,t) being the order parameter at the bifurcation. The parameters p, a, and δ are in general complex, that is of the form χ = χ_{r} + iχ_{i}. This equation contains terms which account for most of the physical effects such as linear growth, nonlinear saturation, dispersion, diffusion, nonlinear renormalization of the frequency of temporal oscillations. By combining all these phenomena, CGL becomes a generalization of an integrable system (nonlinear Schroedinger equation) and of a variational equation (GL with real coefficients).

Since the study of solitons and the related issue of the construction of solution to a wide class of nonlinear equation has become one of the most exciting and extremely active areas of research investigation, the (1+1) dimensional CGL equation has been analytically solved. Using Hirota’s method [11-12], Weiss-Tabor-Carnevale (WTC) [13] and Conte and Musette method [14], it appears that the (1+1) dimensional CGL equation possesses a rich variety of solutions. These solutions include coherent structures such as pulses, fronts, periodic unbounded waves, sinks and sources [11-19] and a transition to chaos [20-24]. The main interest of scientists were restricted to the (1+1) dimensional CGL equation case, because of the difficulty of finding exact analytically high dimensional solutions. Motivated by this, and since it is often put forward that a first step toward understanding complex dynamics in nonequilibrium systems is to find solution describing a basic pattern which play crucial role in long time behavior, in the present work, we investigate analytical solutions of the (2+1) dimensional CGL. Specifically, we focus, on Eq.(1) with r⃗ = (x, y), Δ = ∂^{2}/∂x^{2}+∂^{2}/∂y^{2}. We will use in our analysis, a method which is derived from the Painlevé test for integrability and Hirota’s multilinear formalism. By the use of the method presented in [25-27] in which the process begins with Painlevé analysis

EXACT LOCALIZED SOLUTIONS OF THE (2+1)D CGL EQUATION 3

developed by WTC [28] to obtain in natural way the ansatze which are then used in Hirota’s method. The major contribution of our methodology lies in this use of one method to obtain the ansatze, which application has been intuitive. This gives an algorithmic way of obtaining the ansatze, thus making the use of Hirota’s method considerably easier. The outline of this paper is as follows. In section 2, we present the formulation of the below mentioned problem. The strategy for obtaining special solutions is developed in section 3. Finally, the section 4 is devoted to the degeneracies of the (2+1)dimensional CGL equation.

2. Painlevé Analysis of the (2+1)Dimensional CGL Equation

Let us rewrite Eq.(1) as the following system of equations:

U_{t} + p(U_{xx} + U_{yy}) = aU − δU^{2}V,  (2.a)
V_{t} + p^{∗}(V_{xx} + V_{yy}) = a^{∗}V − δ^{∗}V^{2}U,  (2.b)

where a^{∗}, p^{∗}, δ^{∗} and V are complex conjugates of a, p, δ and U respectively. In this analysis, U and V are assumed to be independent of each other.

The singular manifold expansion is introduced by assuming that U and V can be expanded as Laurent series:

U(x, y, t) = ϕ^{μ}(x, y, t) ∑_{j=0}^{∞} U_{j}(x, y, t)ϕ^{j}(x, y, t),  (3.a)
V (x, y, t) = ϕ^{η}(x, y, t) ∑_{j=0}^{∞} V_{j}(x, y, t)ϕ^{j}(x, y, t)  (3.b)

where ϕ is arbitrary and analytic function which determines a singular manifold. In order to understand the singular structure of Eqs. (2.a) and (2.b), we carry out a leading order analysis by representing U and V as follows:

U(x, y, t) ∼ U_{0}(x, y, t)ϕ^{μ}(x, y, t), V (x, y, t) ∼ V_{0}(x, y, t)ϕ^{η}(x, y, t),  (4)

where μ and η are the leading order exponents. The leading terms are:

p(U_{xx} + U_{yy}) ∼ −δU^{2}V,  (5.a)
p^{∗}(V_{xx} + V_{yy}) ∼ −δ^{∗}V^{2}U,  (5.b)

Inserting Eq.(4) into Eqs.(5.a) and (5.b) and equating the exponents of the dominant terms, we find

μ + η = −2.  (6)

To solve μ and η, we must equate the coefficients of the terms from Eqs.(5.a) and (5.b), we obtain

U_{0}V_{0} = −\frac{μ(μ − 1)p}{δ}(ϕ_{x}^{2} + ϕ_{y}^{2}),  (7.a)
U_{0}V_{0} = −\frac{η(η − 1)p^{∗}}{δ^{∗}} (ϕ_{x}^{2} + ϕ_{y}^{2}).  (7.b)

4 EMMANUEL YOMBA^{1,2} AND TIMOLÉON CRÉPIN KOFANÉ^{3}

Eqs.(7.a) and (7.b) are combined, with help of Eq.(6), the leading order are easily seen to be

μ = −1 − iα, η = −1 + iα,  (8)

where U_{0}, V_{0} and α are solutions of the equations:

α = \frac{3}{2} \frac{R}{I} ± [2 + \frac{9}{4} \frac{R^{2}}{I^{2}} ]^{\frac{1}{2}} , α ≠ 0,  (9.a)
U_{0}V_{0} = \frac{[3αI − (2 − α^{2})R](ϕ_{x}^{2} + ϕ_{y}^{2})}{|δ|^{2}},  (9.b)

and the intermediate variables R and I are defined as

R = R_{e}(pδ^{∗}) = p_{r}δ_{r} + p_{i}δ_{i},  (10.a)
I = I_{m}(pδ^{∗}) = p_{i}δ_{r} − p_{r}δ_{i}.  (10.b)

We consider here the non-degenerate case I ≠ 0. The case I = 0 will be treated at the last section.

3. Special Solutions of the (2+1) Dimensional CGL Equation

The leading-order exponent 1 + iα is not an integer: the non integrable (2+1) dimensional CGL equation has already failed the Painlevé test. As expected, its general solutions can not be expressed in terms of elementary functions. However, partial integrability remains possible. An usual assumption to find these solutions, is to require the Laurent series to truncate after the power ϕ^{0±iα} and to find a relation between (U_{1}, V_{1}) and (U,V). This is known as “WTC truncation procedure”. Thus truncated expansions for Eq.(4) are:

U = U_{0}ϕ^{−1−iα} + U_{1}ϕ^{−iα}, V = V_{0}ϕ^{−1+iα} + V_{1}ϕ^{iα}.  (11)

We can progress by observing that (U_{1}, V_{1}) satisfies the same relation as (U,V) through the equations:

ϕ^{−iα} : U_{1t} + p(U_{1xx} + U_{1yy}) = aU_{1} − δU_{1}^{2}V_{1},  (12.a)
ϕ^{iα} : V_{1t} + p^{∗}(V_{1xx} + V_{1yy}) = a^{∗}V_{1} − δ^{∗}V_{1}^{2}U_{1}.  (12.b)

Equations (12.a) and (12.b) justify the fact that the expansion can be truncated at the level ϕ^{∓iα}. So that by choosing U_{1} = V_{1} = 0, the resulting zeroth order of equation is easier to solve.

Thus the truncated expansions take the form:

U = \frac{U_{0}}{ϕ^{1+iα}} , V = \frac{V_{0}}{ϕ^{1−iα}} .  (13)

By observing that the plane waves

U(x, y, t) = R_{K}e^{i(K_{1}x+K_{2}y−ωt)},  (14)

EXACT LOCALIZED SOLUTIONS OF THE (2+1)D CGL EQUATION 5

with

|R_{K}|^{2} = \frac{p_{r}(K_{1}^{2} + K_{2}^{2}) + a_{r}}{δ_{r}} ,  (15.a)

and the dispersion relation

ω(K_{1}, K_{2}) = −p_{i}(K_{1}^{2} + K_{2}^{2}) − a_{i} + δ_{i}|R_{K}|^{2},  (15.b)

where K_{1} and K_{2} are free parameters. The general solutions of the (2+1) dimensional CGL equation can be built by the help of the trivial solutions (14).

So the expressions (13) may be expressed as:

U = \frac{G}{F^{1+iα}} e^{i(K_{1}x+K_{2}y−Ωt)}, V = \frac{H}{F^{1−iα}} e^{−i(K_{1}x+K_{2}y−Ωt)}.  (16)

If we substitute relation (16) into Eqs.(2.a) and (2.b), then we convert our full set of trilinear equations:

F \{[−iΩ − a − p(K_{1}^{2} + K_{2}^{2}) + D_{α,t} + 2ip(K_{1}D_{α,x} + K_{2}D_{α,y}) + p(D_{α,x}^{2} + D_{α,y}^{2})]\}(GF) − \frac{1}{2}(1 + iα)(2 + iα)pG[D_{0,x}^{2} + D_{0,y}^{2}](F.F) = −δHG^{2}, (17.a)

F \{[iΩ − a^{∗} − p^{∗}(K_{1}^{2} + K_{2}^{2}) + D_{α,t}^{∗} − 2ip^{∗}(K_{1}D_{α,x}^{∗} + K_{2}D_{α,y}^{∗}) + p^{∗}(D_{α,x}^{∗2} + D_{α,y}^{∗2})]\}(HF) − \frac{1}{2}(1 − iα)(2 − iα)p^{∗}H[D_{0,x}^{2} + D_{0,y}^{2}](F.F) = −δ^{∗}GH^{2}, (17.b)

where D_{α,z} is Hirota’s modified bilinear operator defined by

D_{α,z}(GF) = \{[\frac{∂}{∂z} − (1 + iα)\frac{∂}{∂z′} ] G(z)F (z′)\}_{z=z′} .  (18)

Two families of solutions are found depending of the forms of expansions chosen for F, G and H.

3.1. 2D Pulse solutions. Pulse solutions are recovered through the following assumption:

G = γϵg_{1}, H = λϵh_{1}, F = 1 + ϵ^{2}f_{2},  (19)

where γ and λ are complex constant and h_{1}, g_{1} and f_{2} are real functions. Substituting Eq.(19) into Eq.(17.a) and collecting the coefficients of different powers of ϵ we obtain a set of three equations. Solving this set of equations, we find

g_{1} = h_{1} = exp {kx + qy + ωt + ξ^{(0)}} ,  (20)
f_{2} = g_{1}^{2} = h_{1}^{2} = exp {2[kx + qy + ωt + ξ^{(0)}]} ,  (21)

where unknowns in these expressions are defined depending of the value of the coefficient p_{r} of the CGL equation as follows:

Case 1: when p_{r} ≠ 0, we found that:

K_{1}k + K_{2}q = α(k^{2} + q^{2}),  (22)

this leads to two subcases:

Subcase 1: K_{1} = αk, K_{2} = αq.(23)

6 EMMANUEL YOMBA^{1,2} AND TIMOLÉON CRÉPIN KOFANÉ^{3}

Then we obtain the following results:

λγ = 4\frac{[(2 − α^{2})R − 3αI]}{|δ|^{2}}(k^{2} + q^{2}) > 0  (24.a)
ω = 0,  (24.b)
Ω = −a_{i} + [(1 − α^{2})p_{i} + 2αp_{r}](k^{2} + q^{2})  (24.c)

and

k^{2} + q^{2} = R^{2}  (24.d)

with

R^{2} = \frac{a_{r}}{(1 − α^{2})p_{r} − 2αp_{i}} > 0.  (24.e)

All parameters of the solutions are defined for this family of 2D stationary pulse solutions.

Subcase 2:

K_{1} ≠ αk, K_{2} ≠ αq,  (25)

we find a two-parameter family of solutions. The unknowns K_{1}, K_{2}, γ, λ, ω, Ω are parametrized by k, q:

Ω = −a_{i} − p_{i}(K_{1}^{2} + K_{2}^{2}) + 2p_{r}(K_{1}k + K_{2}q) + p_{i}(k^{2} + q^{2}),  (26.a)
K_{2} = \frac{α}{q}(k^{2} + q^{2}) − \frac{k}{q} K_{1},  (26.b)

γλ and ω are given by Eq.(24.a) and Eq.(24.b) respectively. K_{1} is determined by a quadratic equation

[p_{r}(k^{2} + q^{2})]K_{1}^{2} − 2[αp_{r}(k^{2} + q^{2})k]K_{1} + {α^{2}p_{r}(k^{2} + q^{2})^{2} − q^{2}[(p_{r} − 2αp_{i})(k^{2} + q^{2}) − a_{r}]} = 0.(27)

Using Eqs.(16),(19), (20), (21) and (24.b), we obtain the 2D stationary pulse solutions as follows:

U(x, y, t) = \frac{γe^{(kx+qy+ξ^{(0)})}}{[1 + e^{2(kx+qy+ξ^{(0)})}]^{1+iα}} e^{i(K_{1}x+K_{2}y−Ωt)},  (28.a)
V (x, y, t) = \frac{λe^{(kx+qy+ξ^{(0)})}}{[1 + e^{2(kx+qy+ξ^{(0)})}]^{1−iα}} e^{−i(K_{1}x+K_{2}y−Ωt)}.  (28.b)

Case 2: p_{r} = 0

In the non-degenerate case treated here (I = p_{i}δ_{r} − p_{r}δ_{i} ≠ 0), this condition implies that p_{i}δ_{r} ≠ 0. We find a two-parameter family of solutions. The unknowns K_{1}, K_{2}, γ, λ, ω, Ω are given by the following relations:

λγ = \frac{4p_{i}}{|δ|^{2}} [(2 − α^{2})δ_{i} − 3αδ_{r}](k^{2} + q^{2}),  (29.a)
ω = 2p_{i}(kK_{1} + qK_{2}) − 2αp_{i}(k^{2} + q^{2}),  (29.b)
Ω = −a_{i} − p_{i}(K_{1}^{2} + K_{2}^{2}) + p_{i}(k^{2} + q^{2}),  (29.c)
k^{2} + q^{2} = R^{′2},  (29.d)

with

R^{′2} = \frac{−a_{r}}{2αp_{i}} > 0.  (29.e)

K_{1} and K_{2} are free parameters. By using Eqs.(16), (19), (20) and (21), the propagating pulse solutions obtained in this case are recovered as:

U(x, y, t) = \frac{γe^{(kx+qy+ωt+ξ^{(0)})}}{[1 + e^{2(kx+qy+ωt+ξ^{(0)})}]^{1+iα}} e^{i(K_{1}x+K_{2}y−Ωt)},  (30.a)
V (x, y, t) = \frac{λe^{(kx+qy+ωt+ξ^{(0)})}}{[1 + e^{2(kx+qy+ωt+ξ^{(0)})}]^{1−iα}} e^{−i(K_{1}x+K_{2}y−Ωt)}.  (30.b)

After some arrangements: the 2D stationary pulse solutions (28) can be written as:

U(x, y, t) = \frac{γ}{2} sech(kx+qy+ξ^{(0)})e^{i\{[(K_{1}−αk)x+(K_{2}−αq)y−Ωt−αξ^{(0)}]+α ln[ \frac{1}{2} sech(kx+qy+ξ^{(0)})]\}}, (31.a)

U(x, y, t) = \frac{λ}{2}sech(kx + qy + ξ^{(0)})e^{−i\{[(K_{1}−αk)x+(K_{2}−αq)y−Ωt−αξ^{(0)}]+α ln[ \frac{1}{2} sech(kx+qy+ξ^{(0)})]\}},

(31.b)

and the 2D propagating pulse solutions (30) can be written:

U(x, y, t) = \frac{γ}{2} sech(kx+qy+ωt+ξ^{(0)})e^{i\{[(K_{1}−αk)x+(K_{2}−αq)y−(Ω+αω)t−αξ^{(0)}]+α ln[ \frac{1}{2} sech(kx+qy+ωt+ξ^{(0)})]\}},

(32.a)

V (x, y, t) = \frac{λ}{2}sech(kx+qy+ωt+ξ^{(0)})e^{−i\{[(K_{1}−αk)x+(K_{2}−αq)y−(Ω+αω)t−αξ^{(0)}]+α ln[ \frac{1}{2} sech(kx+qy+ωt+ξ^{(0)})]\}}.

(32.b)

3.2. 2D Propagating hole solutions. Next, we move on to the construction of the propagating hole solutions of the (2+1) dimensional CGL equation. To obtain the propagating hole solutions, we assume the following forms for G, H and F:

G = b(1 + ϵ^{2}g_{2}), H = b^{∗}(1 + ϵ^{2}h_{2}), F = 1 + ϵ^{2}f_{2}  (33)

only f_{2} is real function, g_{2} and h_{2} are complexe functions and b is complexe constant. Substituting relation (33) into Eq.(17.a), we obtain four equations at order ϵ^{0}, ϵ^{2}, ϵ^{4} and ϵ^{6}. These equations are solved and we find:

g_{2} = κe^{2(kx+qy+ωt+ξ^{(0)})}, h_{2} = κ^{∗}e^{2(kx+qy+ωt+ξ^{(0)})}, f_{2} = e^{2(kx+qy+ωt+ξ^{(0)})}.  (34)

The unknowns b, κ, k, q, ω, Ω, K_{1} and K_{2} can be computed from

|b|^{2} = −\frac{2e_{1}}{|1 − κ|^{2}|δ|^{2}} (k^{2} + q^{2}),  (35.a)
ω = 2p_{i}(kK_{1} + qK_{2}) − 2αp_{i}(k^{2} + q^{2}) + \frac{δ_{i}}{2α}(|κ|^{2} − 1)|b|^{2},  (35.b)
Ω = −p_{i}(K_{1}^{2} + K_{2}^{2}) − a_{i} + δ_{i}|b|^{2},  (35.c)

8 EMMANUEL YOMBA^{1,2} AND TIMOLÉON CRÉPIN KOFANÉ^{3}

K_{2} = \frac{1}{p_{r}q} [αp_{r} + \frac{e_{1}δ_{r}}{2α} \frac{|κ|^{2} − 1}{|1 − κ|^{2}|δ|^{2}} ] (k^{2} + q^{2}) − \frac{k}{q} K_{1},  (35.d)
e_{1} = 2{δ_{r}[(2 − α^{2})p_{r} − 3αp_{i}] + δ_{i}[(2 − α^{2})p_{i} + 3αp_{r}]} < 0.  (35.e)

K_{1} is determined by a quadratic equation

e_{2}(k, q)K_{1}^{2} − 2e_{3}(k, q)K_{1} + e_{4}(k, q) = 0,  (36)

where

e_{2}(k, q) = p_{r}^{2}|δ|^{2}|1 − κ|^{2}(k^{2} + q^{2}),  (37.a)
e_{3}(k, q) = p_{r}|1 − κ|^{2}|δ|^{2} [αp_{r} + \frac{e_{1}δ_{r}}{2α} \frac{|κ|^{2} − 1}{|1 − κ|^{2}|δ|^{2}} ] k(k^{2} + q^{2}),  (37.b)

e_{4}(k, q) = |1−κ|^{2}|δ|^{2} [αp_{r} + \frac{e_{1}δ_{r}}{2α} \frac{|κ|^{2} − 1}{|1 − κ|^{2}|δ|^{2}} ]^{2} (k^{2}+q^{2})^{2}+q^{2}[2e_{1}δ_{r}p_{r}(k^{2}+q^{2})+p_{r}|1−κ|^{2}|δ|^{2}a_{r}].

(37.c)

k and q are free parameters. The remaining unknowns κ_{r} and κ_{i} should verify the following equations:

d_{2r}κ_{r}^{3} + d_{3r}κ_{r}^{2} + d_{4r}κ_{r} − d_{2i}κ_{i}^{3} + d_{5r}κ_{i}^{2} + d_{6i}κ_{i} − d_{2i}κ_{i}κ_{r}^{2} + d_{2r}κ_{r}κ_{i}^{2} + d_{7i}κ_{r}κ_{i} + d_{8r} = 0 (38.a)
d_{2i}κ_{r}^{3} + d_{3i}κ_{r}^{2} + d_{4i}κ_{r} + d_{2r}κ_{i}^{3} + d_{5i}κ_{i}^{2} + d_{6r}κ_{i} + d_{2r}κ_{i}κ_{r}^{2} + d_{2i}κ_{r}κ_{i}^{2} + d_{7r}κ_{r}κ_{i} + d_{8i} = 0, (38.b)

where

d_{2r} = d_{1r}|δ|^{2} − \frac{e_{1}δ_{i}}{2α} , d_{2i} = d_{1i}|δ|^{2} + \frac{e_{1}δ_{r}}{2α} ,  (39.a)
d_{3r} = (αd_{1i} − 3d_{1r})|δ|^{2} + \frac{e_{1}}{2} (δ_{r} + \frac{δ_{i}}{α}), d_{3i} = −(αd_{1r} + 3d_{1i})|δ|^{2} + \frac{e_{1}}{2} (δ_{i} − \frac{δ_{r}}{α}),  (39.b)
d_{4r} = (3d_{1r} − 2αd_{1i})|δ|^{2} − e_{1}(δ_{r} − \frac{δ_{i}}{2α}), d_{4i} = (3d_{1i} + 2αd_{1r})|δ|^{2} − e_{1}(δ_{i} + \frac{δ_{r}}{2α}),  (39.c)
d_{5r} = (αd_{1i} − d_{1r})|δ|^{2} + \frac{e_{1}}{2} (δ_{r} + \frac{δ_{i}}{α}), d_{5i} = −(αd_{1r} + d_{1i})|δ|^{2} + \frac{e_{1}}{2} (δ_{i} − \frac{δ_{r}}{α}),  (39.d)
d_{6r} = d_{1r}|δ|^{2} + \frac{e_{1}δ_{i}}{2α} , d_{6i} = −d_{1i}|δ|^{2} + \frac{e_{1}δ_{r}}{2α} ,  (39.e)
d_{7r} = −2d_{1r}|δ|^{2}, d_{7i} = 2d_{1i}|δ|^{2},  (39.f)
d_{8r} = (αd_{1i} − d_{1r})|δ|^{2} + \frac{e_{1}}{2} (δ_{r} − \frac{δ_{i}}{α}), d_{8i} = −(d_{1i} + αd_{1r})|δ|^{2} + \frac{e_{1}}{2} (δ_{i} + \frac{δ_{r}}{α}),  (39.g)
d_{1r} = p_{r} − αp_{i}, d_{1i} = p_{i} + αp_{r}.  (39.h)

Another family of 2D propagating hole solutions is obtained, which satisfies the following conditions:

κ_{r} = 1, κ_{i} = α,  (40.a)
|b|^{2} = −\frac{2e_{1}}{α^{2}|δ|^{2}} (k^{2} + q^{2}),  (40.b)
ω = 2p_{i}(kK_{1} + qK_{2}) − 2(\frac{2}{α}p_{i} + 3p_{r})(k^{2} + q^{2}),  (40.c)
Ω = −a_{i} − p_{i}(K_{1}^{2} + K_{2}^{2}) + δ_{i}|b|^{2},  (40.d)

K_{2} = \frac{1}{qp_{r}} [\frac{2}{α}p_{r} − 3p_{i}] (k^{2} + q^{2}) − \frac{k}{q} K_{1},  (40.e)

Here K_{1} satisfies the following quadratic equation

[α^{2}p_{r}^{2}(k^{2} + q^{2})]K_{1}^{2} − 2[α^{2}p_{r}(\frac{2}{α}p_{r} − 3p_{i})k(k^{2} + q^{2})]K_{1} + \{α^{2}(\frac{2}{α}p_{r} − 3p_{i})^{2}(k^{2} + q^{2})^{2} + q^{2}[4p_{r}[(2 − α^{2})p_{r} − 3αp_{i}](k^{2} + q^{2}) + α^{2}p_{r}a_{r}]} = 0.(41)

k and q are free parameters. Using Eqs.(16), (33) and (34), the 2D propagating hole solutions are given as follows

U(x, y, t) = \frac{b[1 + κe^{2(kx+qy+ωt+ξ^{(0)})}]}{[1 + e^{2(kx+qy+ωt+ξ^{(0)})}]^{1+iα}} e^{i(K_{1}x+K_{2}y−Ωt)}  (42.a)
V (x, y, t) = \frac{b^{∗}[1 + κ^{∗}e^{2(kx+qy+ωt+ξ^{(0)})}]}{[1 + e^{2(kx+qy+ωt+ξ^{(0)})}]^{1−iα}}e^{−i(K_{1}x+K_{2}y−Ωt)}  (42.b)

After some arrangements, the expressions (42.a) and (42.b) can be written in this final form

U(x, y, t) = \frac{bκe^{(kx+qy+ωt+ξ^{(0)})} + be^{−(kx+qy+ωt+ξ^{(0)})}}{[e^{(kx+qy+ωt+ξ^{(0)})} + e^{−(kx+qy+ωt+ξ^{(0)})}]^{1+iα}} e^{i\{[K′_{1}x+K′_{2}y−Ω′t−αξ^{(0)}]+α ln[ \frac{1}{2} sech(kx+qy+ωt+ξ^{(0)})]\}},

(43.a)

V (x, y, t) = \frac{b^{∗}κ^{∗}e^{(kx+qy+ωt+ξ^{(0)})} + b^{∗}e^{−(kx+qy+ωt+ξ^{(0)})}}{[e^{−i\{(kx+qy+ωt+ξ^{(0)})} + e^{−(kx+qy+ωt+ξ^{(0)})}]^{1−iα}} e^{−i\{[K′_{1}x+K′_{2}y−Ω′t−αξ^{(0)}]+α ln[ \frac{1}{2} sech(kx+qy+ωt+ξ^{(0)})]\}},

(43.b)

with

K′_{1} = K_{1} − αk, K′_{2} = K_{2} − αq, Ω′ = Ω + αω.  (44)

It can then be easily seen that, by setting, ζ = kx+qy +ωt+ξ^{(0)} we have the following results:

lim_{ζ⟶+∞} U(ζ) = b(κ_{r} + iκ_{i})e^{i[k_{1}x+q_{1}y−Ω′_{1}t−2αξ^{(0)}]},  (45.a)
lim_{ζ⟶+∞} V (ζ) = b^{∗}(κ_{r} − iκ_{i})e^{−i[k_{1}x+q_{1}y−Ω′_{1}t−2αξ^{(0)}]},  (45.b)
lim_{ζ⟶−∞} U(ζ) = be^{i[k_{2}x+q_{2}y−Ω′_{2}t]},  (45.c)
lim_{ζ⟶−∞} V (ζ) = b^{∗}e^{−i[k_{2}x+q_{2}y−Ω′_{2}t]},  (45.d)

with

k_{1} = K′_{1} − αk, q_{1} = K′_{2} − αq, Ω′_{1} = Ω′ + αω,  (46.a)
k_{2} = K′_{1} + αk, q_{2} = K′_{2} + αq, Ω′_{2} = Ω′ − αω.  (46.b)

Each solution connect two waves with different wave vectors. The solution behaves like a homogeneous wave of amplitude b(κ_{r} + iκ_{i}) and wave vector k⃗ = (k_{1}, q_{1}) as ζ ⟶ +∞; for ζ ⟶ −∞, it corresponds to a wave of amplitude b and wave vector k⃗′ = (k_{2}, q_{2}). If we assume that b is real, the parameter bκ can then be written as bκ = b|κ|exp(iθ)

10 EMMANUEL YOMBA^{1,2} AND TIMOLÉON CRÉPIN KOFANÉ^{3}

where

cos(θ) = \frac{κ_{r}}{\sqrt{κ_{r}^{2} + κ_{i}^{2}}} ,  (47.a)
sin(θ) = \frac{κ_{i}}{\sqrt{κ_{r}^{2} + κ_{i}^{2}}} .  (47.b)

Eqs.(47.a) and (47.b) represent the phase shift θ ≡ arg(bκ, b) between the two asymptotic waves.

3.3. 2D Stationary dark solutions. The 2D stationary dark solutions are recovered under these special conditions:

ω = 0, κ_{r} = −1, κ_{i} = 0.  (48)

Then the general solutions which take into account Eqs.(16), (33) and (34) have the form:

U(x, y, t) = \frac{b[1 − e^{2(kx+qy+ξ^{(0)})}]}{[1 + e^{2(kx+qy+ξ^{(0)})}]^{1+iα}} e^{i(K_{1}x+K_{2}y−Ωt)}  (49.a)
V (x, y, t) = \frac{b^{∗}[1 − e^{2(kx+qy+ξ^{(0)})}]}{[1 + e^{2(kx+qy+ξ^{(0)})}]^{1−iα}} e^{−i(K_{1}x+K_{2}y−Ωt)}.  (49.b)

The unknowns in these solutions are determined depending on two cases:

Case 1: K_{1} = αk, K_{2} = αq.(50)

A discrete set of solutions is found with:

|b|^{2} = \frac{3αI − (2 − α^{2})R}{|δ|^{2}}(k^{2} + q^{2}),  (51.a)
Ω = −a_{i} − (2p_{i} + 3αp_{r})(k^{2} + q^{2}),  (51.b)
k^{2} + q^{2} = R^{′′2},  (51.c)
R^{′′2} = −\frac{a_{r}}{2p_{r} − 3αp_{i}} > 0.  (51.d)

Case 2:

K_{1} ≠ αk, K_{2} ≠ αq.  (52)

We find a two-parameter family of solution defined by

K_{2} = \frac{α}{q}(k^{2} + q^{2}) − \frac{k}{q} K_{1},  (53.a)
Ω = −a_{i} − p_{i}(K_{1}^{2} + K_{2}^{2}) − [(2 − α^{2})p_{i} + 3αp_{r}](k^{2} + q^{2}).  (53.b)

Here K_{1} is determined by a quadratic equation

[p_{r}(k^{2}+q^{2})]K_{1}^{2}−2[αp_{r}k(k^{2}+q^{2})]K_{1}+{α^{2}p_{r}(k^{2} + q^{2})^{2} + q^{2} [[(2 − α^{2})p_{r} − 3αp_{i}](k^{2} + q^{2}) + a_{r}]} = 0.

(54)

|b|^{2} still satisfies Eq.(51.a). k and q are free parameters.

EXACT LOCALIZED SOLUTIONS OF THE (2+1)D CGL EQUATION 11

4. Degeneracies of (2+1) Dimensional CGL Equation

In the limiting case of I_{m}(\frac{2p}{δ}) = 0, the 2D CGL equation degenerates. This puts the following constraints on the coefficients of the equation.

I = p_{i}δ_{r} − p_{r}δ_{i} = 0.  (55)

Leading-order analysis then leads to:

U_{0}V_{0} = \frac{2R}{|δ|^{2}} (ϕ_{x}^{2} + ϕ_{y}^{2}).  (56)

As before, looking for special solutions of the (2+1) dimensional CGL equation with the restrictions I = 0, we obtain that its systematic resolution leads to 2D pulses, 2D propagating hole solutions and 2D stationary dark solutions. The method is identical to that presented in the previous section.

4.1. 2D Pulse solutions. The pulse solutions read

U(x, y, t) = \frac{γ}{2} sech(kx + qy + ωt + ξ^{(0)})e^{i[K_{1}x+K_{2}y−Ωt]},  (57.a)
V (x, y, t) = \frac{λ}{2}sech(kx + qy + ωt + ξ^{(0)})e^{−i[(K_{1}x+(K_{2}y−Ωt]},  (57.b)

and necessarily respect p_{r}(K_{1}k + K_{2}q) = 0. We distinguish two cases:

Case 1: P_{r} = 0 The parameters are determined by

K_{2} = −\frac{k}{q} K_{1},  (58.a)
λγ = \frac{8R}{|δ|^{2}} (k^{2} + q^{2}), R > 0,  (58.b)
ω = 0,  (58.c)
Ω = −a_{i} + \frac{p_{i}}{q^{2}} (k^{2} + q^{2})(q^{2} − K_{1}^{2}),  (58.d)
K_{1}^{2} = \frac{p_{r}q^{2}(k^{2} + q^{2}) − a_{r}q^{2}}{p_{r}(k^{2} + q^{2})} > 0.  (58.e)

Here k and q are chosen arbitrarily.

Such restrictions (I=0) include the case of real Ginzburg-Landau equation (2D RGL:p_{i} = δ_{i} = a_{i} = 0):

U_{t} + p_{r}(U_{xx} + U_{yy}) = a_{r}U − δ_{r}|U|^{2}U.  (59)

All parameters are parametrized by k and q.

K_{2} = −\frac{k}{q} K_{1},  (60.a)
λγ = \frac{8p_{r}}{δ_{r}} (k^{2} + q^{2}), p_{r}δ_{r} > 0,  (60.b)
ω = 0,  (60.c)
Ω = 0,  (60.d)

K_{1}^{2} = \frac{p_{r}q^{2}(k^{2} + q^{2}) − a_{r}q^{2}}{p_{r}(k^{2} + q^{2})} , .  (60.e)

Case 2: p_{r} = 0. A four-parameter family of pulses is found for the 2D Schroedinger equation (p_{r} = δ_{r} = a_{r}):

−iU_{t} + p_{i}(U_{xx} + U_{yy}) = a_{i}U − δ_{i}|U|^{2}U.  (61)

The free parameters are chosen to be k, q, K_{1} and K_{2}:

λγ = \frac{8p_{i}}{δ_{i}} (k^{2} + q^{2}), p_{i}δ_{i} > 0,  (62.a)
ω = 2p_{i}(kK_{1} + qK_{2}),  (62.b)
Ω = −a_{i} + p_{i}[k^{2} + q^{2} − (K_{1}^{2} + K_{2}^{2})].  (62.c)

4.2. 2D Propagating hole solutions. Propagating hole solutions are obtained:

U(x, y, t) = \frac{bκe^{(kx+qy+ωt+ξ^{(0)})} + be^{−(kx+qy+ωt+ξ^{(0)})}}{[e^{(kx+qy+ωt+ξ^{(0)})} + e^{−(kx+qy+ωt+ξ^{(0)})}]} e^{i(K_{1}x+K_{2}y−Ωt)},  (63.a)
V (x, y, t) = \frac{b^{∗}κ^{∗}e^{(kx+qy+ωt+ξ^{(0)})} + b^{∗}e^{−(kx+qy+ωt+ξ^{(0)})}}{[e^{(kx+qy+ωt+ξ^{(0)})} + e^{−(kx+qy+ωt+ξ^{(0)})}]} e^{−i(K_{1}x+K_{2}y−Ωt)}.  (63.b)

The unknowns are determined as:

|b|^{2} = −\frac{8R}{|δ|^{2}|1 − κ|^{2}} (k^{2} + q^{2}), κ ≠ 1, R < 0  (64.a)

ω = \frac{8R(κ_{r}−1)\{δ_{r}[(κ_{r}−1)p_{r}−κ_{i}p_{i}]+δ_{i}[(κ_{r}−1)p_{i}+κ_{i}p_{r}]\}−2|δ|^{2}|1−κ|^{2}\{[(κ_{r}−1)p_{r}−κ_{i}p_{i}]^{2}+[(κ_{r}−1)p_{i}+κ_{i}p_{r}]^{2}\}}{|δ|^{2}|1−κ|^{2}p_{r}[(κ_{r}−1)^{2}+κ_{i}^{2}]}(k^{2} + q^{2}), (64.b)

K_{2} = \frac{e}{q}(k^{2} + q^{2}) − \frac{k}{q} K_{1},  (64.c)
Ω = −a_{i} − p_{i}(K_{1}^{2} + K_{2}^{2}) − \frac{8p_{i}}{|1 − κ|^{2}} (k^{2} + q^{2}),  (64.d)
e = \frac{4(κ_{r} − 1)R[δ_{r}(κ_{r} − 1) − κ_{i}δ_{i}] − |δ|^{2}|1 − κ|^{2}p_{i}[(κ_{r} − 1)^{2} + κ_{i}^{2}]}{|δ|^{2}|1 − κ|^{2}p_{r}[(κ_{r} − 1)^{2} + κ_{i}^{2}]}.  (64.e)

Here K_{1} satisfies the following quadratic equation: [|1 − κ|^{2}p_{r}(k^{2} + q^{2})]K_{1}^{2} − 2[|1 − κ|^{2}ep_{r}k(k^{2} + q^{2})]K_{1} + {e^{2}(k^{2} + q^{2})^{2}|1 − κ|^{2}p_{r} + q^{2}[8p_{r}(k^{2} +

q^{2}) + |1 − κ|^{2}a_{r}]} = 0.(65)

κ_{r} and κ_{i} should lie on the following circle:

κ_{r}^{2} + κ_{i}^{2} = 1,  (66)

and k and q are free parameters.

EXACT LOCALIZED SOLUTIONS OF THE (2+1)D CGL EQUATION 13

The case includes the 2D nonlinear Schroedinger equation (Eq.(61)) when p_{r} = δ_{r} = a_{r} = 0. All coefficients can be parametrized by k, q, K_{1} and K_{2}.

|b|^{2} = −\frac{8p_{i}}{δ_{i}|1 − κ|^{2}} (k^{2} + q^{2}), p_{i}δ_{i} < 0, κ ≠ 1,  (67.a)
ω = 2p_{i} [kK_{1} + qK_{2} + \frac{κ_{i}}{κ_{r} − 1}(k^{2} + q^{2})] ,  (67.b)
Ω = −a_{i} − p_{i}(K_{1}^{2} + K_{2}^{2}) − \frac{8p_{i}}{|1 − κ|^{2}} (k^{2} + q^{2}),  (67.c)
κ_{r}^{2} + κ_{i}^{2} = 1.  (67.d)

This case also includes the 2D RGL equation (Eq.(59)) when p_{i} = δ_{i} = a_{i} = 0. A twoparameter family of 2D propagating hole solutions is found. We use k and q as free parameters:

|b|^{2} = −\frac{8p_{r}}{δ_{r}|1 − κ|^{2}} (k^{2} + q^{2}), p_{r}δ_{r} < 0,  (68.a)
ω = −2p_{r} [k^{2} + q^{2} − \frac{κ_{r} − 1}{κ_{i}}(kK_{1} + qK_{2})] ,  (68.b)
Ω = 0,  (68.c)
K_{2} = \frac{e′}{q} (k^{2} + q^{2}) − \frac{k}{q} K_{1},  (68.d)
e′ = \frac{4(κ_{r} − 1)}{[(κ_{r} − 1)^{2} + κ_{i}^{2}]|1 − κ|^{2}} ,  (68.e)
κ_{r}^{2} + κ_{i}^{2} = 1.  (68.f)

Where K_{1} satisfies the following equation

[|1 − κ|^{2}p_{r}(k^{2} + q^{2})]K_{1}^{2} − 2[|1 − κ|^{2}e′p_{r}k(k^{2} + q^{2})]K_{1} + {e^{′2}(k^{2} + q^{2})^{2}|1 − κ|^{2}p_{r} + q^{2}[8p_{r}(k^{2} + q^{2}) + |1 − κ|^{2}a_{r}]} = 0.(69)

4.3. 2D stationary dark solutions. These solutions read

U(x, y, t) = \frac{b[1 − e^{2(kx+qy+ξ^{(0)})}]}{[1 + e^{2(kx+qy+ξ^{(0)})}]}e^{i(K_{1}x+K_{2}y−Ωt)},  (70.a)
V (x, y, t) = \frac{b^{∗}[1 − e^{2(kx+qy+ξ^{(0)})}]}{[1 + e^{2(kx+qy+ξ^{(0)})}]}e^{−i(K_{1}x+K_{2}y−Ωt)}.  (70.b)

A two-parameter of such family of solutions is found

|b|^{2} = −\frac{2R}{|δ|^{2}} (k^{2} + q^{2}), R < 0,  (71.a)
Ω = −a_{i} − p_{i}[K_{1}^{2} + K_{2}^{2} + 2(k^{2} + q^{2})],  (71.b)
K_{2} = −\frac{k}{q} K_{1},  (71.c)
K_{1}^{2} = −\frac{[2p_{r}(k^{2} + q^{2}) + a_{r}]q^{2}}{p_{r}(k^{2} + q^{2})} > 0.  (71.d)

14 EMMANUEL YOMBA^{1,2} AND TIMOLÉON CRÉPIN KOFANÉ^{3}

Here k and q are free parameters. For this kind of solutions, we find that it includes the case of 2D nonlinear Schroedinger equation (Eq.(61)) when p_{r} = δ_{r} = a_{r} = 0. All the unknowns are parametrized by k, q and K_{1}.

|b|^{2} = −\frac{2p_{i}}{δ_{i}} (k^{2} + q^{2}), p_{i}δ_{i} < 0,  (72.a)
K_{2} = −\frac{k}{q} K_{1},  (72.b)
Ω = −p_{i}[K_{1}^{2} + K_{2}^{2} + 2(k^{2} + q^{2})].  (72.c)

These solutions also include the 2D RGL equation (Eq.(59)) when p_{i} = δ_{i} = a_{i} = 0.

|b|^{2} = −\frac{2p_{r}}{δ_{r}} (k^{2} + q^{2}), p_{r}δ_{r} < 0,  (73.a)
K_{2} = −\frac{k}{q} K_{1},  (73.b)
Ω = 0,  (73.c)
K_{1}^{2} = −\frac{2p_{r}(k^{2} + q^{2}) + a_{r}q^{2}}{p_{r}(k^{2} + q^{2})} > 0.  (73.d)

Here k and q are free parameters.

5. conclusion

In summary, for the (2+1) dimensional CGL equation, there are four types of 2D localized solutions namely propagating pulses, stationary pulses, propagating hole solutions and stationary dark solutions obtaining by a combined methodology which derived from Painlevé test for integrability and Hirota’s technique of multilinearization. This method is a practical one of family of complex Ginzburg-Landau equation. The approach is convenient and effective since it permits to build an ansatz in natural way in the two dimensional case. We will try in the forthcoming study to use the same method to solve the (2+1) dimensional complex cubic-quintic Ginzburg-Landau equation. By studying the degeneracies of the (2+1) dimensional cubic CGL equation and its solutions, it appears that this equation is a generalization of an integrable system (nonlinear Schroedinger equation) and of a variational equation (Real Ginzburg-Landau equation), thus exhibits rich variety of phenomena.

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