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

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

Abstract. The (2+1)-dimensional complex cubic-quintic Ginzburg-Landau equation is analytically studied. A method derived from the Painlevé test for integrability and Hirota’s multilinear formalism is used. We have found that 2D propagating pulse, 2D stationary pulse, 2D stationary hole and 2D propagating hole solutions are the exact localized solutions that exist for this equation. The existence of these special solutions shows that the coefficients are constrained by certain relations. The degeneracies of this (2+1)-dimensional Cubic-quintic Ginzburg-Landau equation are examined as well as their solutions. By analyzing these degeneracies, it appears that this equation is a generalization of the nonlinear cubic-quintic Schrödinger equation and the real cubic-quintic Ginzburg-Landau equation.

KEYWORDS

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

There is emergence of stable spatio-temporal patterns when many spatially extended systems are driven sufficiently far from their equilibrium states. In the weakly nonlinear regime, these patterns can be described by the complex cubic-quintic Ginzburg-Landau (CCQGL) equation. These include a variety of nonequilibrium phenomena, such as the dynamics of certain chemical reactions [1], binary fluid convection [2-5], parametric surface waves in fluid [6,7], Taylor-Couette flow between rotating cylinders [8-10], soliton propagation in optical fiber systems with linear and nonlinear gain and spectral filtering [11-16], pulse generation in fiber lasers with additive pulse mode-locking or nonlinear polarization rotation [17-22].

The CCQGL equation was considered in a number of publications using numerical simulations [23-28]. Perturbative analysis of the solitons of the CCQGL equation in the nonlinear Schrödinger equation (NLSE) limit was used in [23,24,29,30]. Analytic approach, which allows to obtain solutions for the (1+1)-dimensional (D) CCQGL equation was developed in [31,32]. These exact solutions include fronts, pulses, sources or sinks, seem to play important role in the selection of a final steady pattern at long times and in the evolution of periodic, quasi periodic or disordered patterns [33-35].

Patterns described by (2+1)-dimensional CCQGL equation are of special interest. In particular steady spiral solutions two dimensional (2D) objects with internal vorticity were studied in recent work [36], other novel types of nonstationary and stationary asymmetric solitons were found in [37]. 2D localized particle like solutions were found in [38]. Recently, the properties of spiral waves and other localized solutions were found [36]. New dynamical behavior that appears in the CCQGL equation as a result of the interplay between the quantized circular phase flow and bending of the spatial localized states through the spatial oscillations of their tails were described in [39]. Thus, it is obvious that numerical approach for the (2+1)- dimensional CCQGL equation has been intensively studied, but to our knowledge, much less is known about analytic solutions of this equation. 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 plays crucial role in long time behavior, in the present work, we investigate analytical solutions of the (2+1)- dimensional CCQGL equation. The (2+1)-dimensional CCQGL equation reads:

U_{t} + pΔU = aU − q|U|^{2}U − γ|U|^{4}U,  (1)

where p, q and γ are in general complex that of χ = χ_{r} + iχ_{i}, U(x,y,t) is the complex wave amplitude. In this paper, we study Eq.(1) by using a method which is derived from the Painlevé test for integrability in the sens of Weiss-Tabor-Carnevale (WTC) [40] and Hirota multi-linear method [41]. We find analytic solutions of Eq.(1) by using the method presented in [42-45] in which the process begins with painlevé analysis developed by WTC to obtain in

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

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, whose application has been intuitive. This gives an algorithmic way of obtaining the ansatze, thus making the use of Hirota’s metod considerably easier. The rest of paper is organized as follows. The Painlevé analysis of the (2+1)-D CCQGL equation is performed in section 2. Exact solutions of the (2+1)-D CCQGL equation are obtained in section 3. The degeneracies of (2+1)-D CCQGL equation are studied in section 4. Finally, we summarize with our conlusions in section 5.

2. PainlevÉ Analysis of the (2+1)-D CCQGL Equation

We follow the WTC procedure [40] to carry out the Painlevé analysis, according to which a given partial differential equation (PDE) is integrable, if its solutions are single valued about the movable singularity manifold. This method requires the following necessary conditions to prove the integrability:

(i) determination of leading orders of Laurent series which should be integers, (ii) identification of powers at which the arbitrary functions can enter into the Laurent series

called resonances,

(iii) verification of the existence of sufficient number of arbitrary functions at the resonance values. In order to carry out the Painlevé analysis, let us assume U^{∗} = V. Eq.(1) becomes:

U_{t} + p(U_{xx} + U_{yy}) = aU − qU^{2}V − γU^{3}V^{2},  (2.a)
V_{t} + p^{∗}(V_{xx} + V_{yy}) = aV − q^{∗}V^{2}U − γ^{∗}U^{2}V^{3},  (2.b)

where * stands for complex conjugation . Leading-order analysis for Eqs.(2a) and (2.b) is achieved by balancing the highest-order derivative with the strongest nonlinearity. The fields U and V are now formally considered as independent variables and obey.

p(U_{xx} + U_{yy}) ∼ −γU^{3}V^{2},  (3.a)
p^{∗}(V_{xx} + V_{yy}) ∼ −γ^{∗}V^{3}U^{2},  (3.b)

To determine the leding-order behavior, we expand

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

where ϕ is arbitrary and analytic function which determines a singular manifold. Here r and s are leading-order exponents. Substituting Eq.(4) into Eqs.(3.a) and (3.b) and equating the dominant terms, we obtain

r = −\frac{1}{2} − iα, s = −\frac{1}{2} + iα,  (5)

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

U_{0}^{2}V_{0}^{2} = \frac{1}{|γ|^{2}} \{2α(p_{i}γ_{r} − p_{r}γ_{i}) − (\frac{3}{4} − α^{2}) (p_{r}γ_{r} + p_{i}γ_{i})\} (ϕ_{x}^{2} + ϕ_{y}^{2}),  (6)

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

α = \frac{p_{r}γ_{r} + p_{i}γ_{i}}{p_{i}γ_{r} − p_{r}γ_{i}} ± \{\frac{3}{4} + (\frac{p_{r}γ_{r} + p_{i}γ_{i}}{p_{i}γ_{r} − p_{r}γ_{i}} )^{2}\}^{\frac{1}{2}} , α ≠ 0.  (7)

We consider first the non degenerate case p_{i}γ_{r} − p_{r}γ_{i} ≠ 0. The leading-order exponents r = −\frac{1}{2} − iα and s = −\frac{1}{2} + iα are not the integers. The non integrable (2+1)-D CCQGL equation has already failed the Painlevé test at the first step of these three steps. However, this does not forbid the existence of particular solutions, provided that they respect the singularity structure derived from the leading-order analysis.

The general solution will be recovered under the Puiseux series as follows.

U(x, y, t) = ∑_{j=0}^{∞} U_{j}(x, y, t)ϕ^{\frac{j}{2} − \frac{1}{2} −iα}(x, y, t),  (8.a),
V (x, y, t) = ∑_{j=0}^{∞} V_{j}(x, y, t)ϕ^{\frac{j}{2} − \frac{1}{2} +iα}(x, y, t).  (8.b).

An usual assumption to find these solutions is to require the Puiseux 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 Eqs.(8.a) and (8.b) are:

U = U_{0}ϕ^{− \frac{1}{2} −iα} + U_{1}ϕ^{−iα},  (9.a),
V = V_{0}ϕ^{− \frac{1}{2} +iα} + V_{1}ϕ^{iα}.  (9.b),

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

ϕ^{−iα} : U_{1t} + p(U_{1xx} + U_{1yy}) = aU_{1} − qU_{1}^{2}V_{1} − γU_{1}^{3}V_{1}^{2},  (10.a)
ϕ^{iα} : V_{1t} + p^{∗}(V_{1xx} + V_{1yy}) = aV_{1} − q^{∗}V_{1}^{2}U_{1} − γ^{∗}U_{1}^{2}V_{1}^{3}.  (10.b)

These Eqs. (10.a) and (10.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^{\frac{1}{2} +iα}} , V = \frac{V_{0}}{ϕ^{\frac{1}{2} −iα}} .  (11)

By observing that the plane waves

U(x, y, t) = R_{K}e^{i(K_{1}x+K_{2}y−ωt)},  (12)

with

γ_{r}|R_{K}|^{4} + q_{r}|R_{K}|^{2} − {p_{r}(K_{1}^{2} + K_{2}^{2}) + a} = 0,  (13.a)

and the dispersion relation

ω(K_{1}, K_{2}) = −p_{i}(K_{1}^{2} + K_{2}^{2}) + q_{i}|R_{K}|^{2} + γ_{i}|R_{K}|^{4},  (13.b)

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

where K_{1} and K_{2} are free parameters. The general solutions of the (2+1)-D CCQGL equation can be built by the help of the trivial solutions (12).

So the expression (12) may be expressed as:

U = \frac{G}{F^{\frac{1}{2} +iα}} e^{i(K_{1}x+K_{2}y−Ωt)}, V = \frac{H}{F^{\frac{1}{2} −iα}} e^{−i(K_{1}x+K_{2}y−Ωt)}.  (14)

Expression (14) is the ansatz which should be used in Hirota method. Next we consider the Hirota bilinear transformation (14). Where G(x,y,t) and H(x,y,t) are the complex functions and F(x,y,t) is a real function. Using Eq.(14) into Eqs.(2,a) and (2.b) we obtain the following multilinearized forms:

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} + iα)(\frac{3}{4} + iα)pG(∧_{x} + ∧_{y})(F.F) = −qHG^{2}F − γG^{3}H^{2}, (15.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} − iα)(\frac{3}{4} − iα)p^{∗}H(∧_{x} + ∧_{y})(F.F) = −q^{∗}GH^{2}F − γ^{∗}G^{2}H^{3}, (15.b)

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

D_{α,z}(GF) = \{[\frac{∂}{∂z} − (\frac{1}{2} + iα)\frac{∂}{∂z′} ] G(z)F (z′)\}_{z=z′} ,  (16.a)

and

∧_{z}(F.F) = F F_{zz} − F_{z}^{2}.  (16.b)

3. Exact solutions of the (2+1)-dimensional CCQGL equation

We now turn to construct some special solutions for the Eqs.(15.a) and (15.b). Different kinds of solutions are found depending of the forms of expansions chosen for F, G and H.

3.1. 2D Pulse solutions . We begin with the pulse like solution. We assume the following forms for G, H, and F:

G = μϵ^{\frac{1}{2}} g_{\frac{1}{2}} , H = ηϵ^{\frac{1}{2}} h_{\frac{1}{2}} , F = 1 + ϵf_{1} + ϵ^{2}f_{2},  (17)

where μ and η are complex constants, and g_{\frac{1}{2}} , h_{\frac{1}{2}} , f_{1} and f_{2} are real valued functions. Substituting Eq.(17) into Eq.(15.a) and collecting the coefficients of different powers of ϵ (ϵ^{\frac{1}{2}} , ϵ^{\frac{3}{2}} , ϵ^{\frac{5}{2}} , ϵ^{\frac{7}{2}} , ϵ^{\frac{9}{2}} ) we obtain a set of five equations. Solving this set of equations, we find

g_{\frac{1}{2}} = h_{\frac{1}{2}} = exp\{\frac{1}{2}(kx + qy + ωt + ξ^{(0)})\},  (18.a)
f_{1} = d exp{(kx + qy + ωt + ξ^{(0)})},  (18.b)
f_{2} = exp{2(kx + qy + ωt + ξ^{(0)})},  (18.c)

where unknowns in these expressions obey different relationships according to the value of the coefficient p_{r}:

Case 1: p_{r} ≠ 0

This first case includes two subcases:

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

Subscase 1: K_{1} ≠ αk, K_{2} ≠ αl. A two-parameter family of solutions is found. K_{1} is determined by a quadratic equation

[p_{r}(k^{2} + l^{2})]K_{1}^{2} − 2[αp_{r}k(k^{2} + l^{2})]K_{1} + \{α^{2}p_{r}(k^{2} + l^{2})^{2} + l^{2}[a − (\frac{1}{4}p_{r} − αp_{i})(k^{2} + l^{2})]\} = 0. (19)

The unknowns μη, K_{2}, ω, and Ω are obtained as follows

μ^{2}η^{2} = e_{1}(k^{2} + l^{2}),  (20.a)
ω = 0,  (20.b)
K_{2} = \frac{α}{l}(k^{2} + l^{2}) − \frac{k}{l} K_{1},  (20.c)
Ω = −p_{i}(K_{1}^{2} + K_{2}^{2}) + (\frac{1}{4}p_{i} + αp_{r})(k^{2} + l^{2}),  (20.d)
k^{2} + l^{2} = R^{2},  (20.e)
R^{2} = \frac{e_{1}q_{r}^{2}}{d^{2} [(1 − 2α^{2})p_{r} − 3αp_{i}]^{2}} > 0, ,  (20.f)

e_{1} = \frac{(4 − d^{2}) \{γ_{r} [(\frac{3}{4} − α^{2}) p_{r} − 2αp_{i}] + γ_{i} [(\frac{3}{4} − α^{2}) p_{i} + 2αp_{r}]\}}{|γ|^{2}} > 0, d ≠ \{2, −2\}. (20.g)

Here d is free parameter.

This solution requires the following constraint to be satisfied

q_{r}[(1 − 2α^{2})p_{i} + 3αp_{r}] − q_{i}[(1 − 2α^{2})p_{r} − 3αp_{i}] = 0.  (21)

Subcase 2:

K_{1} = αk, K_{2} = αl.  (22)

The unknowns μη, ω, k, l and Ω are defined by:

μ^{2}η^{2} = e_{1}(k^{2} + l^{2}),  (23.a)
ω = 0,  (23.b)
k^{2} + l^{2} = R^{2},  (23.c)
Ω = [(\frac{1}{4} − α^{2}) p_{i} + αp_{r}] (k^{2} + l^{2}),  (23.d)

R^{2}, e_{1} are given by Eqs.(20.f) and (20.g) respectively.

This solution is recovered under the following constraints:

q_{r}[(1 − 2α^{2})p_{i} + 3αp_{r}] − q_{i}[(1 − 2α^{2})p_{r} − 3αp_{i}] = 0,  (24.a)
a = [(\frac{1}{4} − α^{2}) p_{r} − αp_{i}] (k^{2} + l^{2}).  (24.b)

Here d is free parameter. This first case in its two subcases defines the stationary pulse solutions which reads:

U(x, y, t) = \frac{μe^{\frac{1}{2} (kx+ly+ξ^{(0)})}}{[1 + de^{(kx+ly+ξ^{(0)})} + e^{2(kx+ly+ξ^{(0)})}]^{\frac{1}{2} +iα}} e^{i(K_{1}x+K_{2}y−Ωt)},  (25.a)

V (x, y, t) = \frac{ηe^{\frac{1}{2} (kx+ly+ξ^{(0)})}}{[1 + de^{(kx+ly+ξ^{(0)})} + e^{2(kx+ly+ξ^{(0)})}]^{\frac{1}{2} −iα}} e^{−i(K_{1}x+K_{2}y−Ωt)},  (25.b)

Case 2: p_{r} = 0.

We find a three parameter-family of solution, parametrized by d, K_{1} and K_{2} where:

μ^{2}η^{2} = e′_{1}(k^{2} + l^{2}),  (26.a)
ω = 2p_{i}[kK_{1} + lK_{2} − α(k^{2} + l^{2})],  (26.b)
Ω = −p_{i}(K_{1}^{2} + K_{2}^{2}) + \frac{1}{4}p_{i}(k^{2} + l^{2}),  (26.c)
k^{2} + l^{2} = R^{′2},  (26.d)
R^{′2} = \frac{e′_{1}q_{r}^{2}}{9α^{2}p_{i}^{2}d^{2}} ,  (26.e)
e′_{1} = \frac{(4 − d^{2})p_{i} [−2αγ_{r} + (\frac{3}{4} − α^{2}) γ_{i}]}{|γ|^{2}} > 0, d ≠ \{2, −2\}.  (26.f)

Then α satisfies in this case the following equation

α^{2} − 2\frac{γ_{i}}{γ_{r}} α − \frac{3}{4} = 0.  (27)

This solution imposes the following constraints:

(1 − 2α^{2})q_{r} + 3αq_{i} = 0,  (28.a)
a = −αp_{i}(k^{2} + l^{2}).  (28.b)

The propagating pulse solutions read:

U(x, y, t) = \frac{μe^{\frac{1}{2} (kx+ly+ωt+ξ^{(0)})}}{[1 + de^{(kx+ly+ωt+ξ^{(0)})} + e^{2(kx+ly+ωt+ξ^{(0)})}]^{\frac{1}{2} +iα}} e^{i(K_{1}x+K_{2}y−Ωt)},  (29.a)

V (x, y, t) = \frac{ηe^{\frac{1}{2} (kx+ly+ωt+ξ^{(0)})}}{[1 + de^{(kx+ly+ωt+ξ^{(0)})} + e^{2(kx+ly+ωt+ξ^{(0)})}]^{\frac{1}{2} −iα}} e^{−i(K_{1}x+K_{2}y−Ωt)},  (29.b)

After some arrangements, the stationary pulse solutions (25.a) and (25.b) leads to:

U(x, y, t) = \frac{μ}{[2 cosh(kx + ly + ξ^{(0)}) + d]^{\frac{1}{2}}} e^{i\{(K_{1}−αk)x+(K_{2}−αl)y−Ωt−αξ^{(0)}−α ln[2 cosh(kx+ly+ξ^{(0)})+d]\}},

(30.a)

V (x, y, t) = \frac{η}{[2 cosh(kx + ly + ξ^{(0)}) + d]^{\frac{1}{2}}} e^{−i\{(K_{1}−αk)x+(K_{2}−αl)y−Ωt−αξ^{(0)}−α ln[2 cosh(kx+ly+ξ^{(0)})+d]\}}.

(30.b)

In the same way, solutions (29.a) and (29.b) can be written:

U(x, y, t) = _{\frac{μ}{[2 cosh(kx+ly+ωt+ξ^{(0)})+d]^{\frac{1}{2}}}} e^{i\{(K_{1}−αk)x+(K_{2}−αl)y−(Ω+αω)t−αξ^{(0)}−α ln[2 cosh(kx+ly+ωt+ξ^{(0)})+d]\}}, (31.a)

V (x, y, t) = _{\frac{η}{[2 cosh(kx+ly+ωt+ξ^{(0)})+d]^{\frac{1}{2}}}} e^{−i\{(K_{1}−αk)x+(K_{2}−αl)y−(Ω+αω)t−αξ^{(0)}−α ln[2 cosh(kx+ly+ωt+ξ^{(0)})+d]\}}.(31.b) The solutions (30.a), (30.b), (31.a) and (31.b) have the following asymptotic behavior:

lim_{ζ⟶±∞} U(ζ) = lim_{ζ⟶±∞} V (ζ) = 0,  (32)

3.2. 2D Hole solutions . We now proceed to construct the 2D propagating hole solutions. To obtain these solutions, we assume the following forms for G, H and F.

G = μ(1 + ϵg_{1}), H = η(1 + ϵh_{1}), F = 1 + ϵf_{1} + ϵ^{2}f_{2},  (33)

where μ, η are complex constants, h_{1}, g_{1} are complex functions and f_{1} and f_{2} are real functions. Substituting Eqs.(33) into Eq.(15.a) and proceeding as in the previous manner, we obtain:

g_{1} = c exp (kx + ly + ωt + ξ^{(0)}), h_{1} = c^{∗} exp (kx + ly + ωt + ξ^{(0)}), (34.a)
f_{1} = exp (kx + ly + ωt + ξ^{(0)}), f_{2} = exp 2(kx + ly + ωt + ξ^{(0)}).(34.b)

Whence: (1 − |c|^{2})(1 − 4c_{r} + |c|^{2}) = 0.(35)

The condition (35) implies two cases:

Case 1: c_{i}^{2} = −c_{r}^{2} + 4c_{r} − 1.(36)

The unknowns are defined as follows:

μ^{2}η^{2} = \frac{−e}{(2c_{r} − 1)^{2}|γ|^{2}} (k^{2} + l^{2}), e < 0,  (37.a)
ω = 2p_{i}(kK_{1} + lK_{2}) − 2αp_{i}(k^{2} + l^{2}) + \frac{q_{i}}{2α}(|c|^{2} − 1)μη + \frac{γ_{i}}{2α}(|c|^{4} − 1)μ^{2}η^{2},  (37.b)
Ω = −p_{i}(K_{1}^{2} + K_{2}^{2}) + q_{i}μη + γ_{i}μ^{2}η^{2},  (37.c)
k^{2} + l^{2} = R^{2},  (37.d)
K_{2} = \frac{1}{l} \{[α + \frac{e(|c|^{4} − 1)γ_{r}}{4αp_{r}(2c_{r} − 1)^{2}|γ|^{2}} ] R^{2} − \frac{\sqrt{−e}(|c|^{2} − 1)}{4αp_{r}|2c_{r} − 1||γ|}R\} − \frac{k}{l} K_{1},  (37.e)
R^{2} = \frac{−e(2c_{r} − 1)^{2}|γ|^{2}[(2c_{r} − 1)q_{i} + 2c_{i}q_{r}]^{2}}{4α^{2}e_{2r}^{2}} ,  (37.f)
e = γ_{r} [(\frac{3}{4} − α^{2}) p_{r} − 2αp_{i}] + γ_{i} [(\frac{3}{4} − α^{2}) p_{i} + 2αp_{r}] .  (37.g)

EXACT LOCALIZED SOLUTIONS OF THE (2+1)D CCQGL EQUATION 9

Here α satisfies Eq.(7) and K_{1} satisfies the following quadratic equation

p_{r}R^{2}K_{1}^{2} − 2kb_{1}K_{1} + b_{2}(k, l) = 0,  (38)

where

b_{1} = p_{r}R \{[α + \frac{e(|c|^{4} − 1)γ_{r}}{4αp_{r}(2c_{r} − 1)^{2}|γ|^{2}} ] R − \frac{\sqrt{−e}(|c|^{2} − 1)q_{r}}{4αp_{r}|2c_{r} − 1||γ|} \} ,  (39.a)

b_{2}(k, l) = p_{r} [α + \frac{e(|c|^{4}−1)γ_{r}}{4αp_{r}(2c_{r}−1)^{2}|γ|^{2}} ]^{2} R^{4}−\frac{\sqrt{−e}(|c|^{2}−1)q_{r}}{2α|2c_{r}−1||γ|} [α + \frac{e(|c|^{4}−1)γ_{r}}{4αp_{r}(2c_{r}−1)^{2}|γ|^{2}} ] R^{3}−\frac{e(|c|^{2}−1)^{2}q_{r}^{2}}{16α^{2}p_{r}(2c_{r}−1)^{2}|γ|^{2}} R^{2}+ l^{2} \{a + \frac{eγ_{r}}{(2c_{r}−1)^{2}|γ|^{2}} R^{2} − \frac{\sqrt{−e}q_{r}}{|2c_{r}−1||γ|}R\} .(39.b)

The unknown c_{r} should satisfies the following system of equations:

\{ \begin{matrix}(3c_{r} − 1)[e_{2r}e_{4i} − e_{2i}e_{4r}] + (−c_{r}^{2} + 4c_{r} − 1)^{\frac{1}{2}} [e_{2r}e_{4r} + e_{2i}e_{4i}] = 0, (40.a) \\ (4c_{r} − 1)e_{4r} + 2α(3c_{r} − 1)e_{2r} + 2α(−c_{r}^{2} + 4c_{r} − 1)^{\frac{1}{2}} e_{2i} = 0.(40.b) \end{matrix}

The unknowns remaining e_{2r}, e_{2i}, e_{4r} and e_{4i} are functions of c_{r} such as defined as follows

e_{2r} = d_{1r}c_{r}^{2} + d_{2r}c_{r} + d_{3r} + (−c_{r}^{2} + 4c_{r} − 1)^{\frac{1}{2}} [−d_{1i}c_{r} + d_{4i}],  (41.a)
e_{2i} = d_{1i}c_{r}^{2} + d_{2i}c_{r} + d_{3i} + (−c_{r}^{2} + 4c_{r} − 1)^{\frac{1}{2}} [d_{1r}c_{r} + d_{4r}],  (41.b)
e_{4r} = j_{1r}c_{r}^{2} + j_{2r}c_{r} + j_{3r} + (−c_{r}^{2} + 4c_{r} − 1)^{\frac{1}{2}} [−j_{1i}c_{r} + j_{3i}],  (41.c)
e_{4i} = j_{1i}c_{r}^{2} + j_{2i}c_{r} + j_{3i} + (−c_{r}^{2} + 4c_{r} − 1)^{\frac{1}{2}} [j_{1r}c_{r} − j_{3r}].  (41.d)

The unknowns are given in Appendix. The following constraint should be respected:

[(2c_{r} − 1)e_{2r} + 2(−c_{r}^{2} + 4c_{r} − 1)^{\frac{1}{2}} e_{2i}]q_{r} − [2(−c_{r}^{2} + 4c_{r} − 1)^{\frac{1}{2}} e_{2r} − (2c_{r} − 1)e_{2i}]q_{i} = 0. (42)

Case 2: c_{i}^{2} = 1 − c_{r}^{2}(43) We find that

kK_{1} + lK_{2} = α(k^{2} + l^{2}),  (44)

which leads to two subcases:

Subcase 1: K_{1} = αk, K_{2} = αl.(45) The unknowns remaining μη, k, l, ω, Ω and c_{r} can be computed from the following equations

μ^{2}η^{2} = \frac{3e}{(2c_{r} − 1)^{2}|γ|^{2}} (k^{2} + l^{2}), e > 0,  (46.a)
ω = 0,  (46.b)
Ω = −α^{2}p_{i}(k^{2} + l^{2}) + q_{i}μη + γ_{i}μ^{2}η^{2},  (46.c)
k^{2} + l^{2} = R^{′2},  (46.d)
R^{′2} = \frac{3e(2c_{r} − 1)^{2}|γ|^{2}q_{r}^{2}}{e_{2r}^{′2}} .  (46.e)

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

where e is given by Eq.(37.g). These two constraints should be respected

e′_{2r}q_{i} − e′_{2i}q_{r} = 0,  (47.a)
a = [\frac{3eγ_{r}}{(2c_{r} − 1)^{2}|γ|^{2}} − α^{2}p_{r}] R^{′2} − \frac{\sqrt{3e}q_{r}}{|2c_{r} − 1||γ|}R′.  (47.b)

Subcase 2: K_{1} ≠ αk, K_{2} ≠ αl. The unknowns are defined as follows:

μ^{2}η^{2} = \frac{3e}{(2c_{r} − 1)^{2}|γ|^{2}} (k^{2} + l^{2}), e > 0,  (48.a)
ω = 0,  (48.b)
Ω = −p_{i}(K_{1}^{2} + K_{2}^{2}) + q_{i}μη + γ_{i}μ^{2}η^{2},  (48.c)
k^{2} + l^{2} = R^{′2},  (48.d)
K_{2} = \frac{α}{l}R^{′2} − \frac{k}{l} K_{1}.  (48.e)

K_{1} satisfies the following quadratic equation:

p_{r}R^{′2}K_{1}^{2} −2αp_{r}R^{′2}kK_{1} +\{α^{2}p_{r}R^{′4} − l^{2} [\frac{3eγ_{r}}{(2c_{r} − 1)^{2}|γ|^{2}} R^{′2} + \frac{\sqrt{3e}q_{r}}{|2c_{r} − 1||γ|}R′ − a]\} = 0. (49)

where e, R^{′2} respect Eq.(37.g) and Eq.(46.e) respectively. This solution is obtained under the constraint (47.a).

The remaining unknown c_{r} in these two subcases should satisfy:

\{ \begin{matrix}(c_{r} − 1)[e′_{2i}e′_{4r} − e′_{2r}e′_{4i}] + (1 − c_{r}^{2})^{\frac{1}{2}} [e′_{2r}e′_{4r} + e′_{2i}e′_{4i}] = 0, (50.a) \\ e′_{4r} − (c_{r} − 1)e′_{2r} + (1 − c_{r}^{2})^{\frac{1}{2}} e′_{2i} = 0.(50.b) \end{matrix}

The unknowns remaining e′_{2r}, e′_{2i}, e′_{4r} and e′_{4i} are functions of c_{r} such as defined as follows

e′_{2r} = d′_{1r}c_{r}^{2} + d′_{2r}c_{r} + d′_{3r} + (1 − c_{r}^{2})^{\frac{1}{2}} [−d′_{1i}c_{r} + \frac{1}{2}d′_{1i}],  (51.a)
e′_{2i} = d′_{1i}c_{r}^{2} + d′_{2i}c_{r} + d′_{3i} + (1 − c_{r}^{2})^{\frac{1}{2}} [d′_{1r}c_{r} − \frac{1}{2}d′_{1r}],  (51.b)
e′_{4r} = j′_{1r}c_{r}^{2} + j′_{2r}c_{r} + j′_{3r} + (1 − c_{r}^{2})^{\frac{1}{2}} [−j′_{1i}c_{r} + j′_{3i}],  (51.c)
e′_{4i} = j′_{1i}c_{r}^{2} + j′_{2i}c_{r} + j′_{3i} + (1 − c_{r}^{2})^{\frac{1}{2}} [j′_{1r}c_{r} − j′_{3r}].  (51.d)

The remaining unknowns are given in Appendix. This second case determines the 2D stationary hole solution. The 2D propagating hole solutions obtained in the first case can be written:

U(x, y, t) = \frac{μ[1 + (c_{r} + ic_{i})e^{(kx+ly+ωt+ξ^{(0)})}]}{[1 + e^{(kx+ly+ωt+ξ^{(0)})} + e^{2(kx+ly+ωt+ξ^{(0)})}]^{\frac{1}{2} +iα}} e^{i(K_{1}x+K_{2}y−Ωt)},  (52.a)

V (x, y, t) = \frac{η[1 + (c_{r} − ic_{i})e^{(kx+ly+ωt+ξ^{(0)})}]}{[1 + e^{(kx+ly+ωt+ξ^{(0)})} + e^{2(kx+ly+ωt+ξ^{(0)})}]^{\frac{1}{2} −iα}} e^{−i(K_{1}x+K_{2}y−Ωt)}.  (52.b)

After some arrangements these solutions lead to:

U(x, y, t) = \frac{μ(c_{r}+ic_{i})e^{\frac{1}{2} (kx+ly+ωt+ξ^{(0)})}+μe^{− \frac{1}{2} (kx+ly+ωt+ξ^{(0)})}}{[1+e^{(kx+ly+ωt+ξ^{(0)})}+e^{−(kx+ly+ωt+ξ^{(0)})}]^{\frac{1}{2}}} ×

e^{i{(K′_{1}x+K′_{2}y−Ω′t)−αξ^{(0)}−α ln[2 cosh(kx+ly+ωt+ξ^{(0)})+1]}}, (53.a)

V (x, y, t) = \frac{η(c_{r}−ic_{i})e^{\frac{1}{2} (kx+ly+ωt+ξ^{(0)})}+ηe^{− \frac{1}{2} (kx+ly+ωt+ξ^{(0)})}}{[1+e^{(kx+ly+ωt+ξ^{(0)})}+e^{−(kx+ly+ωt+ξ^{(0)})}]^{\frac{1}{2}}} ×

e^{−i{(K′_{1}x+K′_{2}y−Ω′t)−αξ^{(0)}−α ln[2 cosh(kx+ly+ωt+ξ^{(0)})+1]}}, (53.b)

Where

K′_{1} = K_{1} − αk, K′_{2} = K_{2} − αl, Ω′ = Ω + αω.  (54)

It can be easily easily seen that the solutions (53.a) and (53.b) have the following asymptotic behavior for ζ = kx + ly + ωt + ξ^{(0)}

lim_{ζ⟶+∞} U(ζ) = μ(c_{r} + ic_{i})e^{i[k_{1}x+l_{1}y−Ω′_{1}t−2αξ^{(0)}]},  (55.a)
lim_{ζ⟶+∞} V (ζ) = η(c_{r} − ic_{i})e^{−i[k_{1}x+l_{1}y−Ω′_{1}t−2αξ^{(0)}]},  (55.b)
lim_{ζ⟶−∞} U(ζ) = μe^{i[k_{2}x+l_{2}y−Ω′_{2}t]},  (55.c)
lim_{ζ⟶−∞} V (ζ) = ηe^{−i[k_{2}x+l_{2}y−Ω′_{2}t]},  (55.d)

with

k_{1} = K′_{1} − αk, l_{1} = K′_{2} − αl, Ω′_{1} = Ω′ − αω,  (56.a)
k_{2} = K′_{1} + αk, l_{2} = K′_{2} + αl, Ω′_{2} = Ω′ + αω.  (56.b)

This result shows that solutions (53.a) and (53.b) connect two waves with different wave vectors. The solution U(x, y, t) (Eq.(53.a)) behaves like a homogeneous wave of amplitude μ(c_{r} + ic_{i}) and wave vector k⃗ = (k_{1}, l_{1}) as ζ ⟶ +∞; for ζ ⟶ −∞, it corresponds to a wave of amplitude μ and wave vector k⃗′ = (k_{2}, l_{2}). If we assume that μ is real, the parameter μ(c_{r} + ic_{i}) can then be written as μ(c_{r} + ic_{i}) = μ|c|exp(iθ)

where

cos(θ) = \frac{c_{r}}{\sqrt{c_{r}^{2} + c_{i}^{2}}} ,  (58.a)
sin(θ) = \frac{c_{i}}{\sqrt{c_{r}^{2} + c_{i}^{2}}} .  (58.b)

Eqs.(58.a) and (58.b) represent the phase shift θ ≡ arg(μc, μ) between the two asymptotic waves. The 2D stationary hole solutions read:

U(x, y, t) = \frac{μ(c_{r}+ic_{i})e^{\frac{1}{2} (kx+ly+ξ^{(0)})}+μe^{− \frac{1}{2} (kx+ly+ξ^{(0)})}}{[1+e^{(kx+ly+ξ^{(0)})}+e^{−(kx+ly+ξ^{(0)})}]^{\frac{1}{2}}} e^{i\{(K′_{1}x+K′_{2}y−Ω′t)−αξ^{(0)}−α ln[2 cosh(kx+ly+ξ^{(0)})+1]\}}, (59.a)

V (x, y, t) = \frac{η(c_{r}−ic_{i})e^{\frac{1}{2} (kx+ly+ξ^{(0)})}+ηe^{− \frac{1}{2} (kx+ly+ξ^{(0)})}}{[1+e^{(kx+ly+ξ^{(0)})}+e^{−(kx+ly+ξ^{(0)})}]^{\frac{1}{2}}} e^{−i\{(K′_{1}x+K′_{2}y−Ω′t)−αξ^{(0)}−α ln[2 cosh(kx+ly+ξ^{(0)})+1]\}}.(59.b)

We should recall that these solutions are obtained in the case 2 with its two subcases.

4. The degeneracies of the (2+1)-dimensional CCQGL equation

The singularity structure of the (2+1)-D CCQGL equation degenerates in the limiting case of α = 0, and correspond to I_{m} (\frac{2p}{γ}) = 0. This put the following restriction on the allowed coefficients in this equation,

p_{r}γ_{i} − p_{i}γ_{r} = 0.  (60)

The constraint (60) has to be solved by respecting the leading-order balance (3.a) and (3.b). Leading-order analysis then leads to:

U_{0}^{2}V_{0}^{2} = −\frac{3}{4|γ|^{2}} (p_{r}γ_{r} + p_{i}γ_{i})(ϕ_{x}^{2} + ϕ_{y}^{2}).  (61)

The special solutions of the degeneracies of the CCQGL equation may be obtained if we proceed in a similar way as presented in the previous section.

4.1. 2D Pulses . The pulse solutions read

U(x, y, t) = \frac{μ}{[2 cosh(kx + ly + ωt + ξ^{(0)}) + d]^{\frac{1}{2}}} e^{i(K_{1}x+K_{2}y−Ωt)},  (62.a)
V (x, y, t) = \frac{η}{[2 cosh(kx + ly + ωt + ξ^{(0)}) + d]^{\frac{1}{2}}} e^{−i(K_{1}x+K_{2}y−Ωt)},  (62.b)

and necessary respect p_{r}(kK_{1} + lK_{2}) = 0. We distinguish two cases:

Case 1: p_{r} ≠ 0 A discrete set of stationary pulses is found in a co-dimension one subspace of parameter space defined by

p_{r}γ_{i} − p_{i}γ_{r} = 0

The unknowns are fixed by:

μ^{2}η^{2} = \frac{3(4 − d^{2})(p_{r}γ_{r} + p_{i}γ_{i})}{4|γ|^{2}}(k^{2} + l^{2}) > 0, d ≠ \{2, −2\},  (63.a)
ω = 0,  (63.b)
K_{2} = −\frac{k}{l} K_{1},  (63.c)
Ω = −p_{i}(K_{1}^{2} + K_{2}^{2}) + \frac{1}{4}p_{i}(k^{2} + l^{2}),  (63.d)
k^{2} + l^{2} = R_{0}^{2},  (63.e)
R_{0}^{2} = \frac{3(4 − d^{2})(p_{r}γ_{r} + p_{i}γ_{i})}{4|γ|^{2}} \frac{q_{r}^{2}}{p_{r}^{2}d^{2}} ,  (63.f)

K_{1}^{2} = l^{2} [\frac{1}{4} − \frac{a}{p_{r}R_{0}^{2}} ] > 0.  (63.g)

d is free parameter.

Restrictions (60) include the case of (2+1)-dimensional real CQ Ginzburg-Landau equation (RCQGL: p_{i} = q_{i} = γ_{i} = 0):

U_{t} + p_{r}(U_{xx} + U_{yy}) = aU − q_{r}|U|^{2}U − γ_{r}|U|^{4}U.  (64)

Unknowns are defined by:

μ^{2}η^{2} = \frac{3(4 − d^{2})p_{r}}{4γ_{r}} (k^{2} + l^{2}), d ≠ \{2, −2\},  (65.a)
ω = 0,  (65.b)
K_{2} = −\frac{k}{l} K_{1},  (65.c)
Ω = 0,  (65.d)
k^{2} + l^{2} = R_{1}^{2},  (63.e)
R_{1}^{2} = \frac{3(4 − d^{2})}{4p_{r}γ_{r}} \frac{q_{r}^{2}}{d^{2}} ,  (65.f)
K_{1}^{2} = l^{2} [\frac{1}{4} − \frac{a}{p_{r}R_{1}^{2}} ] > 0.  (65.g)

with (4 − d^{2})p_{r}γ_{r} > 0. d is free parameter.

Case 2: p_{r} = 0

A three-parameter family of propagating pulses is found for the cubic-quintic nonlinear Schrödinger equation (p_{r} = q_{r} = γ_{r} = a = 0):

−iU_{t} + p_{i}(U_{xx} + U_{yy}) = −q_{i}|U|^{2}U − γ_{i}|U|^{4}U.  (66)

The unknowns ω, Ω, μη, k and l are parametrized by K_{1}, K_{2} and d

μ^{2}η^{2} = \frac{3(4 − d^{2})p_{i}}{4γ_{i}} (k^{2} + l^{2}), d ≠ \{2, −2\},  (67.a)
ω = 2p_{i}(kK_{1} + lK_{2}),  (67.b)
Ω = −p_{i}(K_{1}^{2} + K_{2}^{2}) + \frac{1}{4}p_{i}(k^{2} + l^{2}),  (67.c)
k^{2} + l^{2} = R_{2}^{2},  (67.d)
R_{2}^{2} = \frac{3(4 − d^{2})}{4p_{i}γ_{i}} \frac{q_{i}^{2}}{d^{2}} ,  (67.f)

with (4 − d^{2})p_{i}γ_{i} > 0.

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

4.2. 2D Propagating hole solutions . These solutions read:

U(x, y, t) = \frac{μ(c_{r} + ic_{i})e^{\frac{1}{2} (kx+ly+ωt+ξ^{(0)})} + μe^{− \frac{1}{2} (kx+ly+ωt+ξ^{(0)})}}{[1 + e^{(kx+ly+ωt+ξ^{(0)})} + e^{−(kx+ly+ωt+ξ^{(0)})}]^{\frac{1}{2}}} e^{i\{(K_{1}x+K_{2}y−Ωt)\}},  (68.a)

V (x, y, t) = \frac{η(c_{r} − ic_{i})e^{\frac{1}{2} (kx+ly+ωt+ξ^{(0)})} + ηe^{− \frac{1}{2} (kx+ly+ωt+ξ^{(0)})}}{[1 + e^{(kx+ly+ωt+ξ^{(0)})} + e^{−(kx+ly+ωt+ξ^{(0)})}]^{\frac{1}{2}}} e^{−i\{(K_{1}x+K_{2}y−Ωt)\}}, (68.b)

The unknowns parameters k, l, ω, μη, K_{1}, K_{2}, Ω, c_{r} and c_{i} are defined by

μ^{2}η^{2} = \frac{e′_{1}}{(2c_{r} − 1)^{2}|γ|^{2}} (k^{2} + l^{2}),  (69.a)

ω = −\frac{1}{2c_{i}} \{[(2c_{r} − 1)p_{i} + 2c_{i}p_{r}](k^{2} + l^{2}) + 2[(2c_{r} − 1)p_{r} − 2c_{i}p_{i}](kK_{1} + lK_{2}) + 2(2c_{r} − 1)q_{i}μη + 4(2c_{r} − 1)γ_{i}μ^{2}η^{2}}, (69.b)

Ω = −p_{i}(K_{1}^{2} + K_{2}^{2}) + q_{i}μη + γ_{i}μ^{2}η^{2},  (69.c)
k^{2} + l^{2} = R_{0}^{′2},  (69.d)
K_{2} = \frac{1}{l}(e_{2}R_{0}^{′2} + e_{3}R′_{0}) − \frac{k}{l} K_{1},  (69.e)
c_{r}^{2} + c_{i}^{2} = 1.  (69.f)

K_{1} satisfies the following quadratic equation

p_{r}R_{0}^{′2}K_{1}^{2}−2p_{r}[e_{2}R_{0}^{′2}+e_{3}R′_{0}]kK_{1}+\{p_{r}[e_{2}R_{0}^{′2} + e_{3}R′_{0}]^{2} − l^{2} [\frac{e′_{1}γ_{r}}{(2c_{r} − 1)^{2}|γ|^{2}} R_{0}^{′2} + \frac{\sqrt{e′_{1}}q_{r}}{|2c_{r} − 1||γ|}R′_{0} − a]\} = 0.

(70)

The remaining unknowns are given by

e′_{1} = \frac{9}{4}(p_{r}γ_{r} + p_{i}γ_{i}) > 0,  (71.a)
R_{0}^{′2} = e′_{1} \frac{4(2c_{r} − 1)^{2}|γ|^{2}[(c_{r} − 2)q_{i} + c_{i}q_{r}]^{2}}{\{(5 − 4c_{r})|γ|^{2}[(c_{r} − 2)p_{i} + c_{i}p_{r}] + 4e′_{1}[(c_{r} − 2)γ_{i} + c_{i}γ_{r}]\}^{2}} ,  (71.b)
e_{2} = \frac{2e′_{1}[(2c_{r} − 1)γ_{i} + 2c_{i}γ_{r}]}{3(2c_{r} − 1)|γ|^{2}p_{r}} − \frac{[(4c_{r} − 5)p_{i} + 4c_{i}p_{r}]}{6p_{r}},  (71.c)
e_{3} = \frac{(2c_{r} − 1)\sqrt{e′_{1}}[(2c_{r} − 1)q_{i} + 2c_{i}q_{r}]}{3|2c_{r} − 1||γ|p_{r}}.  (71.d)

These solutions are obtained under the following constraints

p_{r}γ_{i} − p_{i}γ_{r} = 0,  (72.a)
[(5 − 4c_{r})|γ|^{2}p_{i} + 4e′_{1}γ_{i}]q_{r} − [(5 − 4c_{r})|γ|^{2}p_{r} + 4e′_{1}γ_{r}]q_{i} = 0.  (72.b)

EXACT LOCALIZED SOLUTIONS OF THE (2+1)D CCQGL EQUATION 15

Restrictions (60) include the case of CQ nonlinear Schrödinger equation Eq.(66) the parameters are defined as:

μ^{2}η^{2} = \frac{9p_{i}}{4(2c_{r} − 1)^{2}γ_{i}} (k^{2} + l^{2}),  (73.a)

ω = \frac{2c_{i}}{2c_{r} − 1}p_{i}(k^{2} + l^{2}) + 2p_{i}(kK_{1} + lK_{2}),  (73.b)
Ω = −p_{i}(K_{1}^{2} + K_{2}^{2}) + q_{i}μη + γ_{i}μ^{2}η^{2},  (73.c)
k^{2} + l^{2} = \frac{9(2c_{r} − 1)^{2}q_{i}^{2}}{4(7 − 2c_{r})^{2}} \frac{1}{p_{i}γ_{i}} ,  (73.d)
c_{r}^{2} + c_{i}^{2} = 1.  (73.e)

with p_{i}γ_{i} > 0, K_{1} and K_{2} are free parameters. Restrictions (60) also include the case of RCQGL equation (Eq.(64)). The parameters can be computed from:

μ^{2}η^{2} = \frac{9p_{r}}{4(2c_{r} − 1)^{2}γ_{r}} (k^{2} + l^{2}),  (74.a)
ω = 0,  (74.b)
Ω = 0,  (74.c)
K_{2} = \frac{−c_{i}}{l(2c_{r} − 1)}(k^{2} + l^{2}) − \frac{k}{l} K_{1},  (74.d)

k^{2} + l^{2} = R_{1}^{′2},  (74.e)
R_{1}^{′2} = \frac{9(2c_{r} − 1)^{2}q_{r}^{2}}{4(7 − 2c_{r})^{2}} \frac{1}{p_{r}γ_{r}} ,  (74.f)
c_{r}^{2} + c_{i}^{2} = 1.  (74.g)

With p_{r}γ_{r} > 0.

K_{1} is solution of the following quadratic equation

(2c_{r}−1)^{2}p_{r}R_{1}^{′2}K_{1}^{2}+2[p_{r}c_{i}(2c_{r}−1)R_{1}^{′2}]kK_{1}+\{c_{i}^{2}p_{r}R_{1}^{′4} − l^{2} [\frac{9}{4}p_{r}R_{1}^{′2} + \frac{3}{2}q_{r}|2c_{r} − 1|\sqrt{\frac{p_{r}}{γ_{r}}} R1^{′} − (2c_{r} − 1)^{2}a]\}

= 0.(75)

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

5. conclusion

Based on the (2+1)-dimensional complex cubic-quintic Ginzburg-Landau (CCQGL) equation, we have investigated exact analytic localized solutions. Using a method derived from a combination of the methodology of Painlevé test for integrability due to Weiss-Tabor-Carnevale and the Hirota multilinear methodology, we have found that a number of exact solutions exist for this equation provided that coefficients are bound by special relations. The set of solutions include particular types of localized structures such as: 2D propagating pulse, 2D stationary pulse, 2D propagating hole and 2D stationary hole solutions. It appears that this combined methodology is a very powerful method, and we hope that the method may also be used to other (2+1)-dimensional family of Ginzburg-Landau equation like, such as the coupled GL equation with waves that travelled right and left.

By analysing the degeneracies of the (2+1)-D CCQGL equation, it appears that this equation is a generalization of the (2+1)-D CQ NLS and of the (2+1)-D Real CQ GL equation.

6. appendix

The coefficients related to Eqs.(41.a) - (41.d) are:

d_{1r} = 2 [(p_{r} − 2αp_{i})|γ|^{2} − \frac{2eγ_{i}}{α} ] , d_{1i} = 2 [(p_{i} + 2αp_{r})|γ|^{2} + \frac{2eγ_{r}}{α} ] ,  (A.1)
d_{2r} = 2 \{[α(p_{i} + 2αp_{r}) − (p_{r} − 2αp_{i})] |γ|^{2} + e (2γ_{r} + \frac{γ_{i}}{α} )\} ,  (A.2.a)
d_{2i} = −2 \{[α(p_{r} − 2αp_{i}) + (p_{i} + 2αp_{r})] |γ|^{2} − e (2γ_{i} − \frac{γ_{r}}{α} )\} ,  (A.2.b)
d_{3r} = −\frac{1}{2}\{[2α(p_{i} + 2αp_{r}) − (p_{r} − 2αp_{i})]|γ|^{2} + 4eγ_{r}\},  (A.3.a)
d_{3i} = \frac{1}{2}\{[2α(p_{r} − 2αp_{i}) + (p_{i} + 2αp_{r})]|γ|^{2} − 4eγ_{i}\},  (A.3.b)
d_{4r} = −|γ|^{2}(p_{r} − 2αp_{i}), d_{4i} = |γ|^{2}(p_{i} + 2αp_{r}),  (A.4)
j_{1r} = 4α \{−2|γ|^{2} [(\frac{3}{4} − α^{2}) p_{r} − 2αp_{i}] + e (2γ_{r} + \frac{γ_{i}}{α} )\} ,  (A.5.a)
j_{1i} = 4α \{−2|γ|^{2} [(\frac{3}{4} − α^{2}) p_{i} + 2αp_{r}] + e (2γ_{i} − \frac{γ_{r}}{α} )\} ,  (A.5.b)
j_{2r} = 4α \{3|γ|^{2} [(\frac{3}{4} − α^{2}) p_{r} − 2αp_{i}] − e (3γ_{r} − \frac{γ_{i}}{α} )\} ,  (A.6.a)
j_{2i} = 4α \{3|γ|^{2} [(\frac{3}{4} − α^{2}) p_{i} + 2αp_{r}] − e (3γ_{i} + \frac{γ_{r}}{α} )\} ,  (A.6.b)
j_{3r} = 4α \{−|γ|^{2} [(\frac{3}{4} − α^{2}) p_{r} − 2αp_{i}] + eγ_{r}\} ,  (A.7.a)

j_{3i} = 4α \{−|γ|^{2} [(\frac{3}{4} − α^{2}) p_{i} + 2αp_{r}] + eγ_{i}\} .  (A.7.b)

The coefficients related to Eqs.(51.a) - (51.d) are defined by

d′_{1r} = 2(p_{r} − 2αp_{i})|γ|^{2}, d′_{1i} = 2(p_{i} + 2αp_{r})|γ|^{2},  (A.8)
d′_{2r} = 2[α(p_{i} + 2αp_{r}) − (p_{r} − 2αp_{i})]|γ|^{2}, d′_{2i} = −2[α(p_{r} − 2αp_{i}) + (p_{i} + 2αp_{r})]|γ|^{2}, (A.9)

d′_{3r} = −\frac{1}{2}\{[2α(p_{i} + 2αp_{r}) − (p_{r} − 2αp_{i})]|γ|^{2} − 12eγ_{r}\},  (A.10.a)
d′_{3i} = \frac{1}{2}\{[2α(p_{r} − 2αp_{i}) + (p_{i} + 2αp_{r})]|γ|^{2} + 12eγ_{i}\},  (A.10.b)

j′_{1r} = −4|γ|^{2} [(\frac{3}{4} − α^{2}) p_{r} − 2αp_{i}] ,  (A.11.a)
j′_{1i} = −4|γ|^{2} [(\frac{3}{4} − α^{2}) p_{i} + 2αp_{r}] ,  (A.11.b)
j′_{2r} = 6 \{|γ|^{2} [(\frac{3}{4} − α^{2}) p_{r} − 2αp_{i}] + eγ_{r}\} ,  (A.12.a)
j′_{2i} = 6 \{|γ|^{2} [(\frac{3}{4} − α^{2}) p_{i} + 2αp_{r}] + eγ_{i}\} ,  (A.12.b)
j′_{3r} = −2 \{|γ|^{2} [(\frac{3}{4} − α^{2}) p_{r} − 2αp_{i}] + 3eγ_{r}\} ,  (A.13.a)
j′_{3i} = −2 \{|γ|^{2} [(\frac{3}{4} − α^{2}) p_{i} + 2αp_{r}] + 3eγ_{i}\} .  (A.13.b)

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