^{1}

^{*}

^{2}

A subset of traveling wave solutions of the quintic complex Ginzburg-Landau equation (QCGLE) is presented in compact form. The approach consists of the following parts: 1) Reduction of the QCGLE to a system of two ordinary differential equations (ODEs) by a traveling wave ansatz; 2) Solution of the system for two (ad hoc) cases relating phase and amplitude; 3) Presentation of the solution for both cases in compact form; 4) Presentation of constraints for bounded and for singular positive solutions by analysing the analytical properties of the solution by means of a phase diagram approach. The results are exemplified numerically.

As a partial differential equation (PDE) the quintic complex Ginzburg-Landau equation (QCGLE) is one of the most studied nonlinear equations in physics. Apart from many applications in the natural sciences [

Probably, due to this reason, several non-perturbative methods have been proposed to find some particular solutions of (partly) nonintegrable systems (“tanh-method”, “exponential-method”, “Riccati-method”, “Jacobi expansion-method”, ...., see also [

As a starting point for perturbation calculations or stability analysis exact traveling solutions of nonlinear equations such as (2) below are very useful, however rare. Remarkably, if certain constraints for the parameters are satisfied, solutions can be derived. In [

The rest of the paper is organised as follows. In Section 2 we reduce the QCGLE to a system of two ordinary differential equations, specify it for two particular (ad hoc) relations between phase function and amplitude, and describe the procedure to obtain the ansatz parameters c , ω , d , b 0 , b 1 and constraints of these relations. An exact solution together with its dependence on the parameters of the QCGLE is presented in Section 3. For elucidation examples are presented in Section 4. In Section 5, the paper concludes with comments on articles having a certain contact to the present paper and with suggestions for further investigations.

We seek particular (traveling wave) solutions [

ψ ( x , t ) = f ( x − c t ) e i ( ϕ ( x − c t ) − ω t ) , c ∈ R , ω ∈ R , (1)

where f and ϕ are real-valued, of the QCGLE

i ∂ ∂ t ψ ( x , t ) + ( c 3 + i h 3 ) | ψ ( x , t ) | 2 ψ ( x , t ) + ( c 5 + i h 5 ) | ψ ( x , t ) | 4 ψ ( x , t ) + ( c 1 − i h 1 ) ∂ 2 ∂ x 2 ψ ( x , t ) − i ε ψ ( x , t ) = 0, (2)

with complex-valued function ψ ( x , t ) and with real dimensionless constants ε , h 1 , h 3 , h 5 , c 1 , c 3 , c 5 . As mentioned in the Introduction, Equation (2) has a wide range of applications. Thus the meaning of variables and parameters may be quite different (see [

To find real f , ϕ , c , ω , together with corresponding assumptions and constraints for their existence, we insert ansatz (1) into Equation (2), set z = x − c t , introduce F ( z ) = f 2 ( z ) , τ ( z ) = ϕ ′ ( z ) , and separate real and imaginary parts. Hence we obtain the system of two ordinary differential equations for functions F ( z ) and τ ( z )

4 ω F 2 ( z ) + 4 c 3 F 3 ( z ) + 4 c 5 F 4 ( z ) + 4 c τ ( z ) F 2 ( z ) − 4 c 1 τ 2 ( z ) F 2 ( z ) + 4 h 1 τ ( z ) F ( z ) F ′ ( z ) − c 1 ( F ′ ( z ) ) 2 + 4 h 1 τ ′ ( z ) F 2 ( z ) + 2 c 1 F ( z ) F ″ ( z ) = 0, (3)

− 4 ε F 2 ( z ) + 4 h 3 F 3 ( z ) + 4 h 5 F 4 ( z ) + 4 h 1 τ 2 ( z ) F 2 ( z ) − 2 c F ( z ) F ′ ( z ) + 4 c 1 τ ( z ) F ( z ) F ′ ( z ) + h 1 ( F ′ ( z ) ) 2 + 4 c 1 τ ′ ( z ) F 2 ( z ) − 2 h 1 F ( z ) F ″ ( z ) = 0. (4)

As mentioned above, there is a general method in the literature [

τ ( z ) = d F ′ ( z ) F ( z ) , (5)

τ ( z ) = b 0 + b 1 F ( z ) , (6)

where real constants d , b 0 , b 1 are to be determined.

We consider the solutions F ( z ) of the system (3)-(4), with assumptions for τ ( z ) from above, and determine parameters c , ω , d , b 0 , b 1 for both cases (5), (6) separately. In each part (for (5) and for (6)) we derive the solution ψ ( x , t ) and study its properties (disregarding its stability). We note that system (3)-(4) is equivalent to (13)-(14) in [

To simplify formulas we use the abbreviations

D 1 = c 1 2 + h 1 2 , D 2 = c 3 h 1 + c 1 h 3 , D 3 = c 1 c 3 − h 1 h 3 , D 4 = c 5 h 1 + c 1 h 5 , D 5 = c 1 c 5 − h 1 h 5

in what follows.

First, we consider Equations (3), (4) subject to τ ( z ) = d F ′ ( z ) F ( z ) . Equations (3), (4) read in this case

4 ω F 2 ( z ) + 4 c 3 F 3 ( z ) + 4 c 5 F 4 ( z ) + 4 c d F ( z ) F ′ ( z ) − c 1 ( 1 + 4 d 2 ) ( F ′ ( z ) ) 2 + ( 4 h 1 d + 2 c 1 ) F ( z ) F ″ ( z ) = 0 , (7)

− 4 ε F 2 ( z ) + 4 h 3 F 3 ( z ) + 4 h 5 F 4 ( z ) − 2 c F ( z ) F ′ ( z ) + h 1 ( 1 + 4 d 2 ) ( F ′ ( z ) ) 2 + ( 4 c 1 − 2 h 1 ) F ( z ) F ″ ( z ) = 0. (8)

By eliminating F ″ ( z ) from (7) and (8), we obtain for F ′ ( z )

F ′ ( z ) = F ( z ) ( ± a + b F ( z ) + f F 2 ( z ) + c c 1 2 d D 1 ) , (9)

with

a = c 2 c 1 2 4 d 2 D 1 2 + 2 ε ( c 1 + 2 d h 1 ) + 2 ω ( 2 d c 1 − h 1 ) d ( 1 + 4 d 2 ) D 1 , (10)

b = 4 d D 3 − 2 D 2 d ( 1 + 4 d 2 ) D 1 , (11)

f = 4 d D 5 − 2 D 4 d ( 1 + 4 d 2 ) D 1 . (12)

Using (9), we get

F ″ ( z ) = F ′ ( z ) ( ± 2 a + 3 b F ( z ) + 4 f F 2 ( z ) 2 a + b F ( z ) + f F 2 ( z ) + c c 1 2 d D 1 ) . (13)

Substituting (9) and (13) to the system (7)-(8), we obtain the following equivalent system only in terms of F ( z ) and the parameters of QCGLE

4 a c d D 1 ( 2 h 1 d + c 1 ) 2 + [ c 2 c 1 ( c 1 2 + 4 d 2 ( c 1 2 + 2 h 1 2 ) + 4 c 1 h 1 d ) + 4 a d 2 D 1 2 ( c 1 − 4 c 1 d 2 + 4 h 1 d ) + 16 ω d 2 D 1 2 ] a + b F ( z ) + f F 2 ( z ) + F ( z ) [ 2 b c d D 1 ( 8 h 1 2 d 2 + 10 c 1 h 1 d + 3 c 1 2 ) + 8 d 2 D 1 2 ( 2 c 3 + b ( c 1 − 2 c 1 d 2 + 3 h 1 d ) ) a + b F ( z ) + f F 2 ( z ) ] + F 2 ( z ) [ 8 c d D 1 f ( c 1 2 + 3 c 1 h 1 d + 2 h 1 2 d 2 ) + 4 d 2 D 1 2 ( 4 c 5 + f ( 3 c 1 − 4 c 1 d 2 + 8 h 1 d ) ) a + b F ( z ) + f F 2 ( z ) ] = 0 (14)

and

4 a c d D 1 ( 4 c 1 h 1 d 2 + 2 ( c 1 2 − h 1 2 ) d − c 1 h 1 ) + [ c 2 c 1 ( 4 c 1 h 1 d 2 − 4 h 1 2 d − c 1 h 1 ) + 4 a d 2 D 1 2 ( 4 h 1 d 2 + 4 c 1 d − h 1 ) − 16 ε d 2 D 1 2 ] a + b F ( z ) + f F 2 ( z ) + F ( z ) [ 2 b c d D 1 ( 8 c 1 2 d 2 + ( c 1 2 − 4 h 1 2 ) d − 3 c 1 h 1 ) + 8 d 2 D 1 2 ( 2 h 3 + b ( 3 c 1 d − h 1 + 2 h 1 d 2 ) ) a + b F ( z ) + f F 2 ( z ) ] + F 2 ( z ) [ 8 c d D 1 f ( 2 c 1 h 1 d + ( 2 c 1 2 − h 1 2 ) d − c 1 h 1 ) + 4 d 2 D 1 2 ( 4 h 5 + f ( 8 c 1 d + 4 h 1 d 2 − 3 h 1 ) ) a + b F ( z ) + f F 2 ( z ) ] = 0 (15)

In Equations (14)-(15) the coefficients in front of F ( z ) , F 2 ( z ) , a + b F ( z ) + f F 2 ( z ) as well as free coefficients must be equal to zero, leading to equations which imply c = 0 necessarily. With c = 0 the system (14)-(15) can be simplified to the following system of six equations:

4 ω = a ( 4 c 1 d 2 − 4 h 1 d − c 1 ) , 4 ε = a ( 4 h 1 d 2 + 4 c 1 d − h 1 ) , (16)

b ( 2 c 1 d 2 − 3 h 1 d − c 1 ) = 2 c 3 , b ( 2 h 1 d 2 + 3 c 1 d − h 1 ) = − 2 h 3 , (17)

f ( 4 c 1 d 2 − 8 h 1 d − 3 c 1 ) = 4 c 5 , f ( 4 h 1 d 2 + 8 c 1 d − 3 h 1 ) = − 4 h 5 . (18)

Combining the first Equation (16) and (10), we obtain

ω = ε ( 4 c 1 d 2 − 4 h 1 d − c 1 ) 4 h 1 d 2 + 4 c 1 d − h 1 . (19)

Furthermore, Equations (16)-(18) are solved by

d = − 3 D 3 ± 9 D 3 2 + 8 D 2 2 4 D 2 and d = − 2 D 5 ± 4 D 5 2 + 3 D 4 2 2 D 4 . (20)

Consistency of d’s in the Equation (20) leads to the constraints

− 3 D 3 ± 9 D 3 2 + 8 D 2 2 4 D 2 = − 2 D 5 ± 4 D 5 2 + 3 D 4 2 2 D 4 , (21)

necessary for the existence of solutions d.

Thus, parameters d , ω and a , b , f in Equation (9) are expressed according to (19)-(20) and (10)-(12), respectively in terms of the parameters of the QCGLE (with constraint (21)). The solution F ( z ) of Equation (9) is presented below.

Second, considering the case τ ( z ) = b 0 + b 1 F ( z ) , and inserting τ ( z ) into the system (3)-(4), we get

4 ( ω + c b 0 − c 1 b 0 2 ) F 2 ( z ) + 4 ( c 3 + c b 1 − 2 c 1 b 0 b 1 ) F 3 ( z ) + 4 ( c 5 − c 1 b 1 2 ) F 4 ( z ) − c 1 ( F ′ ( z ) ) 2 + 4 h 1 ( b 0 F ( z ) + 2 b 1 F 2 ( z ) ) F ′ ( z ) + 2 c 1 F ( z ) F ″ ( z ) = 0 (22)

and

4 ( h 1 b 0 2 − ε ) F 2 ( z ) + 4 ( h 3 + 2 h 1 b 0 b 1 ) F 3 ( z ) + 4 ( h 5 + h 1 b 1 2 ) F 4 ( z ) + h 1 ( F ′ ( z ) ) 2 + ( ( 4 c 1 b 0 − 2 c ) F ( z ) + 8 c 1 b 1 F 2 ( z ) ) F ′ ( z ) − 2 h 1 F ( z ) F ″ ( z ) = 0. (23)

Eliminating F ″ ( z ) from (22) and (23), F ′ ( z ) can be derived as

F ′ ( z ) = ( h 1 ( ω + c b 0 ) − c 1 ε ) F ( z ) + ( D 2 + c h 1 b 1 ) F 2 ( z ) + D 4 F 3 c c 1 2 − b 0 D 1 − 2 b 1 D 1 F ( z ) . (24)

Assuming for simplicity b 0 = c c 1 2 D 1 , we get

F ′ ( z ) = − h 1 ( ω + c 2 c 1 2 D 1 ) − ε c 1 + ( c h 1 b 1 + D 2 ) F ( z ) + D 4 F 2 ( z ) 2 b 1 D 1 (25)

and hence

F ″ ( z ) = − ( c h 1 b 1 + D 4 + 2 D 2 F ( z ) ) F ′ ( z ) 2 b 1 D 1 . (26)

Substitution of (25) and (26) into system (22)-(23) and considering vanishing coefficients of powers of F ( z ) , leads to

2 D 1 ( h 1 ω − c 1 ε ) + c 2 c 1 h 1 = 0, (27)

− 16 b 1 4 c 1 D 1 2 + 3 c 1 D 4 2 + 16 c 1 b 1 2 D 1 D 5 = 0 , (28)

c 1 D 2 D 4 + 4 b 1 2 c 1 D 1 D 3 = 0 , (29)

− 2 c c 1 h 1 b 1 D 1 D 2 + c 1 ( c 2 c 1 h 1 D 4 + D 1 ( D 2 2 − 2 c 1 ε D 4 + 2 h 1 ω D 4 ) ) − b 1 2 D 1 ( c 2 c 1 ( 4 c 1 2 − 3 h 1 2 ) + 16 D 1 c 1 ( h 1 ε + c 1 ω ) ) = 0, (30)

where c ≠ 0, h 1 ≠ 0, b 1 ≠ 0 have been assumed. Parameters ω , b 1 and c must satisfy system (27)-(30). Equation (27) implies

ω = 2 c 1 ε D 1 − c 2 c 1 h 1 2 h 1 D 1 . (31)

Solutions of (28) and (29) are

b 1 2 = 2 D 5 + 4 D 5 2 + 3 D 4 2 4 D 1 (32)

and

b 1 2 = − D 4 D 2 4 D 1 D 3 , (33)

respectively. Inserting ω into (30) and solving for c we get

c = − h 1 D 2 ± 2 D 1 D 2 2 + 4 ε h 1 b 1 2 D 1 2 ( 4 c 1 2 + 3 h 1 2 ) b 1 ( 3 h 1 2 + 4 c 1 2 ) . (34)

Consistency of (32) and (33) yields the constraint

D 4 D 2 2 + 4 D 2 D 3 D 5 − 3 D 4 D 3 2 = 0. (35)

With (31), (32) (or (33)), (34) all parameters in (25) are determined in terms of the parameters of the QCGLE, so that (25) can be solved for F ( z ) subject to (35).

The nonlinear first order ODEs (9) (with c = 0 ) and (25) can be solved by standard methods yielding F ( z ) as an inverse function of an elliptic integral, but not F ( z ) explicitly. Thus, it is obvious to look for another possibility to find elliptic solutions of (9) and (25). With F ′ according to (9), c = 0 necessarily, and hence, taking into account (10)-(12) and (16)-(21), the solution of system (7)-(8) uniquely can be rewritten as

( F ′ ( z ) ) 2 = α F 4 ( z ) + 4 β F 3 ( z ) + 6 γ F 2 ( z ) (36)

with

α = f , β = b 4 , γ = a 6 . (37)

Thus, Equation (9) (with c = 0 ) and Equation (36) are equivalent.

Following the same line with F ′ according to (25) (using (22) or (23), (27)-(35)) we obtain (36), and hence equivalence of (25) and (36). The coefficients in (36) are given by

α = D 4 2 4 b 1 2 D 1 2 , β = ( c h 1 b 1 + D 2 ) D 4 8 b 1 2 D 1 2 , γ = ( c h 1 b 1 + D 2 ) 2 24 b 1 2 D 1 2 . (38)

The solution of (36) is well known (see [

F ( z ) = F 0 2 α F 0 2 + 4 β F 0 + 6 γ ℘ ′ ( z ) + 4 ℘ 2 ( z ) + ( 8 γ + 4 β F 0 ) ℘ ( z ) − 2 γ β F 0 − 5 γ 2 4 ℘ 2 ( z ) − 4 ℘ ( z ) ( α F 0 2 + 2 β F 0 + γ ) + 4 F 0 2 ( β 2 − α γ ) + 4 β γ F 0 + γ 2 , (39)

where ℘ ( z ; g 2 , g 3 ) denotes Weierstrass’ elliptic function with invariants g 2 = 3 γ 2 , g 3 = − γ 3 and F 0 = F ( 0 ) as an integration constant is the intensity F ( z ) at z = 0 . The invariants g 2 , g 3 can be expressed in terms of the ansatz parameters and the coefficients of the QCGLE. For the case (5) we obtain

g 2 = a 2 12 , g 3 = − a 3 216 (40)

with a given by Equation (10). For case (6) we get

g 2 = 3 ( ( c h 1 b 1 + D 2 ) 2 24 b 1 2 D 1 2 ) 2 , g 3 = − ( ( c h 1 b 1 + D 2 ) 2 24 b 1 2 D 1 2 ) 3 , (41)

with b 1 2 according to Equations (32) or (33) has been used.

Integration of Equations (5) and (6), using Equation (39), yields the phase function ϕ ( z ) . For case (5) we obtain

ϕ ( x ) = d ⋅ log ( F 0 2 α F 0 2 + 4 β F 0 + 6 γ ℘ ′ ( x ) + 4 ℘ 2 ( x ) + ( 8 γ + 4 β F 0 ) ℘ ( x ) − 2 γ β F 0 − 5 γ 2 4 ℘ 2 ( x ) − 4 ℘ ( x ) ( α F 0 2 + 2 β F 0 + γ ) + 4 F 0 2 ( β 2 − α γ ) + 4 β γ F 0 + γ 2 ) , (42)

where α , β , γ , g 2 , g 3 given by (37) and (40), respectively and chirp parameter d by (20). Since c = 0 in this case F and ϕ are stationary. As is known [

x ′ = x + v t , t ′ = t , ψ ′ ( x ′ , t ′ ) = ψ ( x , t ) e i ( v x + v 2 2 t ) , v = c o n s t ,

to solutions (39) and (42). —We disregard this possibility to find solutions of the QCGLE.

For case (6), integral ϕ ( z ) = ∫ ( b 0 + b 1 F ( z ) ) d z with F ( z ) according to (39), cannot be evaluated in closed form (in general). With respect to the example presented below, for particular F 0 , integration yields a closed form result. If F 0 = − β α , α > 0 , β < 0 (see example in Section 4), solution F ( z ) reads

F ( z ) = β 2 α α ℘ ′ ( z ) − 2 β α ℘ 2 ( z ) − 2 β 3 3 α 2 ℘ ( z ) + 4 β 5 9 α 3 2 ℘ 2 ( z ) + 2 β 2 3 α ℘ ( z ) − 4 β 4 9 α 2 (43)

and

ϕ ( z ) = ( b 0 − β α b 1 ) z + b 1 2 α ( log ( ℘ ( z ) − β 2 3 α ) − log ( ℘ ( z ) + 2 β 2 3 α ) ) , (44)

where α , β , g 2 , g 3 are given by (37) ( γ = 2 β 2 3 α ), (40) and by b 0 = c c 1 2 D 1 , b 1 2 = − D 2 D 4 4 D 1 D 3 with c according to (34).

It should be noted that in both cases (40) and (41) the discriminant of ℘ ( z ; , g 2 , g 3 ) vanishes, so that ℘ ( z ) degenerates to trigonometric ( g 2 > 0 , g 3 > 0 ) or hyperbolic ( g 2 > 0 , g 3 < 0 ) functions, thus depending on the sign of γ in (37) and (38) (see [

Summing up, solutions (1) of Equation (2) can be derived if ϕ ′ ( z ) and F ( z ) , z = x − ω t , are related by (5) or (6), with F ( z ) given by (39) subject to certain constraints and conditions for d , c , b 0 , b 1 , ω . It is necessary to check consistency of the results with the initial assumptions { f , ϕ , c , ω , d , b 0 , b 1 } ⊂ R . This will be done in the following.

First, we note for case (5) that α , β , γ , g 2 , g 3 are real (see Equation (10) with c = 0 ). Chirp parameter d is real and hence ω since (20) is satisfied. Constraint (21) is necessary for unique existence of d.

Second, for case (6), α , β , γ , g 2 , g 3 are real (see Equations (37), (40)) if c is real according to Equation (34) and hence ω . Thus, real c implies nonnegative radicand in Equation (34). Constraint (35) is necessary for unique b 1 2 . —We emphasise that real α , β , γ , g 2 , g 3 are important for evaluation of Equations (38), (39).

Third, we note that real g 2 , g 3 imply real ℘ ( z ; , g 2 , g 3 ) and ℘ ′ ( z ; , g 2 , g 3 ) if z is real (see [

Due to the properties of ℘ ( z ; g 2 , g 3 ) (poles and periods) and the dependence of the denominator on α , β , γ , F 0 , a singularity analysis of F ( z ) w.r.t.z on the basis of (39) is very difficult. Indeed, what can be stated is that F ( z ) exhibits only poles w.r.t. z, consistent with a Theorem by Conte and Ng [

As is known [

It is clear that the choice of the intensity F 0 > 0 in relation to the zeros of ( F ′ ) 2 ( F 1 = 0 , F 2 , 3 = − 2 β ± 4 β 2 − 6 α γ α ) is essential for the singularity behaviour of F ( z ) . F 0 > 0 must be chosen such that ( F ′ ) | F = F 0 2 ≥ 0 . In this manner certain

domains are defined (labelled green or red in

The foregoing results can be summarised as follows. Bounded or unbounded solutions ℜ ( ψ ( x , t ) ) = F ( x − c t ) cos ( ϕ ( x − c t ) − ω t ) of Equation (2) exist if ϕ ′ ( z ) and F ( z ) , z = x − c t are related by ϕ ′ ( z ) = d F ′ ( z ) F ( z ) or

ϕ ′ ( z ) = b 0 + b 1 F ( z ) , if the amplitude F ( z ) satisfies the ODE ( F ′ ( z ) ) 2 = α F 4 ( z ) + 4 β F 3 ( z ) + 6 γ F 2 ( z ) , and if the parameters of the QCGLE satisfy the PDCs associated to

To elucidate the foregoing results, we first consider ϕ ′ ( z ) = d F ′ ( z ) F ( z ) with c = 0 . Needless to say, that if all parameters are prescribed, constraints (21) are not satisfied in general. Due to (21) one of the parameters { ε , h j , c j , j = 1 , 3 , 5 } cannot be prescribed. As a solution of (21), this parameter must be inserted into (20) leading to (lengthy) expressions for d, hence for ω , α , β , γ , g 2 , g 3 and, finally, to F ( x ) and ϕ ( x ) according to (39) and (42), respectively. For simplicity, we assume a particular solution of (21) w.r.t. h 3 and h 5 . If

h 3 = 2 c 3 h 1 c 1 , h 5 = 3 c 5 h 1 c 1 , s i g n c 3 c 1 = ± 1 , (45)

the constraint (21)

− 3 D 3 ∓ 9 D 3 2 + 8 D 2 2 4 D 2 = − 2 D 5 ± 4 D 5 2 + 3 D 4 2 2 D 4

(where ∓ and ± corresponds to sign in (44)) is satisfied and c is real (see Equation (34)), we obtain

d = − c 1 2 h 1 , ω = − ε c 1 h 1 (46)

α = 4 c 5 h 1 2 c 1 D 1 , β = c 3 h 1 2 c 1 D 1 , γ = − 2 ε h 1 3 D 1 (47)

g 2 = 4 ε 2 h 1 2 3 D 1 2 , g 3 = 8 ε 3 h 1 3 27 D 1 3 . (48)

Subject to (47) the PDC according to Figures 1(a)-(f) must be evaluated. For instance, parameters

c 1 = − 1 , c 3 = − 1 , c 5 = 1 8 , h 1 = − 1 , h 3 = − 2 , h 5 = 3 8 , ε = − 1 , F 0 = 4 (49)

are consistent with the PDC of

p = π D 1 ε h 1 . (50)

A plot of ℜ ( ψ ( x , t ) ) = F ( x ) cos ( ϕ ( x ) − ω t ) with ϕ ( x ) according to (42) is shown in

The second case ϕ ′ ( z ) = b 0 + b 1 F ( z ) can be exemplified following the line presented before. If

c 5 = h 5 ( 3 c 1 c 3 2 − 4 c 3 h 1 h 3 − c 1 h 3 2 ) c 3 2 h 1 + 4 c 1 c 3 h 3 − 3 h 1 h 3 2 , (51)

constraint (35) is satisfied. Subject to (51), parameters

c 1 = − 3 4 , c 3 = 1 , c 5 = − 2.5 , h 1 = 1 , h 3 = − 1 , h 5 = − 1 , ε = 1 , F 0 = 7 32 (52)

are consistent with the PDC of

ℜ ( ψ ( x , t ) ) are shown in

As is known [

− 3 D 3 − 9 D 3 2 + 8 D 2 2 4 D 2 = − 2 D 5 + 4 D 5 2 + 3 D 4 2 2 D 4

is satisfied by (45). Thus the range { F 0 , ε , c 1 , c 3 , c 5 , h 1 } is defined by the PDC of

in

In conclusion, we presented an approach to obtain closed-form traveling wave solutions of the QCGLE. The central assumptions are restrictions on the dependence between ϕ ′ ( z ) and F ( z ) ( = f 2 ( z ) ) according to Equations (5) and (6). The solution F ( z ) is compactly represented by Equation (39) in terms of Weierstrass elliptic function ℘ ( z , g 2 , g 3 ) (disregarding the fact that ℘ is degenerating due to the vanishing discriminant of ℘ ). As a consequence, the phase function ϕ ( z ) can be represented in closed form analytically (see Equations (42) and (44)). The behavior of F ( z ) is studied by means of a phase diagram approach leading to conditions for “physical” (periodic, pulse-like, kink-like) solutions F ( z ) as well as to conditions for unbounded (“spiky”) solutions F ( z ) . In particular, we obtained the remarkable result that no bounded solution exists if the parameters of the QCGLE satisfy the condition 2 β 2 − 3 γ α < 0 , where α , β , γ are given by (37) or (38) and irrespective of F 0 > 0 . —The phase diagram approach is also suitable to investigate the parameter dependence of solutions (see Section 4).

Finally, we compare our approach with some other methods for getting solutions of the QCGLE.

1) The “simple technique” presented in [

2) Based on numerical simulations, extreme amplitude solutions of the QCGLE are reported in [

3) The particular relations (3.38 a,b), (3.51 a,b), (3.57 a,b) used in [

4) In [

5) In our estimation, even if we take into account recent publications ( [

6) For the cubic Ginzburg-Landau equation the non-existence of elliptic traveling wave solutions has been proved [

above if c 5 = h 5 = 0 . With ϕ ′ ( z ) = d F ″ ( z ) F ( z ) we obtained c = 0 necessarily, consistent with the proposition in [

Summing up, for both cases (5) and (6), our results are consistent with [

Directions in which further investigations can go should be indicated. First, it would be interesting to find more relations than (5) and (6) in order to increase the solution set of the QCGLE. Secondly, it would be important to find generalisations for ansatz (5) as well as for (6) by, for example, including a further parameter. Thirdly, if (5) and (6) are modified by “small corrections”, the solutions presented above may be taken as start solutions (different from solutions of the NLSE) for a perturbation approach. Finally, a stability analysis of the solutions found with respect to the parameters of the QCGLE as well as with respect to “small” perturbation of f seems possible.

The authors declare no conflicts of interest regarding the publication of this paper.

Schürmann, H.W. and Serov, V. (2021) Traveling Wave Solutions of the Quintic Complex One-Dimen- sional Ginzburg-Landau Equation. Applied Mathematics, 12, 598-613. https://doi.org/10.4236/am.2021.127043