1. IntroductionAs is well known, the nonlinear Schrödinger equation is a classical integrable equation. It can arise in diverse areas of physics, such as water wave, nonlinear optics, biophysics, Bose–Einstein condensates, and plasma physics.[1–5] There are a lot of studies about it, like the N-soliton, breather, and rogue wave solutions.[6–9] It is found that the nonlinear Schrödinger equation coupled to some other equations can yield new nonlinear equations,[10–13] one of which is the coupled NLS–Boussinesq system,
where
α,
β,
γ, and
δ are real constants, and
u is the complex function, and
N is the real function. If
, equation (
1) reduces to an NLS equation. Equation (
1) is Painlevé integrable when
.
[9,10] In the magnetized plasma, when the propagation speed of the low-frequency wave is near the magnetosonic speed,
u and
N in system (
1) are respectively the envelop of Langmuir wave (high-frequency wave response of the electrons) and the amplitude of the bidirectional propagation ion–acoustic wave (low-frequency wave response of the ions).
[8] The bright
N-soliton solution, Homoclinic orbit (Akhmediev breather) solution, a general breather and rogue wave solution of Eq. (
1) were studied previously in Refs. [
12], [
14–
16].
Recently, the investigation of multi-component nonlinear system has received much attention in some areas, that is, the vector NLS equation,[17–23] the coupled derivative NLS equation,[24] the multi-component Yajima–Oikawa system,[25] and so on. One has found that the multi-component nonlinear systems have indeed some novel dynamical behaviors, e.g., the two-component NLS (CNLS) equation, whose nonlinear localized wave solutions are more complex than the ones of NLS equation. It has been shown that the bright–bright solitons for the focusing CNLS equation undergo shape changing collisions with energy redistribution.[17–19] Only the focusing and mixed CNLS have rogue wave solutions,[20,21] and the de-focusing CNLS does not have bright–bright solitons.[20] Those inspire us to investigate the vector NLS–Boussinesq equation. In this paper, we focus mainly on the following two-component nonlinear Schrödinger equations coupled to the Boussinesq equation:
 | |
Our main aim is to investigate whether equation (
2) has more interesting properties than Eq. (
1). We will derive the bright–bright, bright–dark, and dark–dark soltion solutions for Eq. (
2). We will study the collision between the two solitons, and show that the collision between bright–bright solitons is inelastic in most of the cases. One interesting phenomenon will be described, i.e., the two solitons oscillating periodically can occur in the two parallel-traveling bright–bright or bright–dark soliton solutions. The general breather including Ma breather, Akhmediev breather, and rogue wave solution will be given.
2. Soliton solutions to Eq. (2)Set the transformations 
and 
, where gj are complex functions, but f is real. Then equation (2) is written as the following bilinear form:
where
λ is an integration constant to be determined, * denotes the complex conjugate.
2.1. Bright–bright soliton solutionWe restrict the power series expansions of f, gj about small parameter ϵ as
In the following, we set
λ = 0,
,
Pj,
ηj0 are arbitrary complex parameters, and denote
by
Substituting expansion (
4) into Eq. (
3) and solving the linear equations of the same power of
ϵ, one can obtain the one bright–bright soliton solution,
where
,
are arbitrary complex parameters, and the subscripts R and I denote the real and imaginary parts respectively. Either of
uj is a bright soliton wave, whose traveling speed is
.
reach their maximum values
at the line
, respectively.
When 
, there exist bright–bright two-soliton solutions 
and 
, where
and
To obtain the nonsingular solution, the denominator
f needs to be nonzero. The expression of
f can be rewritten as
where
. Thus when
, the solution is nonsingular.
Next, we will analyze the asymptotic behavior of the bright–bright 2-soliton.
(i) Along the line 
, and supposing 
, we have
If
, we have
(ii) Along the line 
, and supposing 
, we have
If
, we have
It is obvious that if
, then
, the solution reduces to two bright–bright solitons of Eq. (
1). If
,
u1 and
u2 in Eqs. (
6) and (
7) are just the bright–bright one- and two-soliton solutions of the coupled NLS system. For the bright–bright two-soliton solutions of Eq. (
2), we find that the collisions of the two bright solitons in
u1 and
u2 are inelastic, provided that
. There exists an exchange of energies between the two solitons during the collision. The phenomenon is also observed in the other two-component nonlinear systems.
[25–27] Figures
1(a) and
1(b) show the inelastic collision. But figures
1(d)–
1(e) describe the elastic collision with
. If one of
is zero, we can observe an interesting phenomenon that the energy of one soliton is completely absorbed by the other one when they collide (see Fig.
2(b) where
). In Figs.
2(d) and
2(e), we can see that if
or
, only one soliton with a shift can be observed in both components
u1 and
u2. All the energy of one soliton is gathered in
u1, the other one is gathered in
u2, and the collision is elastic. It is interesting to note that when
, the two solitons travel along a parallel line. If
, the two solitons oscillate periodically (see Fig.
3). The periodic phenomenon is caused by the periodicities of the real and imaginary parts of both solitons. Since the phase of the soliton is determined by
, and
, the collision period
T can be expressed as
.
2.2. Bright–dark soliton solutionOne can check that equation (2) has an exact solution 
, 
, 
, where k2 and ω2 are real and 
. We set a transformation 
, and we still use h to denote 
. Then the bilinear form Eq. (3) becomes
To obtain bright–dark soliton solutions, we suppose
Substituting Eq. (
14) into Eq.(
13) and solving the linear equations of the same power of
ϵ, one can obtain the one bright–dark soliton solution:
,
,
, where
,
, and
, with
When 
, the solution is nonsingular and can be written as
where
and
. It is obvious that
is always below the height
ρ2 of the background plane wave, and reaches its minimum value
at the line
.
When 
, we obtain the bright–dark two-soliton 
, 
, and 
, where
and
Next, we will analyze the asymptotic behavior of the bright–dark two-soliton.
i) Along the line 
and supposing 
, we have
If
,
ii) Along the line 
, if 
, we have
If
,
One can observe that the intensities of the two solitons before and after interactions are the same (elastic) both for bright soliton in u1 and dark soliton in u2 (see Fig. 4). In the expression of u1, the background parameter ρ2 has some influences on the amplitudes of the bright soliton. But by calculations, it is shown that the bright-component parameter aj has no influence on the amplitude 
nor the phase shift 
of the dark soliton during collisions. Thus, aj parameters in the bright soliton solution do not affect the collision of the dark soliton. Like the bright–bright solitons, when 
, the two solitons travel in parallel, and the periodical appearance of the two-soliton shape is shown in Fig. 5.
2.3. Dark–dark soltion solutionEquation (2) has the following plane wave solutions:
if
. Using the transformations
and
, where
gj are complex functions, and
f is a real function, system (
2) is written as
To obtain the dark–dark soliton solution, we suppose
Substituting these into Eq. (
24) and solving the linear equations of the same power of
ϵ, we obtain the one dark–dark soliton solution:
where
,
and
ηj0 are real, but
are complex parameters, and
The frequency
and wave number
Pj satisfy the following equation:
We can see that
and
are always smaller than the heights of their background plane
ρ1 and
ρ2, respectively. They reaches their minimal values
and
at line
.
When 
, the dark–dark two-soliton can be given by
where
and
satisfy Eq. (
28).
Now we discuss the asymptotic behavior of the dark–dark two-soliton.
(I) Along the line 
, and supposing 
, we have
If
,
(II) Along the line 
, and 
, we have
If
,
where
. It is obvious that the collisions of the two dark–dark soliton solutions are elastic (see Figs.
6(a)–
6(c)). But we should notice that the solutions
u1 and
u2 indeed interrelate with each other. The relationship (
28) shows the fact that the background solution
has influences on the amplitude, the wave velocity, and the phase of the soliton solution.
If the speed of wave 
, 
, and Pj 
satisfy the equation,
then there are two parallel-traveling dark–dark solitons. Because
η1 and
η2 are both real, in this case there are not two periodically parallel-travelling dark–dark solitons, which is different from the case of two parallel-travelling bright–bright solitons (see Figs.
6(d)–
6(f)).
As the application of Eq. (1) in an electron–ion plasma, equation (2) has physical application in the electron–ion plasma, where the high-frequency wave of the electrons has two types of electronic oscillations u1 and u2. The 
and 
represent the intensities of the waves, respectively. The term bright (dark) means that the intensity of the wave is always higher (lower) than the one of background plane wave. It should be pointed out that the components u1 and u2 in all of the solutions can degenerate to the solutions of the CNLS system by making 
. The CNLS system has important applications in two-mode nonlinear optical fiber and Bose–Einstein condensate.
2.4. Breather and rogue wave solutions to Eq. (2)The procedure of deriving the breather solution is similar to the case of dark–dark two-soliton. In the expression of dark–dark two-soliton solution Eq. (29), let Pj, 
, ηj0 be complex and 
, 
, then we will obtain the general breather solutions to Eq. (2). It is obvious that the condition 
gives the nonsingular general breather solutions, which can be written in terms of trigonometric and hyperbolic functions as
where
,
,
,
, and
. In Fig.
7, for different values of the parameters, the general breather corresponding to shape
u1 or
u2 has three or four extreme points in one period.
Considering the following equations:
then we have the stationary points,
We classify these points as saddles or extreme points according to their second order derivatives. When
n = 0, if it is a saddle point, then the next saddle point is
n = 2. So the distance between these two points at
x and
t is respectively,
The Akhmedev breather solution(see Figs.
8(a)–
8(c)) which occurs due to the modulation instability
[27] can be obtained by
. Under this condition, the general breather solution (
29) is periodic at
t and localized at
x. If we restrict the imaginary part of the wave number
to 0, i.e.,
, then the general breather solution (
29) is periodic at
x and localized at
t, which is the Ma breather (see Figs.
8(e)–
8(f)).
As well know, the rogue wave can be derived from the general breather solution. Suppose 
, a complex small parameter, then we will have
where
satisfies
If 
, then 
, and
Substituting the above formulae into the expressions of
f,
gj,
fxx,
fx yields
Taking the limit
, and letting
, where
m and
n are real, the expression of the rational solution is written as
It should be noted that condition (
36) is rewritten in the following form
We thus see that for a given initial plane-wave, one can determine
m,
n from Eqs. (
38) and (
39), and then determine the rational solution (
37). Furthermore, we can classify the rogue wave solution as four different types by the dynamical behavior. Solving the following
gives the stationary points,
where
When 
, there are four extreme points, otherwise there are three extreme points. The height of the central point is
So we can classify the rogue wave as four different types as follows.
I) If 
, there are only three stationary points. The central point is a minimal one, and it is lower than the background. In this case, a dark rogue wave is derived.
II) If 
, there are five stationary points. The central point is a saddle one, and it is lower than the background, which is called four-petal rogue wave.
III) If 
, there are five stationary points. The central point is a saddle one, and it is higher than the background. So a two-peak rogue wave is obtained.
IV) If 
, there are only three stationary points. The central point is a maximal one, and its height is at least twice higher than the background, which is called bright rogue wave.
While for Boussinesq equation 
in Eq. (37), its structure is simple. The three stationary points for 
are
and the value of
at the central(minimum) point
is
, which is always lower then the background. The maximum points occur at
, and their values are both
. so the Boussinesq equation
in Eq. (
37) has two small peaks and one deep hole in the center between two peaks, which is just the usual dark rogue wave.
Figure 9 shows the rogue wave with two sets of parameters. In Fig. 9(b) 
and in Fig. 9(d) 
are the rogue waves with four petals, in Fig. 9(a) 
is a bright rogue wave, in Fig. 9(e) 
is a dark rogue wave. Actually, for different values of δ1 and δ2, we can always find appropriate values for the left parameters to make the rogue waves of u1 and u2 are different combinations of bright, dark and four-petal rogue waves. It should be noted that the rogue wave solutions for NLS equation, coupled NLS equation, (2+1)-dimensional NLS equation have been investigated in Refs. [9], [28–35].
