
Citation: Hiroshi Takahashi, Ken-ichi Yoshihara. Approximation of solutions of multi-dimensional linear stochastic differential equations defined by weakly dependent random variables[J]. AIMS Mathematics, 2017, 2(3): 377-384. doi: 10.3934/Math.2017.3.377
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We consider the following nonlinear partial differential equation:
iQt+Qxx+αQ+β|Q|nQ+γ|Q|2nQ+δ|Q|3nQ+λ|Q|4nQ=0, | (1.1) |
where Q(x,t) is a complex function representing the wave amplitude, x is the coordinate, t is the time, n is a rational number indicating the nonlinearity order, and α,β,γ,δ,λ are related to the dispersion and nonlinearity of the medium. We consider the generalized nonlinear Schrödinger equation (1.1) due to its broad applicability in describing pulse propagation in nonlinear optical fibers. This equation extends the standard nonlinear Schrödinger equation by incorporating higher-order nonlinear terms, providing a more accurate model for complex optical systems (see [11,12,13,26] for example). There are many studies about the generalized nonlinear Schrödinger equation; see references in [5,6,10,17,19]. Elsonbaty et al. [5] considered a newly generalized nonlinear Schrödinger equation with a triple refractive index and non-local nonlinearity. In order to obtain optical solitons for a specific case of this innovative model, they employed the improved modified extended tanh function method. And they derived various solutions, including bright solitons, dark solitons, singular solitons, singular periodic solutions, trigonometric solutions, and hyperbolic solutions. Wang and Yang studied [19] a generalized nonlinear Schrödinger equation by constructing the modified generalized Darboux transformation. They analyzed the type-Ⅰ, type-Ⅱ, and type-Ⅲ degenerate solitons for the equation by some semirational solutions. In particular, letting n=2,β≠0,α=γ=δ=λ=0, Eq (1.1) becomes the famous nonlinear Schrödinger equation. In [11], the author proposed a more generalized equation for describing pulse propagation in optical fiber in the form
iQt+Qxx+αQ+βQ|Q|2m−2n+γQ|Q|2m−n+δQ|Q|2m+n+λQ|Q|2m+2n=0, | (1.2) |
where m,n are rational numbers.
In fact, (1.1) is a special case of (1.2) for the case m=n. In addition, letting m=0, (1.2) can be written as follows:
iQt+Qxx+αQ+βQ|Q|−2n+γQ|Q|−n+δQ|Q|n+λQ|Q|2n=0, | (1.3) |
which was introduced in paper [9] and has been widely investigated. However, Eq (1.3) cannot be implemented in practice according to physicists on account of the negative degree terms. Therefore, Eq (1.2) for m≠0 is better than Eq (1.1) for describing the propagation of various types of pulses in an optical fiber. In paper [11], some solutions of Eq (1.2) for two cases, m=n and m=2n, were obtained by applying the variable transformation method.
In addition, Eq (1) generalizes several equations describing propagation pulses in nonlinear optics (see [1,2,7,8,20,22] for example). In [10], the author derived the implicit solitary wave solutions of (1.1) through transformations of variables. However, it is worth noting that there may be some potential errors affecting the correctness of the results in [10].
We hope to find solutions to Eq (1.1) as follows:
Q(x,t)=ϕ(ξ)ei(κx−ωt), ξ=x−vt, | (1.4) |
where κ and ω represent real-valued constants, ξ denotes the wave variable, and ϕ(ξ) stands for the amplitude component. Substituting (1.4) into (1.1), dividing by the complex exponential function ei(κx−ωt), and separating the real and imaginary parts, one obtains two ordinary differential equations
ϕ′′+(−κ2+α+ω)ϕ+βϕn+1+γϕ2n+1+δϕ3n+1+λϕ4n+1=0, | (1.5) |
and
(2κ−v)ϕ′=0, | (1.6) |
where ′ represents differentiation with respect to ξ. Obviously, (1.6) implies that v=2κ.
Let b=−κ2+α+ω. Then (1.5) is equivalent to the following system:
dϕdξ=y, dydξ=−(bϕ+βϕn+1+γϕ2n+1+δϕ3n+1+λϕ4n+1), |
which has a first integral of the form
H0(ϕ,y)=y2+bϕ2+2βn+2ϕn+2+γn+1ϕ2n+2+2δ3n+2ϕ3n+2+λ2n+1ϕ4n+2=h. | (1.7) |
According to [4,21,24], we can make the following transformation:
ϕ=ψ−1n. | (1.8) |
Noting
ϕ′′=1n(1n+1)ψ−1n−2(ψ′)2−1nψ−1n−1ψ′′, | (1.9) |
and substituting (1.8) and (1.9) into (1.5), then one has
ψ3ψ′′−n(bψ4+βψ3+γψ2+δψ+λ)−(1+1n)ψ2(ψ′)2=0. | (1.10) |
Equation (1.10) is equivalent to the planar dynamical system as follows:
dψdξ=y, dydξ=(1+1n)ψ2y2+n(bψ4+βψ3+γψ2+δψ+λ)ψ3, | (1.11) |
where ψ represents the transformed amplitude component, ξ is the wave variable, and n is the nonlinearity order. It is easy to see that system (1.11) has a first integral of the form
H(ψ,y)=y2ψ−2n+2n+n2ψ−4n+2n(λ2n+1+2δ3n+2ψ+γn+1ψ2+2βn+2ψ3+bψ4)=h, | (1.12) |
for n≠−12,−23,−1,−2. If n=−12,−23,−1,−2, then H(ψ,y) contains a term of ln(⋅); we omit them.
Remark 1. In [10], the author did not derive system (1.11). If we only consider H0(ϕ,y)=h given by (1.7), then the transformation (1.8) makes (1.7) become
ψ2(ψ′)2+n2(bψ4+2βn+2ψ3+γn+1ψ2+2δ3n+2ψ+λ2n+1−hψ4+2n)=0. | (1.13) |
Clearly, (1.13) is different from (1.12).
The author of [10] supposed that the first integral of the equation derived by his transformation was
(ψ′)2+(−κ2+α+ω)n2ψ4+2n2βn+2ψ3+n2γn+1ψ2+2n2δ3n+2ψ+n2λ2n+1=0, | (1.14) |
i.e., the formula (12) on page 2 of his paper. As we mentioned above in (1.12), this first integral is incorrect for the Eq (1.11). Therefore, the results in his paper [10] need to be corrected. Actually, they did not derive and study the traveling wave system (1.11). Similar mistakes appeared in Rogers et al. [18] and Zayed et al. [23]. Their results had been corrected by Zhou et al. [27].
We need to say that for an integrable planar differential system, if we make a variable transformation like (1.8), we must derive a new equation with respect to the new variable, and we have to find the new first integral.
System (1.11) is a six-parameter system that relies on the parameter set (b,n,β,γ,δ,λ). It has very abundant dynamical behavior. Furthermore, as defined in [14] and [15], it is classified as the first kind of singular traveling wave system, which has the singular straight line ψ=0. In recent years, many researchers have employed the dynamical systems approach to explore the dynamical behaviors of solutions for the first kind of singular systems and to provide possible parametric representations of solutions (see [16,25] for example).
In this paper, we first assume that δ=λ=0, n=−13, and n=2. We analyze the bifurcations of phase portraits of system (1.11) when there exist two real zeros of f(ψ)=bψ2+βψ+γ. Then, we calculate all possible exact explicit peakons, periodic peakons, smooth periodic solutions, as well as homoclinic orbits of system (1.11). The exact parametric representations of these solutions are presented.
The article is organized as follows: In Section 2, we analyze the bifurcations of phase portraits for (1.11). In Sections 3 and 4, we derive all possible exact explicit parametric representations for some bounded solutions of (1.11). In Section 5, we state the main results of this paper.
When δ=λ=0, system (1.11) is reduced to
dψdξ=y, dydξ=(1+1n)y2+n(bψ2+βψ+γ)ψ. | (2.1) |
The associated system of (2.1) can be written as
dψdζ=yψ, dydζ=(1+1n)y2+n(bψ2+βψ+γ), | (2.2) |
where dξ=ψdζ.
Suppose that Δ=β2−4bγ≥0. Then (2.1) has two equilibrium points E1(ψ1,0) and E2(ψ2,0), where ψ1,2=12b(−β∓√Δ) with ψ1<ψ2.
When (n+1)γ<0, we have Ys=−n2γn+1>0. Then on the straight line ψ=0, system (2.2) has two equilibrium points S∓(0,∓√Ys).
Let M(ψj,0) represent the coefficient matrix for the linearized system of system (2.2), at the point Ej(ψj,0),j=1,2. Let J(⋅,⋅) be their Jacobian determinants. Then
J(ψj,0)=−n(2bψj+β)ψj,j=1,2,J(0,∓√Ys)=2(1+1n)Ys. |
By the theory of planar dynamical systems, for an equilibrium point of a planar integrable system, if J<0, then the equilibrium point is a saddle point; if J>0 and (Trace(M(ψj,0)))2−4J(ψj,0)<0, then it is a center point; if J>0 and (Trace(M(ψj,0)))2−4J(ψj,0)>0, then it is a node; if J=0 and the Poincaré index of the equilibrium point is 0, then it is a cusp.
For H defined by (1.12), we write that hj=H(ψj,0),j=1,2 and hs=H(0,√Ys).
As two examples, we take n=−13 and n=2. Utilizing the aforementioned information for qualitative analysis, we obtain the bifurcations of phase portraits of system (2.1), which are displayed in Figures 1 and 2.
In this case, we know that Ys>0 and J(0,∓√Ys)<0. Therefore, the equilibrium points S∓ are two saddle points. Let bp:=β24γ. Then the function F(ψ)=bψ2+βψ+γ has two different real zeros for bp<b<0 or b>0, which implies system (2.1) has two equilibrium points. Based on the above results, by varying b, we get the bifurcations of phase portraits of (2.1), which are displayed in Figure 1. In the case of γ>0, we do not consider it here.
When γ<0, noting that Ys>0,J(0,∓√Ys)>0, and (Trace(M(0,∓√Ys)))2−4J(0,∓√Ys)>0, one finds that S∓ are two node points. When γ>0, there are no equilibrium points on the singular straight line ψ=0 because Ys<0. By varying b such that Δ>0, i.e., bp<b<0 or b>0 when γ<0 (0<b<bp or b<0 when γ>0, respectively), we obtain the bifurcations of phase portraits of (2.1), which are displayed in Figure 2.
In this section, some parametric representations for the bounded orbits in Figure 1 are derived. In this case, we have the parameter condition n=−13 and β>0,γ<0. We see from (1.12) that y2=hψ4−19(bψ2+65βψ+32γ). By using the first equation of (2.1), one has
ξ=∫ψψ0ψ2dψ√|h−19ψ4(bψ2+65βψ+32γ)|. | (3.1) |
For some orbits shown in Figure 1, if the integral of the right-hand side of (3.1) can be calculated, then we can obtain their parametric representations.
Corresponding to the closed orbit defined by H(ψ,y)=0 in Figure 1(d), enclosing the equilibrium points Ej(ψj,0),j=1,2, and passing through the singular straight line ψ=0, (3.1) can be written as
√b3ξ=∫ψ0dψ√(ψM−ψ)(ψ−ψm), |
where ψM,ψm are determined by
|ψ2+6β5bψ+3γ2b|=(ψM−ψ)(ψ−ψm) |
with ψm<ψ<ψM. Thus, it follows that the parametric representation of a periodic solution (see Figure 3(a)):
ψ(ξ)=12[(ψM−ψm)sin(√b3ξ−ξ01)+(ψM−ψm)], | (3.2) |
where ξ01=arcsin(ψM+ψmψM−ψm).
Corresponding to the arch orbit defined by H(ψ,y)=0 in Figure 1(c), enclosing the equilibrium point E2(ψ2,0), (3.1) can be represented as
√|b|3ξ=∫ψMψdψ√(ψL−ψ)(ψM−ψ), |
where ψM,ψL are determined by
|ψ2+6β5bψ+3γ2b|=(ψL−ψ)(ψM−ψ) |
with ψ<ψM<ψL. Thus, we obtain the following parametric representation of a periodic peakon solution (see Figure 3(b)):
ψ(ξ)=12[−(ψL−ψM)cosh(√|b|3ξ)+(ψL+ψM)], ξ∈(−ξ02,ξ02), | (3.3) |
where ξ02=3√|b|cosh−1(ψL+ψMψL−ψM).
Corresponding to the triangle orbit defined by H(ψ,y)=0 in Figure 1(b), enclosing the singular point E1(ψ1,0), (3.1) can be represented as
√|b|3ξ=∫ψ0dψψ2−ψ. |
It generates the parametric representation of an anti-peakon solution (see Figure 3(c)) as follows:
ψ(ξ)=ψ2(1−e−√|b|3|ξ|). | (3.4) |
Corresponding to the homoclinic orbits to the singular point E2(ψ2,0) defined by H(ψ,y)=h2 in Figure 1(a), enclosing the singular point E1(ψ1,0), (3.1) can be represented as
√|b|3ξ=∫ψψmψ2dψ(ψ2−ψ)√(ψ−ψm)(ψ−ψl)[(ψ−b1)2+a21], |
where ψm,ϕl,b1,a1 are determined by
|9h2b−ψ4(ψ2+6β5bψ+32br)|=(ψ2−ψ)√(ψ−ψm)(ψ−ψl)[(ψ−b1)2+a21] |
with ψl<ψm<ψ<ψ2. It generates the parametric representation of a homoclinic orbit (solitary wave) as follows (see Figure 3(d))
ψ(χ)=ψlA1−ψmB1−(ψmB1+ψlA1)cn(χ,k)(A1−B1)−(A1+B1)cn(χ,k), χ∈(−χ01,χ01),ξ(χ)=3√|b|[−gψ2χ+∫ψψmψdψ√(ψ−ψm)(ψ−ψl)[(ψ−b1)2+a21]+ψ22∫ψψmdψ(ψ2−ψ)√(ψ−ψm)(ψ−ψl)[(ψ−b1)2+a21]], | (3.5) |
where
A21=(ψm−b1)2+a21,B21=(ψl−b1)2+a21,k2=(A1+B1)2−(ψm−ψl)24A1B1,g=1√A1B1,χ01=cn−1(ψ2(A1−B1)−(ψlA1−ψmB1)ψ2(A1+B1)−(ψmB1+ψlA1)), |
sn(⋅,k),cn(⋅,k),dn(⋅,k) are the Jacobian elliptic functions (see Byrd and Fridman [3]). In the right hand of ξ(χ), the formulas of two integrals are too longer; we omit them.
In this section, parametric representations of all bounded orbits in Figure 2 are derived. In this case, we have the parameter conditions: n=2 and β>0,β2>4bγ. We see from (1.12) that y2=hψ3−4(bψ2+12βψ+13γ). By using the first equation of (2.1), we have
ξ=∫ψψ0dψ√|hψ3−4(bψ2+12βψ+13γ)|. | (4.1) |
(i) Corresponding to the periodic orbit family defined by H(ψ,y)=h,h∈(h2,0) in Figure 2(a), enclosing the singular point E2(ψ2,0), (4.1) can be represented as
√|h|ξ=∫ψψbdψ√(ψa−ψ)(ψ−ψb)(ψ−ψc), |
where ψa,ψb,ψc are determined by
|ψ3−4h(bψ2+12βψ+13γ)|=(ψa−ψ)(ψ−ψb)(ψ−ψc),h∈(h2,0) |
with ψa>ψ>ψb>ψc. It follows that the parametric representation of the right family of periodic orbits
ψ(ξ)=ψc+ψb−ψcdn2(12√|h|(ψa−ψc)ξ,k), | (4.2) |
where
k2=ψa−ψcψa−ψc. |
(ⅱ) Corresponding to the periodic orbit family given by H(ψ,y)=h,h∈(0,h1) in Figure 2(a), enclosing the singular point E1(ψ1,0), (4.1) can be represented as
√hξ=∫ψψcdψ√(ψa−ψ)(ψb−ψ)(ψ−ψc), |
where ψa,ψb,ψc are determined by
|ψ3−4h(bψ2+12βψ+13γ)|=(ψa−ψ)(ψ−ψb)(ψ−ψc),h∈(0,h1) |
with ψa>ψb>ψ>ψc. It follows that the parametric representation of the left family of periodic orbits
ψ(ξ)=ψc+(ψb−ψc)sn2(12√h(ψa−ψc)ξ,k), | (4.3) |
where
k2=ψb−ψcψa−ψc. |
(ⅰ) Corresponding to the periodic orbit family given by H(ψ,y)=h,h∈(h1,h2) in Figure 2(b), enclosing the singular point E2(ψ2,0), it has the same parametric representation as (4.2).
(ⅱ) Corresponding to the homoclinic orbit to the singular point E1(ψ1,0) given by H(ψ,y)=h1 in Figure 2(b), enclosing the singular point E2(ψ2,0), (4.1) can be represented as
√|h1|ξ=∫ψMψdψ(ψ−ψ1)√ψM−ψ, |
where ψM is determined by
|ψ3−4h1(bψ2+12βψ+13γ)|=(ψ−ψ1)√ψM−ψ, |
with ψ1<ψ<ψM. Thus, we have the following solitary wave solution:
ψ(ξ)=ψ1+(ψM−ψ1)sech2(12√|h1|(ψM−ψ1)ξ). | (4.4) |
In this case the points S∓ are node points of system (2.2). Now, the changes of the level curves given by H(ψ,y)=h are shown in Figure 4.
(ⅰ) Corresponding to the periodic orbit family defined by H(ψ,y)=h,h∈(h2,0) in Figure 4(a), enclosing the singular point E2(ψ2,0), it has the same parametric representation as (4.2).
(ⅱ) Corresponding to the periodic orbit family given by H(ψ,y)=h,h∈(0,h1) in Figure 4(c), passing through the singular straight line ψ=0 at S∓, it has the same parametric representation as (4.3).
(ⅲ) Corresponding to the homoclinic orbit to the singular E1(ψ1,0) given by H(ψ,y)=h1 in Figure 4(d), passing through the singular straight line ψ=0 at S∓, now, (4.1) can be written as
√h1ξ=∫ψψmdψ(ψ1−ψ)√ψ−ψm, |
where ψm is determined by
|ψ3−4h1(bψ2+12βψ+13γ)|=(ψ1−ψ)√ψ−ψm, |
with ψm<ψ<ψ1. Thus, one obtains the following solitary wave solution:
ψ(ξ)=ψ1−(ψ1−ψm)sech2(12√h1(ψ1−ψm)ξ). | (4.5) |
In this case, the changes of the level curves given by H(ψ,y)=h are displayed in Figure 5.
(ⅰ) Corresponding to the homoclinic orbit to the singular point E1(ψ1,0) given by H(ψ,y)=h1 in Figure 5(a), passing through the singular straight line ψ=0 at S∓, it has the same parametric representation as (4.4).
(ⅱ) Considering the periodic orbit family defined by H(ψ,y)=h,h∈(h1,0) in Figure 5(b), which intersects the singular straight line ψ=0 at S∓, it has the same parametric representation as (4.2).
(ⅲ) Considering the periodic orbit family defined by H(ψ,y)=0 in Figure 5(c), which intersects the singular straight line ψ=0 at S∓, it has the same parametric representation as (3.2).
(ⅳ) Corresponding to the periodic orbit family given by H(ψ,y)=h,h∈(0,h2) in Figure 5(d), passing through the singular straight line ψ=0 at S∓, it has the same parametric representation as (4.3).
(ⅴ) Corresponding to the homoclinic orbit to the singular point E2(ψ2,0) given by H(ψ,y)=h2 in Figure 5 (e), passing through the singular straight line ψ=0 at S∓, now, (4.1) can be written as
√h2ξ=∫ψψmdψ(ψ2−ψ)√ψ−ψm, |
where ψm is determined by
|ψ3−4h2(bψ2+12βψ+13γ)|=(ψ2−ψ)√ψ−ψm, |
with ψm<ψ<ψ2. Thus, we have the following solitary wave solution
ψ(ξ)=ψ2−(ψ2−ψm)sech2(12√h2(ψ2−ψm)ξ). | (4.6) |
In summary, this paper provides a comprehensive analysis of the generalized nonlinear Schrödinger equation by transforming it into a planar dynamical system. We identified several distinct parametric representations for the solutions, including solitary waves, periodic waves, peakons, and periodic peakons. These findings not only enrich the theoretical framework of nonlinear wave propagation but also offer practical guidance for designing optical systems with tailored wave characteristics. Future work will focus on extending this analysis to more complex nonlinear models and exploring their applications in other physical systems.
We will list the main conclusions as follows.
Theorem 1. (1) For the generalized nonlinear Schrödinger equation (1.1), to find the exact explicit solutions with the form q(x,t)=(ψ(ξ))−1nei(κx−ωt), ξ=x−2κt, the amplitude component ϕ(ξ)=(ψ(ξ))−1n satisfies the planar dynamical system (1.11) with respect to ψ(ξ) and has the first integral (1.12).
(2) Assume that δ=λ=0 and n=−13,n=2, respectively. Under different parametric conditions, system (1.11) has the bifurcations of phase portraits, which are shown in Figures 1 and 2.
(3) For n=−13,n=2, system (1.11) has 8 exact explicit parametric representations given by (3.2)–(3.5), and (4.2)–(4.5). The homoclinic orbits give rise to solitary wave solutions of Eq (1.11) with the parametric representations given by (3.5), (4.4), and (4.5). For n=2, the periodic orbit families give rise to periodic wave solutions of Eq (1.11) with the parametric representations (4.2) and (4.3).
(4) Specially, when n=−13, system (1.11) has a periodic solution, a periodic peakon solution, and an anti-peakon solution with parametric representations given by (3.2), (3.3), and (3.4).
Qian Zhang: Writing-original draft, Conceptualization, Data curation, Formal analysis; Ai Ke: Writing-review & editing, Methodology, Software, Validation. All authors have read and approved the final version of the manuscript for publication.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The first author is supported by the Sichuan Science and Technology Program (Grant No. 2024NSFSC1396) and the National Natural Science Foundation of China (12401241). The second author is supported by the National Natural Science Foundation of China (12301214).
The authors declare that they have no conflict of interest.
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