CN103199926A - Complex domain multi-soliton implementation method - Google Patents

Abstract

The invention relates to a complex domain multi-soliton implementation method and belongs to the technical field of fiber optical communication. Topology and nonlinear transformation are introduced, a nonlinearity rupture equation is researched through a functional transformation method, and a number of exact solutions of a variable separation form are formed. Arbitrary function in the exact solutions is properly set, a new oscillation soliton structure is obtained, and multi-soliton with the different numbers is obtained. The problem of difficult generation of the multi-soliton in the fiber optical communication technology is solved. The complex domain multi-soliton implementation method has the advantages of being concise, easy to understand, convenient to implement, strong in practicability, and capable of conducting adjustment of relevant parameters according to actual condition, providing powerful support for intensive study of the field of a fiber optical communication system, and promoting development of the fiber optical communication.

Description

One many orphans of class complex domain implementation method

Technical field

The present invention relates to many orphans of class complex domain implementation method, belong to technical field of optical fiber communication.

Background technology

It is important research content in the non-linear science that orphan's structure excites, if contain the arbitrary function of relevant independent variable in the solution of nonlinear physics equation, by suitably choosing arbitrary function, can excite abundant Local Structure, and these Local Structure can be explained some nonlinear physics phenomenon, because dimension limit in the nonlinear equation, the solution that contains arbitrary function that obtain the low-dimensional equation is very difficult.

In the research for the variable coefficient non linear system, converter technique is class effective ways of structure nonlinear mathematics physical equation exact solution, to the nonlinear transformation of arbitrary function, and construct the accurate off-the-line of some non linear systems, thereby become the strong instrument of research local excitation structure.The research of different many orphans of number production methods is extremely important for the further investigation of fiber optic communication field, yet, generally, being difficult to analysis research, this seriously limits and has hindered corresponding subject development.

Summary of the invention

The method that the present invention proposes has solved the problem that the many orphans of complex domain are difficult to produce in the Fibre Optical Communication Technology, the method that the present invention proposes is understandable, realization convenience, practical succinctly, can carry out the adjustment of relevant parameter according to actual conditions, provide powerful support for for the further investigation in optical fiber telecommunications system field provides, will promote this subject development.

The technical solution adopted for the present invention to solve the technical problems is:

Study the relevant issues that many orphans produce for convenience, the present invention introduces nonlinear equation:

iu _t-u _xx-2(|u| ²-ω)u-iα(|u| ²u) _x-β|u| ⁴u＝0 (1)

In the formula, (x is about (ω, α, β are the non-zero real numbers for x, complex domain function t) t) to u=u.

The function of getting capable waveshape is

In the formula, ξ=x-ct+ ξ ₀,

V (ξ) is real function, ξ ₀Be arbitrary constant, c, θ, k are undetermined constant.

Equation (2) substitution equation (1) formula, by conversion, its real part and imaginary part are respectively:

And then defined function

Equation (5) substitution equation (4), v (ξ) ⁱAll coefficients equal zero, can obtain a ₀=-c/2-k, a ₂=-3 α/4, a ₁=a ₃=...=0, and

(6) formula substitution (3), can obtain

$v^{\&prime;\&prime;}-(\mathrm{\&theta;}+2\mathrm{\&omega;}-\mathrm{ck}-\frac{{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HSA00000856646400011.png"\; state="\{\{state\}\}">c</figure-callout>}}^2}{4})v+(2+\frac{\mathrm{c\&alpha;}}{2})v^3+(\mathrm{\&beta;}+\frac{3}{16}{\mathrm{\&alpha;}}^2)v^5=0---\left(7\right)$

(7) formula is carried out integration, and multiply by v ', can obtain:

${\left(v^{\&prime;}\left(\mathrm{\&xi;}\right)\right)}^2=c_0+(\mathrm{\&theta;}+2\mathrm{\&omega;}-\mathrm{ck}-{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HSA00000856646400011.png"\; state="\{\{state\}\}">c</figure-callout>}}^2/4)v^2-\frac{1}{2}(2+\mathrm{c\&alpha;}/2)v^4-\frac{1}{3}(\mathrm{\&beta;}+\frac{3}{16}{\mathrm{\&alpha;}}^2)v^6---\left(8\right)$

The solution that can get equation is

$u_1(x,t)=e^{i(\mathrm{kx}-\mathrm{\&theta;}-(c/2+k)(x-\mathrm{ct}+{\mathrm{\&xi;}}_0)+\frac{3\sqrt{3}\mathrm{\&alpha;}}{\mathrm{eB}}\arctan \left(\frac{\sqrt{3}}{3}\tanh \left(\frac{\mathrm{eA}}{2B}(x-\mathrm{ct}+{\mathrm{\&xi;}}_0)\right)\right)}\&times;{\left(\frac{1-\tanh \left(\frac{\mathrm{eA}}{2B}(x-\mathrm{ct})\right)}{1+\tanh \left(\frac{\mathrm{eA}}{2B}(c-\mathrm{ct})\right)}\right)}^{i\frac{3\mathrm{\&alpha;}}{2\mathrm{eB}}}{\left(\frac{8A\tanh \left(\frac{\mathrm{eA}}{2B}(x-\mathrm{ct})\right)}{B^2(3+\tan h^2\left(\frac{\mathrm{eA}}{2B}(x-\mathrm{ct})\right))}\right)}^{1/2}---\left(9\right)$

Work as parameter A, when B, c, θ, k, e, α chose different numerical value, nonlinear equation (1) just can produce many orphans structure.

The invention has the beneficial effects as follows: the method that the present invention proposes has solved the problem that many orphans are difficult to produce in the Fibre Optical Communication Technology, the method that the present invention proposes is understandable, realization convenience, practical succinctly, can carry out the adjustment of relevant parameter according to actual conditions, provide powerful support for for the further investigation in optical fiber telecommunications system field provides, will promote this subject development.

Description of drawings

Fig. 1 is the present invention (2 * 4) many orphans structure chart (x ∈ [2,2], t ∈ [3 ,-1]);

Fig. 2 is many orphans of the present invention structure chart (x ∈ [3,3], t ∈ [6,0] U=Reu (t, x));

Embodiment

The technical solution adopted for the present invention to solve the technical problems is:

Study the relevant issues that the many orphans of complex domain produce for convenience, the present invention introduces nonlinear equation:

iu _t-u _xx-2(|u| ²-ω) _u-iα(|u| ²u) _x-β|u| ⁴u＝0； (1)

In the formula, (x is about (ω, α, β are the non-zero real numbers for x, complex domain function t) t) to u=u.

The function of getting capable waveshape is

In the formula, ξ=x-ct+ ξ ₀,

V (ξ) is real function, ξ ₀Be arbitrary constant, c, θ, k are undetermined constant.

Equation (2) substitution equation (1) formula, by conversion, its real part and imaginary part are respectively:

And then defined function

Equation (5) substitution equation (4), v (ξ) ⁱAll coefficients equal zero, can obtain a ₀=-c/2-k, a ₂=-3 α/4, a ₁=a ₃=...=0, and

(6) formula substitution (3), can obtain

$v^{\&prime;\&prime;}-(\mathrm{\&theta;}+2\mathrm{\&omega;}-\mathrm{ck}-\frac{{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HSA00000856646400011.png"\; state="\{\{state\}\}">c</figure-callout>}}^2}{4})v+(2+\frac{\mathrm{c\&alpha;}}{2})v^3+(\mathrm{\&beta;}+\frac{3}{16}{\mathrm{\&alpha;}}^2)v^5=0---\left(7\right)$

(7) formula is carried out integration, and multiply by v ', can obtain:

${\left(v^{\&prime;}\left(\mathrm{\&xi;}\right)\right)}^2=c_0+(\mathrm{\&theta;}+2\mathrm{\&omega;}-\mathrm{ck}-{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HSA00000856646400011.png"\; state="\{\{state\}\}">c</figure-callout>}}^2/4)v^2-\frac{1}{2}(2+\mathrm{c\&alpha;}/2)v^4-\frac{1}{3}(\mathrm{\&beta;}+\frac{3}{16}{\mathrm{\&alpha;}}^2)v^6---\left(8\right)$

The solution that can get equation is

$u_1(x,t)=e^{i(\mathrm{kx}-\mathrm{\&theta;}-(c/2+k)(x-\mathrm{ct}+{\mathrm{\&xi;}}_0)+\frac{3\sqrt{3}\mathrm{\&alpha;}}{\mathrm{eB}}\arctan \left(\frac{\sqrt{3}}{3}\tanh \left(\frac{\mathrm{eA}}{2B}(x-\mathrm{ct}+{\mathrm{\&xi;}}_0)\right)\right)}\&times;{\left(\frac{1-\tanh \left(\frac{\mathrm{eA}}{2B}(x-\mathrm{ct})\right)}{1+\tanh \left(\frac{\mathrm{eA}}{2B}(c-\mathrm{ct})\right)}\right)}^{i\frac{3\mathrm{\&alpha;}}{2\mathrm{eB}}}{\left(\frac{8A\tanh \left(\frac{\mathrm{eA}}{2B}(x-\mathrm{ct})\right)}{B^2(3+\tan h^2\left(\frac{\mathrm{eA}}{2B}(x-\mathrm{ct})\right))}\right)}^{1/2}---\left(9\right)$

Work as parameter A, when B, c, θ, k, e, α chose different numerical value, nonlinear equation (1) just can produce many orphans structure.

When parameter A=2, B=2, c=2, θ=2, k=-1, e=1, α=1, ξ ₀=0, x ∈ [2,2], t ∈ [3 ,-1] can produce 2 * 4 orphans as shown in Figure 1, and when parameter x ∈ [3,3], t ∈ [6,0] can produce the more orphan's structure of number as shown in Figure 2.

In a word, the method that the present invention proposes has solved the problem that the many orphans of complex domain are difficult to produce in the Fibre Optical Communication Technology, the method that the present invention proposes is understandable, realization convenience, practical succinctly, can carry out the adjustment of relevant parameter according to actual conditions, provide powerful support for for the further investigation in optical fiber telecommunications system field provides, will promote this subject development.

Claims (3)

1. the present invention is the many orphans of class complex domain implementation method, belongs to technical field of optical fiber communication, it is characterized in that: according to variable coefficient nonlinear equation, adopt the functional transformation method, draw its accurately decay orphan solution.

2. the method that proposes of aforesaid right 1 is characterized in that: by the arbitrary function in the solution is carried out suitable setting, obtained new vibration orphan structure and can obtain many orphans in the complex domain.

3. the method that proposes of aforesaid right 1, it is characterized in that: the method that the present invention proposes has solved the problem that many orphans are difficult to produce in the Fibre Optical Communication Technology, the method that the present invention proposes is understandable, realization convenience, practical succinctly, can carry out the adjustment of relevant parameter according to actual conditions, provide powerful support for for the further investigation in optical fiber telecommunications system field provides, will promote this subject development.

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