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Dynamics of Non-isospectral Evalution Equations. Zhang Da-jun Dept. Mathematics, Shanghai Univ., 200444, Shanghai, China Email: [email protected] Web: http://www.scicol.shu.edu.cn/siziduiwu/zdj/index.htm. Menu. Lax integrability. Solitons of the NLSE. Non-isospectral NLSEs.

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dynamics of non isospectral evalution equations

Dynamics of Non-isospectral Evalution Equations

Zhang Da-jun

Dept. Mathematics, Shanghai Univ., 200444, Shanghai, China

Email:[email protected]

Web:http://www.scicol.shu.edu.cn/siziduiwu/zdj/index.htm

Dynamics of Non-isospectral Evalution Equations

slide2

Menu

Lax integrability

Solitons of the NLSE

Non-isospectral NLSEs

Double-Wronskian solutions

Gauge transformations

Nonisospectral dynamics

References

Dynamics of Non-isospectral Evalution Equations

1 lax integrablity

KdV equation

Lax pair

Compatible condition

1. Lax integrablity

1.1 KdV equation and its Lax pair

Dynamics of Non-isospectral Evalution Equations

1 lax integrablity1

Evolution equation

Lax pair

Integrable characteristics

Inverse scattering transform

Backlund transformation

Darboux transformation

1. Lax integrablity

1.2 Lax pair

Dynamics of Non-isospectral Evalution Equations

1 lax integrablity2

Compatible condition

isospectral

non-isospectral

1. Lax integrablity

1.3 Isospectral and non-isospectral

Dynamics of Non-isospectral Evalution Equations

1 lax integrablity3

Energy:

Velocity:

Amplitude:

1. Lax integrablity

1-soliton of the KdV

1.4 Meaning of ------

Constant

Dynamics of Non-isospectral Evalution Equations

1 lax integrablity4

If

How do they effect wave’s dynamics? (energy, amplitude, velocity)

?

What are the related equations?

?

Do these equations have connection with the isospectral version?

?

1. Lax integrablity

1.5 Questions for

Menu

Dynamics of Non-isospectral Evalution Equations

2 solitons of the nlse

NLSE

Lax pair

Zero curvature equation

2. Solitons of the NLSE

2.1 Lax pair for the NLSE

GT1

GT2

Dynamics of Non-isospectral Evalution Equations

2 solitons of the nlse1

NLSE

N-soliton solution

2. Solitons of the NLSE

2.2 N-soliton solution to the NLSE

Double-Wronskian

Dynamics of Non-isospectral Evalution Equations

2 solitons of the nlse2

1-soliton (N=1)

Characteristics

Energy:

Velocity:

Amplitude:

Top trace:

2. Solitons of the NLSE

2.3 1-soliton of the NLSE

Dynamics of Non-isospectral Evalution Equations

2 solitons of the nlse3

Head-on collision

Periodic interaction when b1=b2

Period

2. Solitons of the NLSE

2.4 2-soliton of the NLSE (N=2)

Menu

Dynamics of Non-isospectral Evalution Equations

3 non isospectral nlse nnlse

NNLSE-I

Lax pair

3. Non-isospectral NLSE (NNLSE)

3.1 NNLSE-I

GT1

Dynamics of Non-isospectral Evalution Equations

3 non isospectral nlse nnlse1

NNLSE-II

Lax pair

3. Non-isospectral NLSE (NNLSE)

3.2 NNLSE-II

GT2

Dynamics of Non-isospectral Evalution Equations

3 non isospectral nlse nnlse2

NNLSE-III

Lax pair

3. Non-isospectral NLSE (NNLSE)

3.3 NNLSE-III

Menu

Dynamics of Non-isospectral Evalution Equations

4 double wronskian solutions

NNLSE-I

Solution

4. Double-Wronskian solutions

4.1 Solution to the NNLSE-I

Dynamics of Non-isospectral Evalution Equations

4 double wronskian solutions1

NNLSE-II

Solution

4. Double-Wronskian solutions

4.2 Solution to the NNLSE-II

Dynamics of Non-isospectral Evalution Equations

4 double wronskian solutions2

Solution

4. Double-Wronskian solutions

NNLSE-III

4.3 Solution to the NNLSE-III

Menu

Dynamics of Non-isospectral Evalution Equations

5 gauge transformations

NLSE

NNLSE-I

Gauge transformation

5. Gauge transformations

Lax pair

Lax pair

5.1 Transformation between the NLSE and NNLSE-I

Dynamics of Non-isospectral Evalution Equations

5 gauge transformations1

NLSE

NNLSE-II

Gauge transformation

5. Gauge transformations

Lax pair

Lax pair

5.2 Transformation between the NLSE and NNLSE-II

Dynamics of Non-isospectral Evalution Equations

5 gauge transformations2

NLSE

NNLSE-I

5. Gauge transformations

5.3 Applications --- solutions

NNLSE-II

Dynamics of Non-isospectral Evalution Equations

5 gauge transformations3

NLSE

For NNLSE-I

For NNLSE-II

5. Gauge transformations

Conserved density/quantity

5.4.1 Applications --- conserved quantity

Dynamics of Non-isospectral Evalution Equations

5 gauge transformations4

5. Gauge transformations

For NLSE

For NNLSE-I

5.4.2 Applications --- explicit conserved densities

For NNLSE-II

Menu

Dynamics of Non-isospectral Evalution Equations

6 nonisospectral dynamics

1-soliton

Notations

6. Nonisospectral dynamics

6.1.1 NNLSE-I --- 1-soliton

Dynamics of Non-isospectral Evalution Equations

6 nonisospectral dynamics1

1-soliton

Comparison

NLSE

NNLSE-I

Profile:

Energy :

Amplitude:

Velocity:

Top trace:

6. Nonisospectral dynamics

6.1.2 NNLSE-I--- Comparison with the NLES

Dynamics of Non-isospectral Evalution Equations

6 nonisospectral dynamics2

Periodic interaction ( )

Period

6. Nonisospectral dynamics

2-soliton scattering

6.1.3 NNLSE-I--- 2-soliton

Dynamics of Non-isospectral Evalution Equations

6 nonisospectral dynamics3

1-soliton

Comparison

NLSE

NNLSE-II

Profile:

Energy :

Amplitude:

Velocity:

Top trace:

6. Nonisospectral dynamics

6.2.1 NNLSE-II--- Comparison with the NLES

Dynamics of Non-isospectral Evalution Equations

6 nonisospectral dynamics4

Quasi-periodic interaction ( )

Extremum points

6. Nonisospectral dynamics

2-soliton scattering

6.2.2 NNLSE-II--- 2-soliton

Dynamics of Non-isospectral Evalution Equations

6 nonisospectral dynamics5

Notations

6. Nonisospectral dynamics

1-soliton

Top trace

6.3.1 NNLSE-III --- 1-soliton

Dynamics of Non-isospectral Evalution Equations

6 nonisospectral dynamics6

6. Nonisospectral dynamics

2-soliton scattering

No periodic interaction

6.3.2 NNLSE-III--- 2-soliton

Dynamics of Non-isospectral Evalution Equations

conclusions

Nonisospectral evolution equations can describe solitary waves in nonuniform media;

Time-dependent spectral parameter usually leads to time-dependent amplitude, velocity and energy;

Some nonisospectral evolution equations are related to their isospectral counterpart;

Many method for solving isospectral systems can be generalized to nonisospectral systems.

Conclusions

Menu

Dynamics of Non-isospectral Evalution Equations

double wronskian

Wronskian

Compact form

Double-Wronskian

(1). Wronskian

Dynamics of Non-isospectral Evalution Equations

double wronskian1

Double-Wronskian

(M+N)-order column vectors:

(2). Double-Wronskian

If M=0, it is an ordinary N -order Wronskian; if N=0, vice versa.

[Back to 2.2]

Dynamics of Non-isospectral Evalution Equations

conservation law cl of the nlse

NLSE

Lax pair

CL

Riccati equation

Conserved density/quantity

Conservation law (CL) of the NLSE

Dynamics of Non-isospectral Evalution Equations

references

[CL]

H.H. Chen,C.S. Liu, Solitons in nonuniform media, Phys. Rev. lett., 37 (1976) 693-697.

[N]

J.J.C. Nimmo, A bilinear Backlund transformation for the nonlinear Schrodinger equation, Phys. Lett. A, 99 (1983) 279-280.

[FN]

N.C. Freeman, J.J.C. Nimmo, Soliton solutions of the KdV and KP equations: the Wronskian technique, Phys. Lett. A, 95 (1983) 1-3.

[TCZ]

T.K. Ning, D.Y. Chen, D.J. Zhang, The exact solutions for the nonisospectral AKNS hierarchy through the inverse scattering transform, Phys. A, 339 (2004) 248-266.

References

Dynamics of Non-isospectral Evalution Equations

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