Dynamics of Non-isospectral Evalution Equations

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Dynamics of Non-isospectral Evalution Equations. Zhang Da-jun Dept. Mathematics, Shanghai Univ., 200444, Shanghai, China Email: djzhang@mail.shu.edu.cn 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

Zhang Da-jun

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

Email:djzhang@mail.shu.edu.cn

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

Dynamics of Non-isospectral Evalution Equations

Lax integrability

Solitons of the NLSE

Non-isospectral NLSEs

Double-Wronskian solutions

Gauge transformations

Nonisospectral dynamics

References

Dynamics of Non-isospectral Evalution Equations

KdV equation

Lax pair

Compatible condition

### 1. Lax integrablity

1.1 KdV equation and its Lax pair

Dynamics of Non-isospectral Evalution Equations

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

Compatible condition

isospectral

non-isospectral

### 1. Lax integrablity

1.3 Isospectral and non-isospectral

Dynamics of Non-isospectral Evalution Equations

Energy:

Velocity:

Amplitude:

### 1. Lax integrablity

1-soliton of the KdV

1.4 Meaning of ------

Constant

Dynamics of Non-isospectral Evalution Equations

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

Dynamics of Non-isospectral Evalution Equations

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

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

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

Periodic interaction when b1=b2

Period

### 2. Solitons of the NLSE

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

Dynamics of Non-isospectral Evalution Equations

NNLSE-I

Lax pair

### 3. Non-isospectral NLSE (NNLSE)

3.1 NNLSE-I

GT1

Dynamics of Non-isospectral Evalution Equations

NNLSE-II

Lax pair

### 3. Non-isospectral NLSE (NNLSE)

3.2 NNLSE-II

GT2

Dynamics of Non-isospectral Evalution Equations

NNLSE-III

Lax pair

### 3. Non-isospectral NLSE (NNLSE)

3.3 NNLSE-III

Dynamics of Non-isospectral Evalution Equations

NNLSE-I

Solution

### 4. Double-Wronskian solutions

4.1 Solution to the NNLSE-I

Dynamics of Non-isospectral Evalution Equations

NNLSE-II

Solution

### 4. Double-Wronskian solutions

4.2 Solution to the NNLSE-II

Dynamics of Non-isospectral Evalution Equations

Solution

### 4. Double-Wronskian solutions

NNLSE-III

4.3 Solution to the NNLSE-III

Dynamics of Non-isospectral Evalution Equations

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

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

NLSE

NNLSE-I

### 5. Gauge transformations

5.3 Applications --- solutions

NNLSE-II

Dynamics of Non-isospectral Evalution Equations

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 transformations

For NLSE

For NNLSE-I

5.4.2 Applications --- explicit conserved densities

For NNLSE-II

Dynamics of Non-isospectral Evalution Equations

1-soliton

Notations

### 6. Nonisospectral dynamics

6.1.1 NNLSE-I --- 1-soliton

Dynamics of Non-isospectral Evalution Equations

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

Periodic interaction ( )

Period

### 6. Nonisospectral dynamics

2-soliton scattering

6.1.3 NNLSE-I--- 2-soliton

Dynamics of Non-isospectral Evalution Equations

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

Extremum points

### 6. Nonisospectral dynamics

2-soliton scattering

6.2.2 NNLSE-II--- 2-soliton

Dynamics of Non-isospectral Evalution Equations

Notations

### 6. Nonisospectral dynamics

1-soliton

Top trace

6.3.1 NNLSE-III --- 1-soliton

Dynamics of Non-isospectral Evalution Equations

### 6. Nonisospectral dynamics

2-soliton scattering

No periodic interaction

6.3.2 NNLSE-III--- 2-soliton

Dynamics of Non-isospectral Evalution Equations

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

Dynamics of Non-isospectral Evalution Equations

Wronskian

Compact form

### Double-Wronskian

(1). Wronskian

Dynamics of Non-isospectral Evalution Equations

### 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

NLSE

Lax pair

CL

Riccati equation

Conserved density/quantity

### Conservation law (CL) of the NLSE

Dynamics of Non-isospectral Evalution Equations

[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

Thank You!

Thank You!

Dynamics of Non-isospectral Evalution Equations