Dynamics of Nonisospectral Evolution Equations

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Dynamics of Non-isospectral Evalution Equations
Zhang Da-jun Dept. Mathematics, Shanghai Univ.,
200444, Shanghai, China Email
djzhang_at_mail.shu.edu.cn Web
http//www.scicol.shu.edu.cn/siziduiwu/zdj/index.h
tm

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Lax integrability
Solitons of the NLSE
Non-isospectral NLSEs
Double-Wronskian solutions

Gauge transformations
Nonisospectral dynamics
References
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1. Lax integrablity

• 1.1 KdV equation and its Lax pair

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1. Lax integrablity

• 1.2 Lax pair

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1. Lax integrablity

• 1.3 Isospectral and non-isospectral

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1. Lax integrablity

• 1.4 Meaning of ------

1-soliton of the KdV
Constant
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1. Lax integrablity

• 1.5 Questions for

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2. Solitons of the NLSE

• 2.1 Lax pair for the NLSE

GT1
GT2
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2. Solitons of the NLSE

• 2.2 N-soliton solution to the NLSE

Double-Wronskian
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2. Solitons of the NLSE

• 2.3 1-soliton of the NLSE

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2. Solitons of the NLSE

• 2.4 2-soliton of the NLSE (N2)

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3. Non-isospectral NLSE (NNLSE)

• 3.1 NNLSE-I

GT1
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3. Non-isospectral NLSE (NNLSE)

• 3.2 NNLSE-II

GT2
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3. Non-isospectral NLSE (NNLSE)

• 3.3 NNLSE-III

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4. Double-Wronskian solutions

• 4.1 Solution to the NNLSE-I

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4. Double-Wronskian solutions

• 4.2 Solution to the NNLSE-II

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4. Double-Wronskian solutions

• 4.3 Solution to the NNLSE-III

NNLSE-III
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5. Gauge transformations

• 5.1 Transformation between the NLSE and NNLSE-I

Lax pair
Lax pair
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5. Gauge transformations

• 5.2 Transformation between the NLSE and NNLSE-II

Lax pair
Lax pair
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5. Gauge transformations

• 5.3 Applications --- solutions

NNLSE-II
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5. Gauge transformations

• 5.4.1 Applications --- conserved quantity

Conserved density/quantity
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5. Gauge transformations

• 5.4.2 Applications --- explicit conserved
  densities

For NLSE
For NNLSE-I
For NNLSE-II
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6. Nonisospectral dynamics

• 6.1.1 NNLSE-I --- 1-soliton

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6. Nonisospectral dynamics

• 6.1.2 NNLSE-I --- Comparison with the NLES

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6. Nonisospectral dynamics

• 6.1.3 NNLSE-I --- 2-soliton

2-soliton scattering
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6. Nonisospectral dynamics

• 6.2.1 NNLSE-II --- Comparison with the NLES

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6. Nonisospectral dynamics

• 6.2.2 NNLSE-II --- 2-soliton

2-soliton scattering
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6. Nonisospectral dynamics

• 6.3.1 NNLSE-III --- 1-soliton

1-soliton
Top trace
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6. Nonisospectral dynamics

• 6.3.2 NNLSE-III --- 2-soliton

2-soliton scattering
No periodic interaction
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Conclusions
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Double-Wronskian

• (1). Wronskian

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Double-Wronskian

• (2). Double-Wronskian

(MN)-order column vectors
If M0, it is an ordinary N -order Wronskian if
N0, vice versa.
Back to 2.2
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Conservation law (CL) of the NLSE
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References
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Thank You!
Thank You!