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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
Menu 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 Menu 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
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
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 Menu 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 Menu 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 Menu 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
Quasi-periodic interaction ( ) 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 Menu 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
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