23/4/19 Wronskian Solutions to Soliton Equations
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Wronskian Solutions to Soliton Equations
Zhang Da-jun
Dept. Mathematics, Shanghai Univ., 200444, Shanghai, China
email: [email protected]
www: http://www.scicol.shu.edu.cn/siziduiwu/zdj/index.htm
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Bilinear Derivatives
Hirota method
Wronskian technique
Classification of Wronskian solutions
References
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1. Bilinear derivatives 1.1 Definition [H]
or
examples
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1. Bilinear derivatives 1.2 Simple Properties
(1) (2)
then
and
(3) If
Hirota method
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2. Hirota method [H] 2.1 Bilinear equation
Korteweg-de Vries (KdV) equation
Bilinear equation
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2. Hirota method 2.2 Perturbation expansion
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2. Hirota method 2.3 Truncate the expansion: 1-soliton
1-soliton
JUST TAKE!
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2. Hirota method 2.4 N-soliton
2-soliton
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N-soliton
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3. Wronskian technique
This technique is developed by Freeman and Nimmo for directly verifying solutions to bilinear equations. [FN]
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3. Wronskian technique 3.1 Wronskian
Compact form
Wronskian
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3. Wronskian technique 3.2 Properties
Examples
jth column is the derivative of (j-1)th column
Derivatives of a Wronskian has simple forms
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3. Wronskian technique 3.3 Needed equalities (I)
Example
Equality (1)
(1)
thenif
usage of equality (1)
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3. Wronskian technique 3.3 Needed equalities (II)
Equality (2)
In fact, using Laplace’s expansion rule, we have
If
then
(2)
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3. Wronskian technique 3.4 Wronskian technique
Equality (I)
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Now we have two forms for N-soliton, Hirota form and Wronskian form. Are they same?
3. Wronskian technique
3.5 N-soliton in Hirota form and in Wronskian form
If take
then
They are same!
Hirota
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3. Wronskian technique 3.5 Generalizaion [SHR]
same
generalizationdiagonal arbitrary
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4. Classification of solutions in Wronskian form
4.1 Normalization of A
(1). A and lead to same solution.
(2). Consider to be the normal form of A .
(3). A determines kinds of solutions.
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4. Classification of solutions in Wronskian form
4.2 Classification of solutions 4.2.1 Case I, A has N distinct negative eigenvalues:
Wronskian entries
When we get N-soliton solutions.
Solutions obtained in Case I are called negatons.
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4. Classification of solutions in Wronskian form
4.2 Classification of solutions 4.2.2 Case II, A has N distinct positive eigenvalues:
Wronskian entries
Solutions obtained in Case II are called positons.
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4. Classification of solutions in Wronskian form
4.2 Classification of solutions 4.2.3 Case III, A has N same negative eigenvalues:
Another choice (*2)
Wronskian entries
(*1)
Note: (*1) and (*2) lead to same solution due to their coefficient
matrixes having same Jordan form, and we call the solution high-order negatons.
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4. Classification of solutions in Wronskian form
4.2 Classification of solutions 4.2.4 Case IV, A has N same positive eigenvalues:
or
Wronskian entries
Name: high-order positons
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4. Classification of solutions in Wronskian form
4.2 Classification of solutions 4.2.5 Case V, A has N zero eigenvalues:
Name: rational solution
or
Wronskian entries
Note: sink and cosk do not lead new results due to
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4. Classification of solutions in Wronskian form
4.3 NotesIf real coefficient matrix A has N=2M distinct complex eigenvalues, then thses eigenvalues appear in conjugate couple, and we can still get real solutions to the KdV equation;[M]
Solutions obtained in Case III, IV, and V are called Jordan block solutions or multipoles solutions in IST sense;
Jordan block solutions can be obtained from a limit of Case I or II solutions;
Other examples
Wronskian solution can also be derived based on Sato Theory and Darboux transformation.
Conditions for Wronskian entries are usually related to Lax pair;
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Usage of equality (1)
[Back to 3.3.2]
From the identity
Equality (1)
[N-soliton]
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Name of solution
KdV equation
Lax pair
conditions for Wronskian entries
Lax pair (u=0):
[negatons]
[positons]
[Mat]
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Limit of solitons
[Back to 4.3]
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Other examples --- Toda lattice
1. Bilinear form
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Other examples --- Toda lattice
2. Casoratian solution
Condition:
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Other examples --- Schrodinger equation
1. Bilinear form
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Other examples --- Schrodinger equation
2. Double-Wronskian
If M=0, it is an ordinary N -order Wronskian; if N=0, vice versa.
(M+N)-order column vectors:
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Other examples --- Schrodinger equation
3. Double-Wronskian solution to the NLSE
Bilinear NLSE
Conditions:
and
complex matrix independent of x
[Back to 4.3]
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References
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.
[FN]
R. Hirota, The Direct Method in Soliton Theory (in English), Cambridge University Press, 2004.
[H]
W.Y. Ma, Solving the KdV equation by its bilinear form: Wronskian solutions, Transaction Americ. Math. Soc., 357 (2005) 1753-1778.
[M]
V.B. Matveev, Generalized Wronskian formula for solutions of the KdV equations: first applications, Phys. Lett. A, 166 (1992) 205-208.
[Mat]
J.J.C. Nimmo, A bilinear Backlund transformation for the nonlinear Schrodinger equation, Phys. Lett. A, 99 (1983) 279-280.
[N]
J. Satsuma, A Wronskian representation of n-soliton solutions of nonlinear evolution equations, J. Phys. Soc. Jpn., 46 (1979) 359-360.
[S]
D.J. Zhang, Singular solutions in Casoratian form for two differential-difference equations, Chaos, Solitons and Fractals, 23 (2005) 1333-1350.
[Z]
[Back to 1.1] [Back to 2.1] [Back to 3.5] [Back to Name of solution]
S. Sirianunpiboon, S.D. Howard, S.K. Roy, A note on the Wronskian form of solutions of the KdV equation, Phys. Lett. A, 134 (1988) 31-33.
[SHR]
[Back to 3]
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Thank You!Thank You!