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: djzhang@mail.shu.edu.cn

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!