1. Multi-solitons of a (2+1)- dimensional vector soliton system Ken-ichi Maruno Department of Mathematics, University of Texas -- Pan American Joint work with Y. Ohta & M. Oikawa Kobe Univ. Kyushu Univ. Mini-Meeting: "Nonlinear Waves and More" August, 15, 2007 University of Colorado, Boulder

2. UTPA (Edinburg) Population 55,297 Population 700,634 McAllen-Edinburg-Mission Area

4. KP-II Line Soliton Solution Nonlinear wave in Plasma and water wave

5. Hirota D-operator Example: Hirota D-operator

6. Wronskian Solution

7. N-soliton solution f is a solution of the dispersion relations We can choose another kind of functions.

9. Wronskian

10. Web Structure Non-stationary complex patterns These are made from Wronskian Solutions of KP (Biondini & Kodama 2003) Classification of all soliton solutions of KP (Kodama; Biondini & Chakravarty)

11. Web structure web structure ＫＰ： (2003) Biondini & Kodama coupled KP: (2002) Isojima, Willox &Satsuma 2D-Toda: (2004) Maruno & Biondini

12. Theory of KP hierarchy (Jimbo-Miwa, 1983) AKP (= KP): Wronskian BKP ： Pfaffian CKP: Wronskian DKP: Pfaffian Semi simple Lie algebra Solution Extention of determinant

13. Solutions are written by Pfaffian

14. Square root of determinant of even antisymmetric matrix

15. Hirota & Ohta; Kodama & KM

16. Four A-solitonTwo D-solitonThree D-soliton Patterns of DKP equation are very complicated. Patterns of DKP are classified using Pfaffian. A-type Weyl group A-type soliton related to A-type Weyl group D-type Weyl group D-type soliton related to D-type Weyl group. (See Kodama & KM, 2006) Patterns of DKP are made from A and D-type Weyl group Weyl group !

17. N-soliton interaction Equations having determinant type solutions KP, 2D-Toda, fully discrete 2D-Toda (Biondini, Kodama, Chakravarty, KM) Equations having pfaffian type solutions DKP (coupled KP) (Kodama & KM)

18. Question Analysis of N-soliton interaction of equations having other types of solutions e.g. Multi-component determinant Vector NLS-type solitons

19. Vector NLS (coupled NLS) equation

20. Vector soliton interaction (vector NLS equation) –R Radhakrishnan, M Lakshmanan, J Hietarinta 1997

21. Remark ● Bright soliton solutions of NLS are written in the form of two-component Wronskian (Nimmo; Date,Jimbo,Miwa, Kashiwara) ● Bright soliton solutions of two-component vector NLS are written in the form of 3- component Wronskian (Ohta)

22. Multi-component determinant solution of NLS type equations Component 1Component 2

23. Two component KP hierarchy (cf. Jimbo & Miwa)

25. 2-component Wronskian Bilinear forms in 2-component KP hierarchy

26. Reduction to NLS

27. Gauge factor NLS 2-component Wronskian n-component NLS (n+1)-component Wronskian

28. Physical Difference between KdV and NLS KdV equation Long wave (e.g. Shallow water wave) NLS equation Short wave (e.g. Deep water wave) Is there any physical phenomenon having both long wave and short wave? Long wave Short wave

29. Resonance Interaction between long wave and short wave Resonance Interaction

30. Example: Surface wave and internal wave (Oikawa & Funakoshi) Yajima-Oikawa System (Long wave- short wave resonance interaction eq.) S L S: Short wave L: Long wave

31. Long wave-short wave resonance interaction: History N. Yajima & M. Oikawa(1976) Interaction of langumuir waves with ion-acoustic waves in plasma, Lax pair (3x3 matrix), Inverse Scattering Transform, Bright soliton D.J. Benney (1976) Water wave Y.C. Ma & L.G. Redekopp (1979) Dark soliton V. K. Melnikov (1983) Extension to multi-component and 2-dimensional case using Lax pair M. Oikawa, M. Funakoshi & M. Okamura: 2- dimensional system in stratified flow, Bright and Dark soliton solutions T. Kikuchi, T. Ikeda and S. Kakei (2003) Painleve V equation Nistazakis, Frantzeskakis, Kevrekidis, Malomed, Carretero-Gonzakez (2007): Spinor BEC

32. 2-dimensional 2-component Yajima-Oikawa system (2-dimensional 2-component long wave-short wave resonance interaction equations) Melnikov: On EQUATIONS FOR WAVE INTERACTION, Lett. Math. Phys. 1983 Lax form

33. 2-dimensional vector Yajima-Oikawa System (2-component) Vector form

34. Bilinear Equations c : a constant, c=0 Bright soliton

35. Solution of 2-dimensional 2-component Yajima-Oikawa system Belongs to 3-component KP hierarchy Theory of multi-component KP hierarchy (T. Date, M. Jimbo, M. Kashiwara, T. Miwa 1981; V. Kac, J. W. van de Leur 2003) Bilinear identities of 3-component Wronskians 3-component Wronskian with constraints of reality and complex conjugacy of complex functions

36. 3-component KP hierarchy 3-component Wronskian

38. Short wave Long wave

41. Phase shift - L S1S1 S2S2 S1S1 S2S2

42. S1S1 S2S2

43. Interaction of 2-line soliton and periodic soliton V-shape - L S1S1 S2S2

44. 2-dimensional vector Yajima-Oikawa System (n-component) tau-function: N-component Wronskian

45. 2D Matrix Yajima-Oikawa system Multi-soliton (Wronskian) solution? BEC?

46. (DKP hierarchy: Jimbo-Miwa; Hirota-Ohta; Kodama-KM) Hietarinta Physical interpretation, Multi-soliton solution Vector and matrix generalization??? Multi-component Pfaffian? Complex variables

47. Summary We constructed Wronskian solutions of 2-dimensional vector YO system Soliton interaction of vector YO system has some unusual properties. Y. Ohta, KM, M. Oikawa: J. Phys. A 40 7659-7672 (2007) KM, Y. Ohta, M. Oikawa, in preparation Analysis of multi-soliton interaction Dark soliton? Dromion? Lump? Soliton interaction of matrix generalization? Future Problems