PERTURBED NONLINEAR EVOLUTION EQUATIONS AND ASYMPTOTIC INTEGRABILITY

1. PERTURBED NONLINEAR EVOLUTION EQUATIONS AND ASYMPTOTIC INTEGRABILITY Yair Zarmi Physics Department &Jacob Blaustein Institutes for Desert Research Ben-Gurion University of the Negev Midreshet Ben-Gurion, Israel

2. INTEGRABLE EVOLUTION EQUATIONS • APPROXIMATIONS TO MORE COMPLEX SYSTEMS • ∞ FAMILY OF WAVE SOLUTIONS CONSTRUCTED • EXPLICITLY • LAX PAIR • INVERSE SCATTERING • BÄCKLUND TRANSFORMATION • ∞ HIERARCHY OF SYMMETRIES • HAMILTONIAN STRUCTURE (SOME, NOT ALL) • ∞ SEQUENCE OF CONSTANTS OF MOTION • (SOME, NOT ALL)

3. ∞ FAMILY OF WAVE SOLUTIONS -BURGERS EQUATION WEAK SHOCK WAVES IN: FLUID DYNAMICS, PLASMA PHYSICS: PENETRATION OF MAGNETIC FIELD INTO IONIZED PLASMA HIGHWAY TRAFFIC: VEHICLE DENSITY WAVE SOLUTIONS: FRONTS

4. SINGLE FRONT BURGERS EQUATION up CHARACTERISTIC LINE um x up DISPERSION RELATION: u(t,x) x um t

5. M WAVES  (M + 1) SEMI-INFINITE  SINGLE FRONTS BURGERS EQUATION TWO “ELASTIC” SINGLE FRONTS: M1 “INELASTIC” SINGLE FRONTS k4 k3 k2 t k1 0 x k1

6. ∞ FAMILY OF WAVE SOLUTIONS - KDV EQUATION SHALLOW WATER WAVES PLASMA ION ACOUSTIC WAVES ONE-DIMENSIONAL LATTICE OSCILLATIONS (EQUIPARTITION OF ENERGY? IN FPU) WAVE SOLUTIONS: SOLITONS

7. SOLITONS ALSO CONSTRUCTED FROMEXPONENTIAL WAVES: “ELASTIC” ONLY KDV EQUATION x t DISPERSION RELATION:

8. ∞ FAMILY OF WAVE SOLUTIONS - NLS EQUATION NONLINEAR OPTICS SURFACE WAVES, DEEP FLUID + GRAVITY + VISCOSITY NONLINEAR KLEIN-GORDON EQN. ∞ LIMIT WAVE SOLUTIONS SOLITONS

9. NLS EQUATION TWO-PARAMETER FAMILY N SOLITONS: ki, vii, Vi SOLITONS ALSO CONSTRUCTED FROMEXPONENTIAL WAVES: “ELASTIC” ONLY

10. SYMMETRIES LIE SYMMETRY ANALYSIS PERTURBATIVE EXPANSION - RESONANT TERMS SOLUTIONS OF LINEARIZATION OF EVOLUTION EQUATION

11. SYMMETRIES BURGERS KDV NLS EACH HAS AN ∞ HIERARCHY OF SOLUTIONS - SYMMETRIES

12. SYMMETRIES BURGERS KDV NOTE: S2 = UNPERTURBED EQUATION!

13. PROPERTIES OF SYMMETRIES LIE BRACKETS SAME SYMMETRY HIERARCHY

14. PROPERTIES OF SYMMETRIES SAME WAVE SOLUTIONS ? (EXCEPT FOR UPDATED DISPERSION RELATION)

15. PROPERTIES OF SYMMETRIES SAME!!!! WAVE SOLUTIONS, MODIFIED kv RELATION BURGERS KDV NF BURGERS KDV

16. ∞ CONSERVATION LAWS KDV & NLS E.G., NLS

17. EVOLUTION EQUATIONS AREAPPROXIMATIONS TO MORE COMPLEX SYSTEMS NIT NF UNPERTURBED EQN. RESONANT TERMS AVOID UNBOUNDED TERMS IN u(n) IN GENERAL, ALL NICE PROPERTIES BREAK DOWN EXCEPT FOR u - ASINGLE WAVE

18. BREAKDOWN OF PROPERTIES FOR PERTURBED EQUATION CANNOT CONSTRUCT • ∞ FAMILY OF CLOSED-FORM WAVE SOLUTIONS • ∞ HIERARCHY OF SYMMETRIES • ∞ SEQUENCE OF CONSERVATION LAWS EVEN IN A PERTURBATIVE SENSE (ORDER-BY-ORDER IN PERTURBATION EXPANSION) “OBSTACLES” TO ASYMPTOTIC INTEGRABILITY

19. OBSTACLES TO ASYMPTOTIC INTEGRABILITY - BURGERS (FOKAS & LUO, KRAENKEL, MANNA ET. AL.)

20. OBSTACLES TO ASYMPTOTIC INTEGRABILITY - KDV KODAMA, KODAMA & HIROAKA

21. OBSTACLES TO ASYMPTOTIC INTEGRABILITY - NLS KODAMA & MANAKOV

22. OBSTCACLE TO INTEGRABILITY - BURGERS EXPLOIT FREEDOM IN EXPANSION

23. OBSTCACLE TO INTEGRABILITY - BURGERS NIT NF

24. OBSTCACLE TO INTEGRABILITY - BURGERS TRADITIONALLY: DIFFERENTIAL POLYNOMIAL

25. OBSTCACLE TO INTEGRABILITY - BURGERS IN GENERAL  ≠0 PART OF PERTURBATION CANNOT BE ACOUNTED FOR “OBSTACLE TO ASYMPTOTIC INTEGRABILITY” TWO WAYS OUT BOTH EXPLOITING FREEDOM IN EXPANSION

26. WAYS TO OVERCOME OBSTCACLES I. ACCOUNT FOR OBSTACLE BY ZERO-ORDER TERM OBSTACLE GAIN: HIGHER-ORDER CORRECTION BOUNDED POLYNOMIAL LOSS: NF NOT INTEGRABLE, ZERO-ORDER UNPERTURBED SOLUTION KODAMA, KODAMA & HIROAKA - KDV KODAMA & MANAKOV - NLS

27. WAYS TO OVERCOME OBSTCACLES II. ACCOUNT FOR OBSTACLE BY FIRST-ORDER TERM ALLOW NON-POLYNOMIAL PART IN u(1) GAIN: NF IS INTEGRABLE, ZERO-ORDER UNPERTURBED SOLUTION LOSS: HIGHER-ORDER CORRECTION IS NOT POLYNOMIAL  HAVE TO DEMONSTRATE THAT BOUNDED VEKSLER + Y.Z.: BURGERS, KDV Y..Z.: NLS

28. HOWEVER I PHYSICAL SYSTEM EXPANSION PROCEDURE EVOLUTION EQUATION + PERTURBATION EXPANSION PROCEDURE II APPROXIMATE SOLUTION

29. FREEDOM IN EXPANSION STAGE I - BURGERS EQUATION USUAL DERIVATION ONE-DIMENSIONAL IDEAL GAS c = SPEED of SOUND 0 = REST DENSITY

30. I - BURGERS EQUATION SOLVE FOR 1 IN TERMS OF u FROM EQ. 1 : POWER SERIES IN  EQUATION FOR u: POWER SERIES IN  FROM EQ.2 RESCALE

31. STAGE I - BURGERS EQUATION OBSTACLE TO ASYMPTOTIC INTEGRABILITY

32. STAGE I - BURGERS EQUATION HOWEVER, EXPLOIT FREEDOM IN EXPANSION SOLVE FOR 1 IN TERMS OF u FROM EQ. 1 : POWER SERIES IN  EQUATION FOR u: POWER SERIES IN  FROM EQ.2

33. STAGE I - BURGERS EQUATION RESCALE

34. STAGE I - BURGERS EQUATION FOR NO OBSTACLE TO INTEGRABILITY MOREOVER

35. STAGE I - BURGERS EQUATION REGAIN “CONTINUITY EQUATION” STRUCTURE

36. STAGE I - KDV EQUATION ION ACOUSTIC PLASMA WAVE EQUATIONS SECOND-ORDER OBSTACLE TO INTEGRABILITY

37. STAGE I - KDV EQUATION EXPLOIT FREEDOM IN EXPANSION: CAN ELIMINATE SECOND-ORDER OBSTACLE IN PERTURBED KDV EQUATION MOREOVER, CAN REGAIN “CONTINUITY EQUATION” STRUCTURE THROUGH SECOND ORDER

38. OBSTACLES TO ASYMPTOTIC INTEGRABILITY - KDV

39. SUMMARY STRUCTURE OF PERTURBED EVOLUTION EQUATIONS DEPENDS ON FREEDOM IN EXPANSION IN DERIVING THE EQUATIONS IF RESULTING PERTURBED EVOLUTION EQUATION CONTAINS AN OBSTACLE TO ASYMPTOTIC INTERABILITY DIFFERENT WAYS TO HANDLE OBSTACLE: FREEDOM IN EXPANSION