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
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)
∞ 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
BURGERS EQUATION up um DISPERSION RELATION: SINGLE FRONT CHARACTERISTIC LINE x up u(t, x) x um t
BURGERS EQUATION M WAVES (M + 1) SEMI-INFINITE SINGLE FRONTS TWO “ELASTIC” SINGLE FRONTS: M 1 “INELASTIC” SINGLE FRONTS k 4 k 3 k 2 t 0 x k 1
∞ 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
KDV EQUATION SOLITONS ALSO CONSTRUCTED FROM EXPONENTIAL WAVES: “ELASTIC” ONLY x t DISPERSION RELATION:
∞ FAMILY OF WAVE SOLUTIONS - NLS EQUATION NONLINEAR OPTICS SURFACE WAVES, DEEP FLUID + GRAVITY + VISCOSITY NONLINEAR KLEIN-GORDON EQN. ∞ LIMIT WAVE SOLUTIONS SOLITONS
NLS EQUATION TWO-PARAMETER FAMILY N SOLITONS: ki, vi i, Vi SOLITONS ALSO CONSTRUCTED FROM EXPONENTIAL WAVES: “ELASTIC” ONLY
SYMMETRIES LIE SYMMETRY ANALYSIS PERTURBATIVE EXPANSION - RESONANT TERMS SOLUTIONS OF LINEARIZATION OF EVOLUTION EQUATION
SYMMETRIES BURGERS KDV NLS EACH HAS AN ∞ HIERARCHY OF SOLUTIONS - SYMMETRIES
SYMMETRIES BURGERS KDV NOTE: S 2 = UNPERTURBED EQUATION!
PROPERTIES OF SYMMETRIES LIE BRACKETS SAME SYMMETRY HIERARCHY
PROPERTIES OF SYMMETRIES SAME WAVE SOLUTIONS ? (EXCEPT FOR UPDATED DISPERSION RELATION)
PROPERTIES OF SYMMETRIES SAME!!!! WAVE SOLUTIONS, MODIFIED k v RELATION BURGERS KDV NF BURGERS KDV
∞ CONSERVATION LAWS KDV & NLS E. G. , NLS
EVOLUTION EQUATIONS ARE APPROXIMATIONS 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 - A SINGLE WAVE
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
OBSTACLES TO ASYMPTOTIC INTEGRABILITY - BURGERS (FOKAS & LUO, KRAENKEL, MANNA ET. AL. )
OBSTACLES TO ASYMPTOTIC INTEGRABILITY - KDV KODAMA, KODAMA & HIROAKA
OBSTACLES TO ASYMPTOTIC INTEGRABILITY - NLS KODAMA & MANAKOV
OBSTCACLE TO INTEGRABILITY - BURGERS EXPLOIT FREEDOM IN EXPANSION
OBSTCACLE TO INTEGRABILITY - BURGERS NIT NF
OBSTCACLE TO INTEGRABILITY - BURGERS TRADITIONALLY: DIFFERENTIAL POLYNOMIAL
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
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
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
HOWEVER I PHYSICAL SYSTEM II EXPANSION PROCEDURE APPROXIMATE SOLUTION EVOLUTION EQUATION + PERTURBATION
FREEDOM IN EXPANSION STAGE I - BURGERS EQUATION USUAL DERIVATION ONE-DIMENSIONAL IDEAL GAS c = SPEED of SOUND 0 = REST DENSITY
I - BURGERS EQUATION 1. SOLVE FOR 1 IN TERMS OF u FROM EQ. 1 : POWER SERIES IN 2. EQUATION FOR u: POWER SERIES IN 3. FROM EQ. 2 RESCALE
STAGE I - BURGERS EQUATION OBSTACLE TO ASYMPTOTIC INTEGRABILITY
STAGE I - BURGERS EQUATION HOWEVER, EXPLOIT FREEDOM IN EXPANSION 1. SOLVE FOR 1 IN TERMS OF u FROM EQ. 1 : POWER SERIES IN 2. EQUATION FOR u: POWER SERIES IN 3. FROM EQ. 2
STAGE I - BURGERS EQUATION RESCALE
STAGE I - BURGERS EQUATION FOR NO OBSTACLE TO INTEGRABILITY MOREOVER
STAGE I - BURGERS EQUATION REGAIN “CONTINUITY EQUATION” STRUCTURE
STAGE I - KDV EQUATION ACOUSTIC PLASMA WAVE EQUATIONS SECOND-ORDER OBSTACLE TO INTEGRABILITY
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
OBSTACLES TO ASYMPTOTIC INTEGRABILITY - KDV
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
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