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
SINGLE FRONT umum upup t x u(t,x)u(t,x) x upup umum CHARACTERISTIC LINE DISPERSION RELATION: BURGERS EQUATION
M WAVES (M + 1) SEMI-INFINITE SINGLE FRONTS TWO “ELASTIC” SINGLE FRONTS: M 1 “INELASTIC” SINGLE FRONTS 0k1k1 k2k2 k3k3 k4k4 x t k1k1 BURGERS EQUATION
SHALLOW WATER WAVES PLASMA ION ACOUSTIC WAVES ONE-DIMENSIONAL LATTICE OSCILLATIONS (EQUIPARTITION OF ENERGY? IN FPU) WAVE SOLUTIONS: SOLITONS ∞ FAMILY OF WAVE SOLUTIONS - KDV EQUATION
SOLITONS ALSO CONSTRUCTED FROM EXPONENTIAL WAVES: “ELASTIC” ONLY t x DISPERSION RELATION: KDV EQUATION
∞ 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: k i, v i i, V i 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 NOTE: S 2 = UNPERTURBED EQUATION! KDV
PROPERTIES OF SYMMETRIES LIE BRACKETS SAME SYMMETRY HIERARCHY
PROPERTIES OF SYMMETRIES SAME WAVE SOLUTIONS ? (EXCEPT FOR UPDATED DISPERSION RELATION)
PROPERTIES OF SYMMETRIES BURGERS KDV SAME!!!! WAVE SOLUTIONS, MODIFIED k v RELATION BURGERS KDV NF
∞ CONSERVATION LAWS KDV & NLS E.G., NLS
EVOLUTION EQUATIONS ARE APPROXIMATIONS TO MORE COMPLEX SYSTEMS NIT NF IN GENERAL, ALL NICE PROPERTIES BREAK DOWN EXCEPT FORu - A SINGLE WAVE UNPERTURBED EQN.RESONANT TERMS AVOID UNBOUNDED TERMS IN u (n)
BREAKDOWN OF PROPERTIES ∞ FAMILY OF CLOSED-FORM WAVE SOLUTIONS ∞ HIERARCHY OF SYMMETRIES ∞ SEQUENCE OF CONSERVATION LAWS FOR PERTURBED EQUATION CANNOT CONSTRUCT 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
NF NIT OBSTCACLE TO INTEGRABILITY - BURGERS
TRADITIONALLY: DIFFERENTIAL POLYNOMIAL
PART OF PERTURBATION CANNOT BE ACOUNTED FOR “OBSTACLE TO ASYMPTOTIC INTEGRABILITY” TWO WAYS OUT BOTH EXPLOITING FREEDOM IN EXPANSION IN GENERAL ≠ 0 OBSTCACLE TO INTEGRABILITY - BURGERS
WAYS TO OVERCOME OBSTCACLES I. ACCOUNT FOR OBSTACLE BY ZERO-ORDER TERM GAIN: HIGHER-ORDER CORRECTION BOUNDED POLYNOMIAL LOSS: NF NOT INTEGRABLE, ZERO-ORDER UNPERTURBED SOLUTION KODAMA, KODAMA & HIROAKA - KDV KODAMA & MANAKOV - NLS OBSTACLE
WAYS TO OVERCOME OBSTCACLES II. ACCOUNT FOR OBSTACLE BY FIRST-ORDER TERM LOSS: HIGHER-ORDER CORRECTION IS NOT POLYNOMIAL HAVE TO DEMONSTRATE THAT BOUNDED GAIN: NF IS INTEGRABLE, ZERO-ORDER UNPERTURBED SOLUTION ALLOW NON-POLYNOMIAL PART IN u (1) VEKSLER + Y.Z.: BURGERS, KDV Y..Z.: NLS
HOWEVER PHYSICAL SYSTEM EXPANSION PROCEDURE EVOLUTION EQUATION + PERTURBATION EXPANSION PROCEDURE APPROXIMATE SOLUTION II I
FREEDOM IN EXPANSION STAGE I - BURGERS EQUATION USUAL DERIVATIONONE-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 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 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 ION 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