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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 ut 2 u u x u xx WAVE SOLUTIONS: FRONTS vc c BURGERS EQUATION u p um u t, x 1 k um DISPERSION RELATION: um 0 vk SINGLE FRONT um u p ek x v t x0 1 ek x v t x0 v u p um , k u p um up CHARACTERISTIC LINE x vt x0 x up u(t,x) x um t BURGERS EQUATION M WAVES (M + 1) SEMI-INFINITE SINGLE FRONTS u t, x ki e 1 e M 0 k1 k2 ... kM TWO “ELASTIC” SINGLE FRONTS: 0 k1 , 0 kM vi ki ki x ki t xi , 0 i 1 M ki x ki t xi , 0 i 1 M1 “INELASTIC” SINGLE FRONTS k1 k2 k2 k 3 ... k4 k kj 1 kj k3 v kj 1 kj kM 1 kM k2 t 0 x k1 k1 ∞ FAMILY OF WAVE SOLUTIONS - KDV EQUATION SHALLOW WATER WAVES PLASMA ION ACOUSTIC WAVES a ONE-DIMENSIONAL LATTICE OSCILLATIONS (EQUIPARTITION OF ENERGY? IN FPU) ut 6 u u x u xxx WAVE SOLUTIONS: SOLITONS KDV EQUATION SOLITONS ALSO CONSTRUCTED FROM EXPONENTIAL WAVES: “ELASTIC” ONLY x t 2 2k u t, x 2 cosh k x vt x0 DISPERSION RELATION: v 4k 2 ∞ FAMILY OF WAVE SOLUTIONS - NLS EQUATION 0 NONLINEAR OPTICS SURFACE WAVES, DEEP FLUID + GRAVITY + VISCOSITY NONLINEAR KLEIN-GORDON EQN. ∞ LIMIT t i xx 2i 2 WAVE SOLUTIONS SOLITONS NLS EQUATION TWO-PARAMETER FAMILY t, x k exp i t V x cosh k x vt 2 v v 2 k 4 , V 2 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 ut F0 u t Sn F0 u Sn 0 SYMMETRIES BURGERS t Sn 2 x u Sn x Sn KDV t Sn 6 x u Sn x Sn NLS 2 3 t Sn i x Sn 2i 2 Sn Sn 2 * 2 * EACH HAS AN ∞ HIERARCHY OF SOLUTIONS - SYMMETRIES SYMMETRIES S1 u x BURGERS S2 2u u x u xx S3 3u u x 3u u xx 3u x u xxx 2 2 S1 u x S2 6 u u x u xxx KDV S3 30 u u x 10 u u xxx 20 u x u xx u5 x 2 S4 140 u u x 70 uu xxx 280 u u x u xx 3 14 u u5 x 70 u x 42 u x u4 x 70 u xx u xxx u7 x 3 NOTE: S2 = UNPERTURBED EQUATION! PROPERTIES OF SYMMETRIES LIE BRACKETS S ,S S n m n u Sm u Sm u Sn u 0 0 ut F0 u ut Sm u S u n Sn Ŝn SAME SYMMETRY HIERARCHY Ŝ u n PROPERTIES OF SYMMETRIES ut F0 u F0 u Sn u ut Sn u SAME WAVE SOLUTIONS ? (EXCEPT FOR UPDATED DISPERSION RELATION) PROPERTIES OF SYMMETRIES ut S2 u ut Sn u SAME!!!! WAVE SOLUTIONS, MODIFIED kv RELATION BURGERS S2 Sn v k v kn 1 2 ut S2 u S3 u S4 u ... NF S2 Sn KDV BURGERS v 4k v 4k 2 2 n 1 v k v k k k ... 2 2 3 KDV 4 k ... v 4 k v 4 k 4 k 2 2 2 2 2 2 3 ∞ CONSERVATION LAWS KDV & NLS E.G., NLS In n dx 2 0 * i 1 x 4 x 2 2 EVOLUTION EQUATIONS ARE APPROXIMATIONS TO MORE COMPLEX SYSTEMS wt F w F0 w F1 w F2 w ... 2 F w S w 0 NIT NF 2 w u u 1 u 2 2 ... ut S2 u U1 U 2 ... UNPERTURBED EQN. 2 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 wt 2 w wx wxx 31 w wx 3 2 w wxx 2 3 3 wx 4 wxxx 2 2 1 2 2 3 4 0 (FOKAS & LUO, KRAENKEL, MANNA ET. AL.) OBSTACLES TO ASYMPTOTIC INTEGRABILITY - KDV wt 6 w wx wxxx 30 1 w 2 wx 10 2 w wxxx 20 w w w 3 x xx 4 5x 140 1 w 3 wx 70 2 w 2 wxxx 280 3 w wx wxx 2 3 14 4 w w5 x 70 5 wx 42 6 wx w4 x 70 7 wxx wxxx 8 w7 x 100 9 3 1 2 4 2 2 18 1 3 60 2 3 24 3 2 18 1 4 67 2 4 24 4 2 140 3 3 1 4 2 18 3 17 4 12 5 18 6 12 7 4 8 0 KODAMA, KODAMA & HIROAKA OBSTACLES TO ASYMPTOTIC INTEGRABILITY - NLS t i xx 2i 2 1 xxx 2 x 3 x 2 2 * 1 xxxx 2 2 xx 3 * x 2 2 i 2 2 * 4 4 xx 5 x 6 18 1 31 2 2 3 2 3 24 1 2 2 4 3 8 4 2 5 4 6 0 2 2 KODAMA & MANAKOV OBSTCACLE TO INTEGRABILITY - BURGERS wt 2 w wx wxx 31 w wx 3 2 w wxx 2 3 3 wx 4 wxxx 2 EXPLOIT FREEDOM IN EXPANSION OBSTCACLE TO INTEGRABILITY - BURGERS NIT w u u 1 ... NF ut S2 u 4 S3 u ... 2u u x u xx 4 3u u x 3u u xx 3ux uxxx 2 2 OBSTCACLE TO INTEGRABILITY - BURGERS u 1 t u 2 uu 1 1 xx x 31 4 u u x 2 3 2 4 u u xx 3 3 4 u x TRADITIONALLY: DIFFERENTIAL POLYNOMIAL 1 2 u au b qu x cu x 2 q u 1 x 2 1 2 2 3 4 0 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 ut S2 u 4 S3 u ut S2 u 4 S3 u R u 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) u au bqux cux t, x 1 2 GAIN: NF IS INTEGRABLE, ZERO-ORDER UNPERTURBED SOLUTION ut S2 u 4 S3 u 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 EXPANSION PROCEDURE APPROXIMATE SOLUTION EVOLUTION EQUATION + PERTURBATION FREEDOM IN EXPANSION STAGE I - BURGERS EQUATION USUAL DERIVATION 1. ONE-DIMENSIONAL IDEAL GAS v 0 c = SPEED of SOUND 2. v v P v 0 0 = REST DENSITY c 0 P 0 2 2 cp c v t x 0 1 v u 2 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 u cw 1 c 0 t t 2 8 2 1 c0 x x 2 STAGE I - BURGERS EQUATION wt 2 w wx wxx 31 w wx 3 2 w wxx 2 3 3 wx 4 wxxx 2 1 0 1 2 3 1 3 4 12 1 4 8 8 1 7 2 1 2 2 3 4 0 24 24 OBSTACLE TO ASYMPTOTIC INTEGRABILITY STAGE I - BURGERS EQUATION HOWEVER, EXPLOIT FREEDOM IN EXPANSION 0 1 2 v u u2 2 2 1. SOLVE FOR 1 IN TERMS OF u FROM EQ. 1 : POWER SERIES IN 2. EQUATION FOR u: POWER SERIES IN FROM EQ.2 u2 au bux 2 STAGE I - BURGERS EQUATION RESCALE u cw 1 c 0 t t 2 1 c0 x x 2 8 2 wt 2 w wx wxx 31 w wx 3 2 w wxx 2 3 3 wx 4 wxxx 2 STAGE I - BURGERS EQUATION 2 1 a 2 1 2 2 3 4 0 3 2 1 FOR 2 b 3 3 1 2 2 1 7 1 3 b b 4 3 3 12 24 24 1 4 1 b NO OBSTACLE TO INTEGRABILITY 8 MOREOVER 1 7 a 2 3 8 8 STAGE I - BURGERS EQUATION wt 2 w wx wxx 2 3 31 w wx 3 2 w wxx 2 3 3 wx 4 wxxx 2 w wx x 3 1 w 2 w wx 4 wxx 2 REGAIN “CONTINUITY EQUATION” STRUCTURE STAGE I - KDV EQUATION ION ACOUSTIC PLASMA WAVE EQUATIONS n n v 0 v v 0 2 2 e n 2 t x 3 n 1 n1 2 1 2 v 1 u 2 SECOND-ORDER OBSTACLE TO INTEGRABILITY STAGE I - KDV EQUATION EXPLOIT FREEDOM IN EXPANSION: n 1 n1 n2 n3 2 4 6 1 2 3 2 4 6 v 1 u u2 u 3 2 4 6 CAN ELIMINATE SECOND-ORDER OBSTACLE IN PERTURBED KDV EQUATION MOREOVER, CAN REGAIN “CONTINUITY EQUATION” STRUCTURE THROUGH SECOND ORDER OBSTACLES TO ASYMPTOTIC INTEGRABILITY - KDV wt 6 w wx wxxx 30 1 w 2 wx 10 2 w wxxx 20 w w w 3 x xx 4 5x 140 1 w 3 wx 70 2 w 2 wxxx 280 3 w wx wxx 2 3 14 4 w w5 x 70 5 wx 42 6 wx w4 x 70 7 wxx wxxx 8 w7 x 2 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