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 ε = v c c HIGHWAY TRAFFIC: VEHICLE DENSITY u t = 2uu x + u xx WAVE SOLUTIONS: FRONTS
( u p + u ) m u m DISPERSION RELATION: : 1 k BURGERS EQUATION u p x SINGLE FRONT u( t, x)= u m + u p e k ( x + v t + x0 ) 1 + e k ( x + v t + x 0 ) v = u p + u m, k = u p u m u(t,x) CHARACTERISTIC LINE ( ) x = vt + x 0 u p u m = 0 v = k u m t x
M WAVES (M + 1) SEMI-INFINITE SINGLE FRONTS 0 < k 1 < k 2 <... < k M TWO ELASTIC SINGLE FRONTS: 0 k 1, 0 k M M 1 INELASTIC SINGLE FRONTS BURGERS EQUATION v i = k i M i=1 u( t, x)= k i e k i ( x + k i t + x i, 0 ) M i=1 1 + e k i ( x + k i t + x i, 0 ) k 4 k 1 k 2 k = k j + 1 k j k k 2 k 3... v = k j + 1 + k j k M 1 k M 0 x k t 3 k 2 k 1
FAMILY OF WAVE SOLUTIONS - KDV EQUATION SHALLOW WATER WAVES PLASMA ION ACOUSTIC WAVES ONE-DIMENSIONAL LATTICE OSCILLATIONS (EQUIPARTITION OF ENERGY? IN FPU) ε = a λ u t = 6uu x + u xxx WAVE SOLUTIONS: SOLITONS
KDV EQUATION SOLITONS ALSO CONSTRUCTED FROM EXPONENTIAL WAVES: ELASTIC ONLY x t u( t, x)= 2 k 2 ( { }) cosh 2 k x + vt + x 0 DISPERSION RELATION: v = 4 k 2
FAMILY OF WAVE SOLUTIONS - NLS EQUATION NONLINEAR OPTICS ε = δω ω 0 SURFACE WAVES, DEEP FLUID + GRAVITY + VISCOSITY NONLINEAR KLEIN-GORDON EQN. λ LIMIT ϕ t = iϕ xx + 2i ϕ 2 ϕ WAVE SOLUTIONS SOLITONS
NLS EQUATION TWO-PARAMETER FAMILY ( ) ϕ( t, x)= k exp i ω t + V x ω = k 2 v2 cosh k x + vt 4, V = v 2 ( ) N SOLITONS: k i, v i ω 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 u t = F 0 [ u] S = F [ u + ν S ] 0 n t n ν ν =0
SYMMETRIES BURGER S t S n = 2 x ( u S )+ 2 S n x n KDV S = 6 ( u S )+ 3 S t n x n x n NLS S = i 2 S + 2i( 2ϕϕ * S + ϕ 2 S *) t n x n n n EACH HAS AN HIERARCHY OF SOLUTIONS - SYMMETRIES
SYMMETRIES BURGERS S 1 = u x S 2 = 2uu x + u xx S 3 = 3u 2 u x + 3uu xx + 3u x 2 + u xxx KDV S 1 = u x S 2 = 6u u x + u xxx S 3 = 30u 2 u x + 10uu xxx + 20u x u xx + u 5 x S 4 = 140u 3 u x + 70uu xxx + 280uu x u xx + 14uu 5 x + 70u 3 x + 42u x u 4 x + 70u xx u xxx + u 7 x NOTE: S 2 = UNPERTURBED EQUATION!
PROPERTIES OF SYMMETRIES LIE BRACKETS ( [ ] ) = 0 λ =0 [ S n,s m ] λ S n u + λ S [ m u] S u + λ S m u n u t = F 0 S n [ u] { [ u] } u t = S m [ u] { Ŝ [ u] } n { } { S } n Ŝn SAME SYMMETRY HIERARCHY
PROPERTIES OF SYMMETRIES u t = F 0 [ u] F 0 [ u] S [ u] n SAME WAVE SOLUTIONS? u t = S n [ u] (EXCEPT FOR UPDATED DISPERSION RELATION)
PROPERTIES OF SYMMETRIES u t = S 2 [ u] u = S [ u] t n SAME!!!! WAVE SOLUTIONS, MODIFIED k v RELATION BURGERS S 2 S n v = k v = k n 1 KDV S 2 S n v = 4 k 2 v = 4 k 2 ( ) n 1 NF u t = S 2 [ u]+ εα S [ u]+ ε 2 β S [ u]+... 3 4 BURGERS v = k v = k + εα k 2 + ε 2 β k 3 +... KDV v = 4 k 2 v = 4 k 2 + εα( 4 k 2 ) 2 + ε 2 β( 4 k 2 ) 3 +...
CONSERVATION LAWS KDV & NLS I n = + ρ n dx E.G., NLS ρ 0 = ϕ 2 ρ 1 = iϕϕ * x ρ 2 = ϕ 4 2 ϕ x M
EVOLUTION EQUATIONS ARE APPROXIMATIONS TO MORE COMPLEX SYSTEMS F 0 w = F[ w]= t [ w]+ ε F [ w]+ ε 2 F [ w]+... 1 2 ( [ w]= S [ w] ) 2 F 0 NIT w = u + ε u ( 1) + ε 2 u ( 2) +... NF u = S [ u]+ εu + ε 2 U +... t 2 1 2 UNPERTURBED EQN. RESONANT TERMS AVOID UNBOUNDED TERMS IN u (n) IN GENERAL, ALL NICE PROPERTIES BREAK DOWN EXCEPT FOR u - A SINGLE W
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 w t = 2ww x + w xx + ε 3α w 2 w + 3α w w 1 x 2 xx + 3α w 2 + α w 3 x 4 xxx 2α 1 α 2 2α 3 + α 4 = 0 (FOKAS & LUO, KRAENKEL, MANNA ET. AL.)
OBSTACLES TO ASYMPTOTIC INTEGRABILITY - KDV w t = 6ww x + w xxx + ε 30α 1 w 2 w x + 10α 2 ww xxx + 20α 3 w x w xx + α 4 w 5 x 140 β 1 w 3 w x + 70β 2 w 2 w xxx + 280β 3 w w x w xx + ε 2 + 14 β 4 w w 5 x + 70β 5 w 3 x + 42β 6 w x w 4 x + 70 β 7 w xx w xxx + β 8 w 7 x ( ) 100 9 3α 1 α 2 + 4α 2 2 18α 1 α 3 + 60α 2 α 3 24α 2 2 3 + 18α 1 α 4 67α 2 α 4 + 24α 4 + 140 3 ( 3β 1 4 β 2 18β 3 + 17β 4 + 12β 5 18β 6 + 12β 7 4 β 8 )= 0 KODAMA, KODAMA & HIROAKA
OBSTACLES TO ASYMPTOTIC INTEGRABILITY - NLS ψ t = iψ xx + 2i ψ 2 ψ + ε( α 1 ψ xxx + α 2 ψ 2 ψ x + α 3 ψ 2 * ψ ) x β 1 ψ xxxx + β 2 ψ 2 ψ xx + β 3 ψ * ( 2 ψ ) x + ε 2 i + β 4 ψ 2 ψ * 2 xx + β 5 ψ ψ x 18α 1 2 3α 1 α 2 + α 2 α 3 2α 3 2 + β 6 ψ 4 ψ + 24 β 1 2 β 2 4 β 3 8 β 4 + 2 β 5 + 4 β 6 = 0 KODAMA & MANAKOV
OBSTCACLE TO INTEGRABILITY - BURGERS w t = 2ww x + w xx + ε 3α w 2 w + 3α w w 1 x 2 xx + 3α w 2 + α w 3 x 4 xxx EXPLOIT FREEDOM IN EXPANSION
OBSTCACLE TO INTEGRABILITY - BURGERS NIT w = u + ε u ( 1) +... NF u t = S 2 [ u]+ εα S [ u]+... 4 3 = 2uu x + u xx ( ) + εα 4 3u 2 u x + 3uu xx + 3u x 2 + u xxx
OBSTCACLE TO INTEGRABILITY - BURGERS u 1 ( ) t = 2 uu 1 ( ( ) ) + u ( 1) xx x + 3( α α )u 2 u 1 4 x + 3( α α )u u 2 4 xx + 3( α α )u 2 3 4 x TRADITIONALLY: DIFFERENTIAL POLYNOMIAL u ( 1) = au 2 + bqu + cu x x ( q = 1 u) x γ = 2α 1 α 2 2α 3 + α 4 = 0
OBSTCACLE TO INTEGRABILITY - BURGERS IN GENERAL γ 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 u t = S 2 u t = S 2 [ u] + εα S [ u] 4 3 [ u]+ εα S u 4 3 ( [ ]+ γ R[ u] ) 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 ALLOW NON-POLYNOMIAL PART IN u (1) u ( 1) = au 2 + bqu + cu + ξ( t, x) x x GAIN: NF IS INTEGRABLE, ZERO-ORDER UNPERTURBED SOLUTION u t = S 2 [ u]+ εα S u 4 3 [ ] LOSS: HIGHER-ORDER CORRECTION IS NOT POLYNOMIAL HAVE TO DEMONSTRATE THAT BOUNDED VEKSLER + Y.Z.: BURGERS, KDV Y..Z.: NLS
HOWEVER I PHYSICAL SYSTEM EXPANSION PROCEDURE II EXPANSION PROCEDURE EVOLUTION EQUATION + PERTURBATION APPROXIMATE SOLUTION
FREEDOM IN EXPANSION STAGE I - BURGERS EQUATION USUAL DERIVATION 1. τ ρ + ( ξ ρ v)= 0 ONE-DIMENSIONAL IDEAL GAS c = SPEED of SOUND 2. ( τ ρ v)+ ( ξ ρ v 2 + P µ ξ v)= 0 ρ 0 = REST DENSITY P = c2 ρ 0 γ ρ ρ 0 γ γ = c p c v τ t = ε 2 τ ξ x = ε ξ ρ = ρ 0 + ε ρ 1 v = ε u
I - BURGERS EQUATION 1. SOLVE FOR ρ 1 IN TERMS OF u FROM EQ. 1 : POWER SERIES IN ε 2. EQUATION FOR u: POWER SERIES IN ε FROM EQ.2 RESCALE u = cw ( t 1 + γ ) 2 c 2 ρ 0 8 µ t ( x 1 + γ )cρ 0 2 µ x
STAGE I - BURGERS EQUATION w t = 2ww x + w xx α 1 = 0 + ε 3α w 2 w + 3α w w 1 x 2 xx + 3α w 2 + α w 3 x 4 xxx α 2 = 1 3 α 3 = 1 4 γ 12 α 4 = 1 8 + γ 8 2α 1 α 2 2α 3 + α 4 = 1 24 + 7γ 24 0 OBSTACLE TO ASYMPTOTIC INTEGRABILITY
STAGE I - BURGERS EQUATION HOWEVER, EXPLOIT FREEDOM IN EXPANSION ρ = ρ 0 + ε ρ 1 + ε 2 ρ 2 v = ε u + ε 2 u 2 1. SOLVE FOR ρ 1 IN TERMS OF u FROM EQ. 1 : POWER SERIES IN ε 2. EQUATION FOR u: POWER SERIES IN ε FROM EQ.2 u 2 = au 2 + bu x
STAGE I - BURGERS EQUATION RESCALE u = cw ( t 1 + γ ) 2 c 2 ρ 0 8 µ t ( x 1 + γ )cρ 0 2 µ x w t = 2ww x + w xx + ε 3α w 2 w + 3α w w 1 x 2 xx + 3α w 2 + α w 3 x 4 xxx
STAGE I - BURGERS EQUATION α 1 = 2 3 a α 2 = 2 3 b 1 3 α 3 = 1 4 + 2 3 + 2 3 b 1 12 γ α 4 = 1 ( 8 γ + 1 )+ b 2α 1 α 2 2α 3 + α 4 = 0 FOR b = 7 24 γ 1 24 NO OBSTACLE TO INTEGRABILITY MOREOVER a = 1 8 γ 7 8 α 2 = α 3
STAGE I - BURGERS EQUATION w t = 2ww x + w xx α 2 = α 3 + ε 3α 1 w 2 w + 3α ww x 2 xx + 3α w 2 + α w 3 x 4 xxx w 2 + w x = x ( ) + ε α 1 w 3 + α 2 ww x + α 4 w xx REGAIN CONTINUITY EQUATION STRUCTURE
STAGE I - KDV EQUATION ION ACOUSTIC PLASMA WAVE EQUATIONS τ n + ( ξ n v)= 0 v 2 τ v + ξ 2 + ϕ = 0 ϕ ξ2 = e ϕ n τ t = ε 3 τ ξ x = ε ξ n = 1 + ε 2 n 1 ϕ = ε 2 ϕ 1 v = ±1 + ε 2 u SECOND-ORDER OBSTACLE TO INTEGRABILITY
STAGE I - KDV EQUATION EXPLOIT FREEDOM IN EXPANSION: n = 1 + ε 2 n 1 + ε 4 n 2 + ε 6 n 3 ϕ = ε 2 ϕ 1 + ε 4 ϕ 2 + ε 6 ϕ 3 v = ±1 + ε 2 u + ε 4 u 2 + ε 6 u 3 CAN ELIMINATE SECOND-ORDER OBSTACLE IN PERTURBED KDV EQUATION MOREOVER, CAN REGAIN CONTINUITY EQUATION STRUCTURE THROUGH SECOND ORDER
OBSTACLES TO ASYMPTOTIC INTEGRABILITY - KDV w t = 6ww x + w xxx + µ 30α 1 w 2 w x + 10α 2 ww xxx + 20α 3 w x w xx + α 4 w 5 x 140β 1 w 3 w x + 70β 2 w 2 w xxx + 280β 3 w w x w xx + µ 2 + 14 β 4 w w 5 x + 70β 5 w 3 x + 42β 6 w x w 4 x + 70β 7 w xx w xxx + β 8 w 7 x ( µ = ε 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