Direct and Inverse Methods in Nonlinear Evolution Equations: by Robert Conte (auth.)

By Robert Conte (auth.)

Many actual phenomena are defined through nonlinear evolution equation. those who are integrable offer quite a few mathematical tools, provided through specialists during this educational publication, to discover certain analytic suggestions to either integrable and in part integrable equations. The direct technique to construct options contains the research of singularities � los angeles Painlevé, Lie symmetries leaving the equation invariant, extension of the Hirota technique, building of the nonlinear superposition formulation. the most inverse process defined right here depends upon the bi-hamiltonian constitution of integrable equations. The booklet additionally offers a few extension to equations with discrete autonomous and based variables.
The varied chapters face from varied issues of view the idea of tangible strategies and of the entire integrability of nonlinear evolution equations. numerous examples and purposes to concrete difficulties let the reader to event at once the facility of different machineries involved.

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Extra info for Direct and Inverse Methods in Nonlinear Evolution Equations: Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, September 5-12, 1999

Example text

To summarize this first stage, the original system is now equivalent to 2 µ 3 5µ S + 2 (λ + 22), Sxx = − S 2 − 2 2ν 8ν 2 µ 5ν 5µ Cx = − S 2 + S+ (3λ + 22), 2 2 8ν v2,x = C − 15νSx , 3 3 (354) (355) (356) 2 v2,t µ µ 1 1243µ + 16 2 λ + 10 = − G(t) + ( (357) 2 ν ν 2ν ν ν lambda − 2)S 1 2 125ν 2 (358) +110µ S 2 − C + 15CSx − Sx , 2ν 2 3 2 55µ 2 5ν 3 5µ µ St = − (λ − 116)S − S − S (λ + 22) − 2 16ν 8ν 4 2 −CSx + 5νSx2 . (359) One equation, and only one, namely (354), is an ODE. Integrating it as an elliptic ODE for S [30] would create useless subcases and complications and should, according to rule 2, not be done.

The process is successful [64, 126, 7] and V is found to be a second solution of pSK (notation U = Vx as usual) b = λ, a = −6U/α, U L1 = ∂x3 + 6 ∂x − λ, α Ux U2 Uxx L2 = β∂t + 18 − 9λ ∂x2 + 36 2 − 6 α α α (303) (304) (305) ∂x − 36λ U , (306) α [L1 , L2 ] = 6β −1 α−1 SK(U ). (307) This is the Lax pair given by Satsuma and Kaup [53]. The BT results from the elimination of Y2 , which provides Eqs. (152)–(153) for Y1 = Y , Yxx + 3Y Yx + Y 3 + 6(U/α)Y − λ = 0, βYt − 9[(λ − 2Ux /α)(Yx + Y 2 ) + 4(λU/α − (U/α) Y ) + (2/3)(Uxx /α)Y ]x = 0, β((Yxx )t − (Yt )xx )/Y = −(6/α)SK(U ), (308) (309) 2 (310) followed by the substitution Y = (v − V )/α, (v − V )xx /α + 3(v − V )(v + V )x /α2 + (v − V )3 /α3 − λ = 0, β(v − V )t /α − (3/2)[(v − V )xxxx /α +(5(v − V )(v + V )xxx + 15(v + V )x (v − V )xx )/α2 +(15(v − V )2 (v − V )xx + 30(v − V )(v + V )2x )/α3 +30(v − V )3 (v + V )x /α4 + 6(v + V )5 /α5 ]x = 0, (311) (312) a result due to Satsuma and Kaup [53].

Conte St CSt , 6Wt = a , Cx Cx St − 2S + = 0, Cx t x 6Wx = 2aS + a M23 ≡ CSt Cx (238) (239) M4 ≡ 1 − 6a−1 F (t) − Cx−1 (4CSSt + CSxxt + 2Cxx St ) 2 St = 0, (240) − Cx−2 (2CSt2 + C 2 St Sx − 2CCxx Sxt ) − 2Cx−3 CCxx and their possible functional dependence is unsettled. Anyhow, the field W cannot be a second solution of (235) [38]. The third order assumption (117)–(118), with the link (146) and the truncated expansion w = W + a∂x Log τ, (241) generates seven determining equations (98). They are easily solved [38] and their unique solution defines the Lax pair (268)–(269), with W a second solution of (236).

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