By Victor A. Galaktionov

ISBN-10: 1584886633

ISBN-13: 9781584886631

"Exact ideas and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics is the 1st publication to supply a scientific development of tangible ideas through linear invariant subspaces for nonlinear differential operators. performing as a advisor to nonlinear evolution equations and types from physics and mechanics, the booklet makes a speciality of the life of recent designated ideas on linear invariant subspaces for nonlinear operators."--BOOK JACKET. learn more... content material: creation : nonlinear partial differential equations and designated options -- 1. Linear invariant subspaces in quasilinear equations : simple examples and types -- 2. Invariant subspaces and modules : arithmetic in a single size -- three. Parabolic equations in a single measurement : skinny movie, Kuramoto-Sivashinsky, and magma versions -- four. Odd-order one-dimensional equations : Korteweg-de Vries, compaction, nonlinear dispersion, and Harry Dym versions -- five. Quasilinear wave and Boussinesq versions in a single size. platforms of nonlinear equations -- 6. purposes to nonlinear partial differential equations in IR[superscript N] -- 7. partly invariant subspaces, invariant units, and generalized separation of variables summary: provides examples of fascinating suggestions on linear invariant subspaces for nonlinear operators. This e-book develops numerous concepts for developing detailed options that describe singularity habit for numerous nonlinear PDEs, together with fuel dynamics types, free-boundary difficulties, and Green-Naghdi equations. learn more...

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Pavlov in 1972; see details and references in [509, p. 21]. 23) using the invariant subspaces. Introducing the pressure variable from filtration theory, u = v σ , yields a PDE with the quadratic differential operator and a constant sink, u t = F[u] ≡ uu x x + σ1 (u x )2 − σ. 24) Clearly, operator F[u] preserves the 2D subspace W2 = L{1, x 2 }, since F[C1 + C2 x 2 ] = 2C1 C2 − σ + 2 1 + 2 σ C22 x 2 ∈ W2 . 25) with the expansion coefficients C1 (t) and C2 (t) satisfying the dynamical system C1 = 2C1 C2 − σ, C2 = 2 1 + σ2 C22 .

71) yields a single PDE √ ut c4 +c2 (u x )2 −(u t )2 t + c2 √ ux c4 +c2 (u x )2 −(u t )2 x = 0, which gives a cubic polynomial equation for u(x, t), c2 c2 + (u x )2 u t t + c2 c4 + (c2 − 1)(u x )2 − (u t )2 u x x − (c2 − 1)u t u x u xt = 0. , u(x, t) = C1 (t) + C2 (t)x. 72) where G[u] is an arbitrary (possibly elliptic) operator. This equation is invariant under the Galilean transformation of the reference frame, x → x + vt for any v = constant. 72), u(x + vt, t) is also a solution. Galilean transformations and invariance under generalized Galilean algebras play an important role in the analysis of various PDEs, and especially in many physical and mechanical models; see Remarks for references.

Therefore, we will need two hypotheses that guarantee that Wn is still invariant. 81), and the second one is slightly more complicated but still easy to derive in general. This analysis can be continued to include three or more hypotheses on the operator. We finish with the following application to more general equations. 26 (Polynomial PDEs) Now let ν = νi, j be a K×K matrix of nonnegative integers, with 2 ≤ K ≤ n−1, and let the corresponding monomial Fν [u] = i j νi, j (i, j ) (Dt D x u) be defined in the space of sufficiently smooth functions u = u(x, t) of two independent variables x and t.

### Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics by Victor A. Galaktionov

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