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The set of such operators is a 6D linear space spanned by F1 [u] = u x x [(u x x )2 + (u x )2 ], F2 [u] = u x [(u x x )2 + (u x )2 ], F3 [u] = u[(u x x )2 + (u x )2 ], F4 [u] = u x x [2uu x x − (u x )2 + u 2 ], F5 [u] = u x [2uu x x − (u x )2 + u 2 ], F6 [u] = u[2uu x x − (u x )2 + u 2 ]. Next, consider the 5D trigonometric subspace W5 = L{1, cos x, sin x, cos 2x, sin 2x}. 56) iff the coefficients {b j } satisfy b2 − 2b5 = 0, b2 − 4b5 = 0, b1 − 4b3 − 4b4 + 16b6 = 0, 2b1 − 4b3 − 5b4 + 8b6 = 0. © 2007 by Taylor & Francis Group, LLC 1 Linear Invariant Subspaces: Examples 31 The set of such operators is a 2D linear space spanned by F1 [u] = uu x x − 34 (u x )2 + u 2 and F2 [u] = (u x x + 4u)2 .

72), then u(x + f (t), t) is a solution for any C 1 -function f (t). 74) for given initial data, it admits an infinite-dimensional set of solutions. 72) are fully nonlinear PDEs with unknown concepts of proper solutions and local regularity properties. Exact solutions may help to clarify some evolution characteristics and possible singularities of such PDEs. 45. 72) may admit exact solutions on various 2D invariant modules of F, such as L{1, cos γ x} and L{1, cosh γ x} for any γ = 0. Consider the Galilean invariant equation with the porous medium operator on the right-hand side (for convenience, the original equation was divided by u x x ) ut − 1 ux x u x u t x = (uu x )x in IR × IR + .

This possesses exact King’s second solution u(x, t) = C1 (t) + C2 (t)x + C3 (t)x 2 + C4 (t)x 3 . 39) 1 Linear Invariant Subspaces: Examples 15 Plugging this into the PDE yields the DS  C1 = 2C3 C1 − 23 C22 ,    C2 = 6C1 C4 − 23 C2 C3 , 2 2    C3 = 2C2 C4 − 3 C3 , C4 = 0. 39) admits the 4D subspace W4 = L{1, x, x 2, x 3 } and F : W4 → W3 = L{1, x, x 2 } . The final two examples represent some remarkable invariant subspaces of the maximal dimension (a crucial theoretical aspect to be studied in the next chapter).

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