(1) { ut + f(u)x + uxxx = 0 for x ∈ R, t > 0, u(x, 0) = u0(x) for x ∈ R, where f(u) = |u|p−1u/p (3 ≤ p < 5). I will show that if the speed of the solitary waves are sufficiently close at the initial time, the wave going ahead becomes larger and the wave going behind becomes smaller and the distance between two solitary waves becomes larger as t→∞. This gives an example of multi-pulse solution o...

The Fourier restriction norm method is used to show local wellposedness for the Cauchy-Problem ut + uxxx + (u 4)x = 0, u(0) = u0 ∈ H s x(R), s > − 1 6 for the generalized Korteweg-deVries equation of order three, for short gKdV3. For real valued data u0 ∈ L 2 x(R) global wellposedness follows by the conservation of the L-norm. The main new tool is a bilinear estimate for solutions of the Airy-e...

We prove the local well-posedness for generalized Korteweg–de Vries equation in $H^s(\mathbb{R})$, $s>1/2$, under general assumptions on nonlinearity $f(x)$, background of an $L^\infty\_{t,x}$-function $\Psi(t,x)$, with $\Psi(t,x)$ satisfying some suitable conditions. As a consequence our estimates, we also obtain unconditional uniqueness solution $H^s(\mathbb{R})$. This result not only gives u...