Asymptotic \(N\)-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. (English) Zbl 1090.35158

The author considers the generalized Korteweg-de Vries equation \[ u_t+ (u_{tt}+ u^p)_x= 0,\qquad t,x\in\mathbb{R},\tag{1} \] in the cases \(p= 2,3,4,5\). Let \(R_j(t, x)= Q_{c_j}(x- c_jt- x_j)\), be \(N\)-soliton solutions of this equation for \(j= 1,\dots,N\). The corresponding speeds are \(0< c_1< c_2<\cdots< c_N\).
In this paper the author constructs a solution \(u(t)\) of (1) such that \[ \lim_{t\to+\infty}\,\Biggl\| w(t)- \sum^N_{j=1} R_j(t)\Biggr\|_{H^1(R)}= 0. \] This solution behaves asymptotically as \(t\to\infty\) as the sum on \(N\) solitons without loss of mass by dispersion. This is an exceptional behavior. Indeed for the given parameters \(c_j\), \(j= 1,\dots,N\), \(x_j\), \(j= 1,\dots,N\) the uniqueness of such a solution is proved.
In the integrable cases for \(p= 0\), such solutions are explicitly known as \(N\)-soliton solutions. The existence results are new only for nonintegrable cases. However, the uniqueness results are new for all cases.


35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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