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.

MSC:

 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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