Asymptotic ????-soliton-like solutions of the Zakharov-Kuznetsov type equations

We study here the Zakharov-Kuznetsov equation in dimension 2 2 , alttext="3"> 3 encoding="application/x-tex">3 and alttext="4"> 4 encoding="application/x-tex">4 modified . Those equations admit solitons, characterized by their velocity shift. Given parameters of alttext="upper K"> K encoding="application/x-tex">K solitons R Superscript k"> R k encoding="application/x-tex">R^k (with distinct velocities), we prove existence uniqueness a multi-soliton alttext="u"> u encoding="application/x-tex">u such that ? stretchy="false">( t stretchy="false">) ? . encoding="application/x-tex">\| u(t) - \sum _{k=1}^K R^k(t) \|_{H^1} \to \quad \text {as} +\infty . \] The convergence takes place s"> s encoding="application/x-tex">H^s with an exponential rate for all alttext="s greater-than-or-equal-to 0"> ?<!-- ? encoding="application/x-tex">s \ge 0 construction is made successive approximations multi-soliton. use classical arguments to control 1"> encoding="application/x-tex">H^1 -norms errors (inspired Martel [Amer. J. Math. 127 (2005), pp. 1103–1140]), introduce new ingredient -norm alttext="d 2"> d encoding="application/x-tex">d\geq 2 technique close monotonicity.