Collision of two solitary waves of ZK

We consider the collision of two co-propagating solitary waves for the Zakharov-Kuznetsov equation, a two-dimensional asymptotic model from plasma physics. Consider a pure two-solitary waves at time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , with nearly equal velocity and nearly equal ordinate. The emanating solution stays close in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> to the sum of two modulated waves on the whole time interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℝ</mml:mi> </mml:math> . This collision is characterized by a minimal distance between the waves and a transfer of mass. We review in this note the description of the collision phenomenon obtained in [29] from time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , inspired from the seminal article of Martel and Merle in [17] on the collision of two solitary waves for the quartic Korteweg-de Vries. The result in higher dimension requires substantial modifications to the proof. First, the solution is approximated by the sum of two modulated waves and a non-local term induced by the quadratic non-linearity, with an adjustment in three intrinsic directions related to the translation invariances and scaling. The evolution of the error induced by this approximation and the modulation parameters is controlled by bootstrap techniques, with a new energy functional, a refined argument to control the transverse direction and a non-explicit approximated ODE system to control the height and the distance between the waves. We also review some asymptotic stability results from [28].