Soonsik Kwon

In this paper, we revisit the infinite iteration scheme of normal form\nreductions, introduced by the first and second authors (with Z. Guo), in\nconstructing solutions to nonlinear dispersive PDEs. Our main goal is to\npresent a simplified approach to this method. More precisely, we study normal\nform reductions in an abstract form and reduce multilinear estimates of\narbitrarily high degrees to successive applications of basic trilinear\nestimates. As an application, we prove unconditional wel

We consider the initial value problem of the fifth-order modified KdV equation on the Sobolev spaces. $$displaylines{ partial_t u - partial_x^5u + c_1partial_x^3(u^3) + c_2upartial_x upartial_x^2 u + c_3uupartial_x^3 u =0cr u(x,0)= u_0(x) }$$ where $u:mathbb{R}imesmathbb{R} o mathbb{R} $ and $c_j$'s are real. We show the local well-posedness in $H^s(mathbb{R})$ for $sgeq 3/4$ via the contraction principle on $X^{s,b}$ space. Also, we show that the solution map from data to the solutions fails to

We study the asymptotic behavior of dispersing solutions to the Vlasov-Poisson system. Due to long interaction range, we do not expect linear scattering (Choi S-H and Ha S-Y 2011 SIAM J. Math. Anal. 43 2050-77). Instead, we prove a modified scattering result (or long range scattering result) of small and dispersing solutions. We find a quasi-free forward trajectory so that along the trajectory, the solution has an asymptotic limit. We extract the logarithmic growth part of the Duhamel term, and

In this paper, we show the orbital stability of solitons arising in the cubic derivative nonlinear Schrodinger equations. We consider the zero mass case that is not covered by earlier works [8, 3]. As this case enjoys L^2 scaling invariance, we expect the orbital stability in the sense up to scaling symmetry, in addition to spatial and phase translations. For the proof, we are based on the variational argument and extend a similar argument in [21]. Moreover, we also show a self-similar type blow

In this paper, we revisit the infinite iteration scheme of normal form reductions, introduced by the first and second authors (with Z. Guo), in constructing solutions to nonlinear dispersive PDEs. Our main goal is to present a simplified approach to this method. More precisely, we study normal form reductions in an abstract form and reduce multilinear estimates of arbitrarily high degrees to successive applications of basic trilinear estimates. As an application, we prove unconditional well-pose

We consider the self-dual Chern-Simons-Schrödinger equation (CSS), also known as a gauged nonlinear Schrödinger equation (NLS). CSS is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">L^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula

In this paper we prove that the fifth order equation arising from the KdV hierarchy $ \partial_tu + \partial_x^5u + c_1\partial_x u\partial_x^2u + c_2u\partial_x^3u = 0 $ is locally well-posed in $ H^s(\mathbb{R}) $ for $ s&gt; 5/2. Also, we prove the solution map of the equation is not uniformly continuous for $s&gt;0$.

Bourgain [2] proved that the periodic modified Korteweg–de Vries (mKdV) equation is locally well-posed in ⁠, ⁠, by introducing new weighted Sobolev spaces Xs,b, where the uniqueness holds conditionally, namely in ⁠. In this paper, we establish unconditional well-posedness of mKdV in ⁠, ⁠, that is, in addition we establish unconditional uniqueness in C([0,T];Hs), ⁠, of solutions to mKdV. We prove this result via differentiation by parts. For the endpoint case ⁠, we perform careful quinti- and sep

In this short note, we prove a refinement of bilinear local smoothing estimates of Airy solutions, when the frequency support of two wave are separated. As an application we prove a smoothing property of a bilinear form.

We consider the initial value problem of the fifth order modified KdV equation on the Sobolev spaces. \partial_t u - \partial_x^5u + c_1\partial_x^3(u^3) + c_2u\partial_x u\partial_x^2 u + c_3uu\partial_x^3 u =0, u(x,0)= u_0(x) where $ u:R\timesR \to R $ and $c_j$'s are real. We show the local well-posedness in H^s(R) for s \geq 3/4 via the contraction principle on $X^{s,b}$ space. Also, we show that the solution map from data to the solutions fails to be uniformly continuous below $H^{3/4}(R)$.