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We consider the Cauchy problem for the fifth-order modified Korteweg-de Vries equation (mKdV) under the periodic boundary condition. The fifth-order mKdV is an asymptotic model for shallow surface waves, and (in the perspective of integrable systems) the second equation in the mKdV hierarchy as well. In strong contrast with the non-periodic case, periodic solutions for dispersive equations do not have a (local) smoothing effect, and this observation becomes a major obstacle to considering the Cauchy problem for dispersive equations under the periodic condition, consequently, the periodic fifth-order mKdV shows a quasilinear phenomenon, while the non-periodic case can be considered as a semilinear equation. In this paper, we mainly establish the global well-posedness of the fifth-order mKdV in the energy space ($H^2(\mathbb T)$), which is an improvement of the former result by the first author (2018). The main idea to overcome the lack of (local) smoothing effect is to introduce a suitable (frequency dependent) short-time space originally motivated by the work by Ionescu, Kenig, and Tataru (2008). The new idea is to combine the (frequency) localized modified energy with additional weight in the spaces, which eventually handles the logarithmic divergence appearing in the energy estimates. Moreover, by using examples localized in low and very high frequencies, we show that the flow map of the fifth-order mKdV equation is not $C^3$, which implies that the Picard iterative method is not available for the local theory. This weakly concludes the quasilinear phenomenon of the periodic fifth-order mKdV., Comment: 38 pages

The local and global control results for a general higher-order KdV-type operator posed on the unit circle are presented. Using spectral analysis, we are able to prove local results, that is, the equation is locally controllable and exponentially stable. To extend the local results to the global one we captured the smoothing properties of the Bourgain spaces, the so-called propagation of singularities, which are proved with a new perspective. These propagation, together with the Strichartz estimates, are the key to extending the local control properties to the global one, precisely, higher-order KdV-type equations are globally controllable and exponentially stabilizable in the Sobolev space $H^{s}(\mathbb{T})$ for any $s \geq 0$. Our results recover previous results in the literature for the KdV and Kawahara equations and extend, for a general higher-order operator of KdV-type, the Strichartz estimates as well as the propagation results, which are the main novelties of this work., Comment: 38 pages, 2 figues

In this paper, we consider the higher-order linear Schr\"odinger equations, that is, a formal finite Taylor expansion of the linear pseudo-relativistic equation. We establish the global-in-time Strichartz estimates for these higher-order equations which hold uniformly in the speed of light. As nonlinear applications, we show that the higher-order Hartree(-Fock) equation approximates the corresponding pseudo-relativistic equation on an arbitrarily long time interval, with higher accuracy than the non-relativistic equation. We also prove small data scattering for the higher-order nonlinear Schr\"odinger equations., Comment: v1: Minor errors are corrected

In this study, we consider the nonlinear Sch\"odinger equation (NLS) with the zero-boundary condition on a two- or three-dimensional large finite cubic lattice. We prove that its solution converges to that of the NLS on the entire Euclidean space with simultaneous reduction in the lattice distance and expansion of the domain. Moreover, we obtain a precise global-in-time bound for the rate of convergence. Our proof heavily relies on Strichartz estimates on a finite lattice. A key observation is that, compared to the case of a lattice with a fixed size [Y. Hong, C. Kwak, S. Nakamura, and C. Yang, \emph{Finite difference scheme for two-dimensional periodic nonlinear {S}chr\"{o}dinger equations}, Journal of Evolution Equations \textbf{21} (2021), no.~1, 391--418.], the loss of regularity in Strichartz estimates can be reduced as the domain expands, depending on the speed of expansion. This allows us to address the physically important three-dimensional case., Comment: 30 pages, Some comments are added, Any comments are welcome

In this paper, we consider the Cauchy problem for $(abcd)$-Boussinesq system posed on one- and two-dimensional Euclidean spaces. This model, initially introduced by Bona, Chen, and Saut, describes a small-amplitude waves on the surface of an inviscid fluid, and derived as a first-order approximation of incompressible, irrotational Euler equations. We mainly establish the ill-posedness of the system under various parameter regimes, which generalize the result of the one-dimensional BBM-BBM case by Chen and Liu. Most of results established here, we obtain the optimal result for two-dimensional BBM-BBM system. The proof follows from an observation of the \emph{high to low-frequency cascade} present in nonlinearity, motivated by Bejenaru and Tao., Comment: 23 pages

In this paper, we develop dispersive PDE techniques for the Fermi--Pasta--Ulam (FPU) system with infinitely many oscillators, and we show that general solutions to the infinite FPU system can be approximated by counter-propagating waves governed by the Korteweg--de Vries (KdV) equation as the lattice spacing approaches zero. Our result not only simplifies the hypotheses but also reduces the regularity requirement in the previous study [45]., Comment: 48page

A nonlinear Schr\"odinger equation (NLS) on a periodic box can be discretized as a discrete nonlinear Schr\"odinger equation (DNLS) on a periodic cubic lattice, which is a system of finitely many ordinary differential equations. We show that in two spatial dimensions, solutions to the DNLS converge strongly in $L^2$ to those of the NLS as the grid size $h>0$ approaches zero. As a result, the effectiveness of the finite difference method (FDM) is justified for the two-dimensional periodic NLS., Comment: 25 pages

This paper is concerned with the Cauchy problem of the modified Kawahara equation (posed on $\mathbb T$), which is well-known as a model of capillary-gravity waves in an infinitely long canal over a flat bottom in a long wave regime \cite{Hasimoto1970}. We show in this paper some well-posedness results, mainly the \emph{global well-posedness} in $L^2(\mathbb T)$. The proof basically relies on the idea introduced in Takaoka-Tsutsumi's works \cite{TT2004, NTT2010}, which weakens the non-trivial resonance in the cubic interactions (a kind of smoothing effect) for the local result, and the global well-posedness result immediately follows from $L^2$ conservation law. An immediate application of Takaoka-Tsutsumi's idea is available only in $H^s(\mathbb T)$, $s > 0$, due to the lack of $L^4$-Strichartz estimate for arbitrary $L^2$ data, a slight modification, thus, is needed to attain the local well-posedness in $L^2(\mathbb T)$. This is the first low regularity (global) well-posedness result for the periodic modified Kwahara equation, as far as we know. A direct interpolation argument ensures the \emph{unconditional uniqueness} in $H^s(\mathbb T)$, $s > \frac12$, and as a byproduct, we show the weak ill-posedness below $H^{\frac12}(\mathbb T)$, in the sense that the flow map fails to be uniformly continuous., Comment: 39 pages, revised the proof of the local well-posedness in $L^2$, accepted for publication in Annales de l'Institut Henri Poincare / Analyse non lineaire

Consider the Hamiltonian $abcd$ system in one dimension, with data posed in the energy space $H^1\times H^1$. This model, introduced by Bona, Chen and Saut, is a well-known physical generalization of the classical Boussinesq equations. The Hamiltonian case corresponds to the regime where $a,c<0$ and $b=d>0$. Under this regime, small solutions in the energy space are globally defined. A first proof of decay for this $2\times 2$ system was given by the two authors and Poblete and Pozo, in a strongly dispersive regime, i.e. under essentially the conditions \[ b=d > \frac29, \quad a,c<-\frac1{18}. \] Additionally, decay was obtained inside a proper subset of the light cone $(-|t|,|t|)$. In this paper, we improve the last result in three directions. First, we enlarge the set of parameters $(a,b,c,d)$ for which decay to zero is the only available option, considering now the so-called weakly dispersive regime $a,c\sim 0$: we prove decay if now \[ b=d > \frac3{16}, \quad a,c<-\frac1{48}. \] This result is sharp in the case where $a=c$, since for $a,c$ bigger, some $abcd$ linear waves of nonzero frequency do have zero group velocity. Second, we sharply enlarge the interval of decay to consider the whole light cone, that is to say, any interval of the form $|x|\sim |v|t$, for any $|v|<1$. This result rules out, among other things, the existence of nonzero speed solitary waves in the regime where decay is present. Finally, we prove decay to zero of small $abcd$ solutions in exterior regions $|x|\gg |t|$, also discarding super-luminical small solitary waves. These three results are obtained by performing new improved virial estimates for which better decay properties are deduced., Comment: 59 pp., 11 figs, Minor mistake was corrected, no changes in the result

In this work, we deal with the initial value problem of the 5th-order Gardner equation in $\mathbb{R}$, presenting the local well-posedness result in $H^2(\mathbb{R})$. As a consequence of the local result, in addition to $H^2$-energy conservation law, we are able to prove the global well-posedness result in $H^2(\mathbb{R})$. Finally, we present a stability result for 5th order Gardner breather solution in the Sobolev space $H^2(\mathbb{R})$., Comment: 34 pages, typos corrected, references updated, title of the latter version changed

In this paper we consider globally defined solutions of Camassa-Holm (CH) type equations outside the well-known nonzero speed, peakon region. These equations include the standard CH and Degasperis-Procesi (DP) equations, as well as nonintegrable generalizations such as the $b$-family, elastic rod and BBM equations. Having globally defined solutions for these models, we introduce the notion of \emph{zero-speed and breather solutions}, i.e., solutions that do not decay to zero as $t\to +\infty$ on compact intervals of space. We prove that, under suitable decay assumptions, such solutions do not exist because the identically zero solution is the global attractor of the dynamics, at least in a spatial interval of size $|x|\lesssim t^{1/2-}$ as $t\to+\infty$. As a consequence, we also show scattering and decay in CH type equations with long range nonlinearities. Our proof relies in the introduction of suitable Virial functionals \`a la Martel-Merle in the spirit of the works by one of us and Ponce, and Kowalczyk-Martel adapted to CH, DP and BBM type dynamics, one of them placed in $L^1_x$, and a second one in the energy space $H^1_x$. Both functionals combined lead to local in space decay to zero in $|x|\lesssim t^{1/2-}$ as $t\to+\infty$. Our methods do not rely on the integrable character of the equation, applying to other nonintegrable families of CH type equations as well., Comment: 27 pp

This paper is a continuation of authors' previous work \cite{CK2018-1}. We extend the argument \cite{CK2018-1} to fifth-order KdV-type equations with different nonlinearities, in specific, where the scaling argument does not hold. We establish the $X^{s,b}$ nonlinear estimates for $b < \frac12$, which is almost optimal compared to the standard $X^{s,b}$ nonlinear estimates for $b > \frac12$ \cite{CGL2010, JH2009}. As an immediate conclusion, we prove the local well-posedness of the initial-boundary value problem (IBVP) for fifth-order KdV-type equations on the right half-line and the left half-line., Comment: 60 pages. More expositions are complemented in the introduction section, and minor typos are corrected. arXiv admin note: text overlap with arXiv:1805.05229

This paper concerns the initial-boundary value problem (IBVP) of the Kawahara equation posed on the right and left half-lines. We prove the local well-posedness in the low regularity Sobolev space. We introduce the Duhamel boundary forcing operator, which is introduced by Colliander - Kenig \cite{CK} in the context of Airy group operators, to construct solutions on the whole line. We also give the bilinear estimate in $X^{s,b}$ space for $b < \frac12$, which is almost sharp compared to IVP of Kawahara equation \cite{CLMW2009, JH2009}., Comment: 47 pages, Corrected typos and added some comments

In this note we show that all small solutions of the BBM equation must decay to zero as $t\to +\infty$ in large portions of the physical space, extending previous known results, and only assuming data in the energy space. Our results also include decay on the left portion of the physical line, unlike the standard KdV dynamics., Comment: 9 pp., We found a small error in the proof and corrected

The Boussinesq $abcd$ system is a 4-parameter set of equations posed in $\mathbb{R}_t \times \mathbb{R}_x$, originally derived by Bona, Chen and Saut as first order 2-wave approximations of the incompressible and irrotational, two dimensional water wave equations in the shallow water wave regime, in the spirit of the original Boussinesq derivation. Among many particular regimes, depending each of them in terms of the value of the parameters $(a,b,c,d)$ present in the equations, the "generic" regime is characterized by the setting $b,d>0$ and $a,c<0$. The system is hamiltonian if also $b=d$. The equations in this regime are globally well-posed in the energy space $H^1\times H^1$, provided one works with small solutions. In this paper, we investigate decay and the scattering problem in this regime, which is characterized as having (quadratic) long-range nonlinearities, very weak linear decay $O(t^{-1/3})$ because of the one dimensional setting, and existence of non scattering solutions (solitary waves). We prove, among other results, that for a sufficiently dispersive $abcd$ systems (characterized only in terms of parameters $a, b$ and $c$), all small solutions must decay to zero, locally strongly in the energy space, in proper subset of the light cone $|x|\leq |t|$. We prove this result by constructing three suitable virial functionals in the spirit of works by Kowalczyk, Martel and the second author, and more recently by the last three authors (valid for the simpler scalar "good Boussinesq" model), leading to global in time decay and control of all local $H^1\times H^1$ terms. No parity nor extra decay assumptions are needed to prove decay, only small solutions in the energy space., Comment: 36 pp., 4 figures. Corrected typos and added some new comments and references

In this paper, we consider the cubic fourth-order nonlinear Schr\"odinger equation (4NLS) under the periodic boundary condition. We prove two results. One is the local well-posedness in $H^s$ with $-1/3 \le s < 0$ for the Cauchy problem of the Wick ordered 4NLS. The other one is the non-squeezing property for the flow map of 4NLS in the symplectic phase space $L^2(\mathbb{T})$. To prove the former we used the ideas introduced in [Takaoka and Tsutsumi 2004] and [Nakanish et al 2010], and to prove the latter we used the ideas in [Colliander et al 2005]., Comment: 38 pages, final version, some typos and an error in (3.40) are revised, accepted for publication in JMAA

In this paper, we prove that the periodic higher-order KdV-type equation \[\left\{\begin{array}{ll} \partial_t u + (-1)^{j+1} \partial_x^{2j+1}u + \frac12 \partial_x(u^2)=0, \hspace{1em} &(t,x) \in \mathbb{R} \times \mathbb{T}, \\ u(0,x) = u_0(x), &u_0 \in H^s(\mathbb{T}). \end{array} \right.\] is globally well-posed in $H^s$ for $s \ge -\frac{j}{2}$, $j \ge 3$. The proof of the global well-posedness is based on "I-method" introduced by Colliander et al. \cite{CKSTT1}. To apply "I-method", we factorize the resonant functions by using the different ways from Hirayama \cite{Hirayama}. Furthermore, we prove the nonsqueezing property of the periodic higher-order KdV-type equation as well. The proof relies on Gromov's nonsqueezing theorem for the finite dimensional Hamiltonian system and approximation for the solution flow. More precisely, after taking the frequency truncation to the solution flow, we applied the nonsqueezing theorem and then the result is transferred to the infinite dimensional original flow. This argument was introduced by Kuksin \cite{Kuksin:1995ue} and made concretely by Bourgain \cite{Bourgain:1994tr} for 1D cubic NLS flow, and Colliander et. al. \cite{CKSTT3} for the KdV flow. One of our observation is that the higher-order KdV-type equation has the better modulation effect from the non-resonant interaction than KdV equation. Hence, unlike the work of Colliander et. al. \cite{CKSTT3}, we can get the nonsqueezing property for the solution flow without the Miura transform., Comment: 30 pages, some typos are modified, the final version, to appear in Journal of Mathematical Analysis and Applications

This paper is a continuation of the paper \emph{Low regularity Cauchy problem for the fifth-order modified KdV equations on $\mathbb{T}$}. In this paper, we consider the fifth-order equation in the Korteweg-de Vries (KdV) hierarchy as following: \begin{equation*} \begin{cases} \partial_t u - \partial_x^5 u + 30u^2\partial_x u + 20 u\partial_x u \partial_x^3u + 10u \partial_x^3 u = 0, \hspace{1em} (t,x) \in \mathbb{R} \times \mathbb{T}, u(0,x) = u_0(x) \in H^s(\mathbb{T}) \end{cases}. \end{equation*} We prove the local well-posedness of the fifth-order KdV equation for low regularity Sobolev initial data via the energy method. This paper follows almost same idea and argument as in the paper \emph{Low regularity Cauchy problem for the fifth-order modified KdV equations on $\mathbb{T}$}. Precisely, we use some conservation laws of the KdV Hamiltonians to observe the direction which the nonlinear solution evolves to. Besides, it is essential to use the short time $X^{s,b}$ spaces to control the nonlinear terms due to \emph{high $\times$ low $\Rightarrow$ high} interaction component in the non-resonant nonlinear term. We also use the localized version of the modified energy in order to obtain the energy estimate. As an immediate result from a conservation law in the scaling sub-critical problem, we have the global well-posedness result in the energy space $H^2$., Comment: 46 pages. Final version. Introduction and typos are modified. Accepted for publication in J. Differential Equations. arXiv admin note: substantial text overlap with arXiv:1510.00464

In this paper, we consider the fifth-order modified Korteweg-de Vries (modified KdV) equation under the periodic boundary condition. We prove the local well-posedness in $H^s(\mathbb T)$, $s > 2$, via the energy method. The main tool is the short-time Fourier restriction norm method, which was first introduced in its current form by Ionescu, Kenig and Tataru [Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math. 173 (2) (2008) 265--304]. Besides, we use the frequency localized modified energy to control the high-low interaction component in the energy estimate. We remark that under the periodic setting, the integrable structure is very useful (but not necessary) to remove harmful terms in the nonlinearity and this work is the first low regularity well-posedness result for the fifth-order modified KdV equation., Comment: 73 pages, final version, accepted for publication in Journal of Hyperbolic Differential Equations

We consider the Cauchy problem of the fifth-order equation arising from the Korteweg-de Vries (KdV) hierarchy u_t + u_{xxxxx} + c_1u_{x} u_{xx} + c_2u u_{x} = 0 x,t \in \R We prove a priori bound of solutions for H^s(\R) with s >= 5/4 and the local well-posedness for s >= 2. The method is a short time X^{s,b} space, which is first developed by Ionescu-Kenig-Tataru in the context of the KP-I equation. In addition, we use a weight on localized X^{s,b} structures to reduce the contribution of high-low frequency interaction where the low frequency has large modulation. As an immediate result from a conservation law, we have the fifth-order equation in the KdV hierarchy is globally well-posed in the energy space H^2., Comment: 35pages. We fixed error in the first version. Main theorem is changed. Coincidentally, Kenig and Pilod have also worked on the same problem and obtained a similar result [arXiv:1205.0169]