nonlinear dispersive equations - ime.unicamp.brnde/abstracts.pdf · 6 stationary kdv equation is a...

First Workshop on

Nonlinear Dispersive Equations30/10/2013 to 1/11/2013

IMECC-UNICAMP, Campinas, Brazil

Scientific Committee

Jaime Angulo - USP, Brazil

Jerry Bona - UIC, USA

Thierry Cazenave - UPMC, France

Rafael Iório - IMPA, Brazil

Felipe Linares - IMPA, Brazil

Gustavo Ponce - UCSB, USA

Jean-Claude Saut - U. Paris-Sud, France

Sponsors:

Organizing Committee

Xavier Carvajal - IM/UFRJ

Mahendra Panthee -IMECC/UNICAMP

Ademir Pastor - IMECC/UNICAMP

Márcia Scialom - IMECC/UNICAMP

For detailed information please visit

www.ime.unicamp.br/~nde/

or send e-mail to [email protected]

CC

mailto:[email protected]

Day -1: Wednesday, 30th October

Auditorio – Hebe BiagioniIMECC

Time Speaker Title

8:00-9:00 Organizers Registration / Opening

Chair Felipe Linares

9:00-9:40 J. Bona Long-crested wave propagation

9:45-10:15 F. Natali Orbital stability of periodic waves for the Klein-Gordon type

equations

10:20-10:40 Coffee Break

Chair Axel Grunrock

10:45-11:15 A. Corcho On the dynamics of symmetric solutions for the

Schrodinger-Korteweg de Vries system in the energy space

11:20-11:50 L.G. Farah On the inhomogeneous nonlinear Schrodinger equation

12:55-12:25 L. C. F. Ferreira On the Schrodinger equation with singular potentials

12:30-14:30 Lunch Break

Chair Thierry Cazenave

14:30-15:10 J. Angulo Instability of cnoidal-peak solutions for the NLS equation

with a periodic δ–interaction

15:15-15:45 C. Hernandez Stability of standing waves of a nonlinear Schrodinger equation

with a Dirac delta potential


16:20-17:00 J. Albert Uniqueness of solutions to equations for KdV multisolitons

17:05-17:35 J. M. Jimenez On the persistence properties of solutions of a fifth order KdV

equation in weighted Sobolev spaces

20:00-22:00 Football - Match Escola de Futebol Boca Juniors

Rua Carlos Martins, 17, Barao Geraldo (starts at 21:00)

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Day -2: Thursday, 31st October


Time Speaker Title

Chair Jaime Angulo

9:00-9:40 R. Iorio On the Cauchy problem associated to the Birnkman flow in Rn

9:45-10:15 A. Pazoto Controllability of a 1-D tank containing a fluid modeled by a

Boussinesq system


Chair Adan Corcho

10:45-11:15 N. Godet Blow-up for the nonlinear Schrodinger equation on manifolds

11:20-11:50 N. Visciglia On the inhomogeneous nonlinear Schrodinger equation

12:55-12:25 R. Pastran On a perturbation of the Benjamin-Ono equation


Chair John Albert

14:30-15:10 A. Grunrock The Cauchy problem for the Zakharov-Kuznetsov equation in 2 D

15:15-15:45 A-S Suzzoni Almost sure global well-posedness for the cubic wave equation

15:50-16:20 Coffee Break Poster Session

16:20-17:00 F. Linares Dispersive perturbations of Burgers and hyperbolic equations I:

local theory

17:05-17:35 - Poster Session

20:00-23:00 Dinner Estancia Grill

Av. Albino J.B. de Oliveira 271, Barao Geraldo

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Day -3: Friday, 1st November


Time Speaker Title

Chair Jerry Bona

9:00-9:40 G. Ponce Recent results on uniqueness and decay properties of solutions

to the KdV and BO equations

9:45-10:15 A. Esfahani Scattering of solutions and stability of solitary waves for the

generalized BBM-ZK equation


Chair Luis Gustavo Farah

10:45-11:15 J. Ramos On the existence of minimal blow-up solutions for the

nonlinear H12 × H− 1

2 wave equation

11:20-11:50 V. Barros Infinite energy solutions for Schrodinger-type equations

with a nonlocal term

12:55-12:25 E. Alarcon The quasi-parabolic nature of the KdV equation in the

asymmetrically weighted Sobolev space


Chair Gustavo Ponce

14:30-15:10 S. Mancas Fifth order BBM type equation: Weierstrass traveling

wave solutions

15:15-15:45 G. Doronin Decay of solutions and critical size of spatial domains for

the ZK equation


16:20-17:00 T. Cazenave Finite-time blowup and global existence for the complex

Ginzburg-Landau equation

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Abstracts – Conference Talks

(1) Eduardo Arbieto Alarcon, Universidade Federal de Goias, Brazil, [email protected]

Title: The quasi-parabolic nature of the KdV equation in the asymmetrically

weighted Sobolev space

Abstract: In 1983, Tosio Kato in the paper, On the Cauchy Problem for the

(Generalized) Korteweg-de Vries Equation, considers the initial value problem for

KdV for initial data in asymmetric spaces, with the resulting irreversibility in time.

We consider the Cauchy problem for the forced Kortewegde Vries equation

∂u

∂t+D3 u+ uDu = f, t > 0, x ∈ R, u(0) = φ, (0.1)

where D =∂

∂x, with initial data in Y = Xs ∩L2

b where L2b = L2(e2bx dx) for s ≥ 0,

b > 0 and Xs is the Sobolev space or the Zhidkov space, of order s. Moreover

f ∈ L2b ∩H∞(R). Results for the globally well posed is obtained for (0.1).

——————————————————————————————————–

(2) John Albert, University of Oklahoma, USA, [email protected]

Title: Uniqueness of solutions to equations for KdV multisolitons

Abstract: Solitary-wave profiles of the Korteweg-de Vries (KdV) equation are so-

lutions of a first-order nonlinear ordinary differential equation. It is an elementary

exercise, often assigned in undergraduate differential equations courses, to integrate

this equation and show that the only nonsingular solutions which vanish at infinity

are the familiar hyperbolic secant solitary-wave profiles. In studying the stability of

multi-soliton solutions of the KdV equation, an analogous problem comes up for the

higher-order nonlinear ordinary differential equations (stationary KdV equations)

which multi-soliton profiles satisfy. That is, are there any nonsingular solutions

of the stationary KdV equations which vanish at infinity, besides the well-known

multi-soliton profiles? We answer this question in the negative, at least for the

case of 2-solitons; and the method used should generalize to the case of arbitrarily

many solitons. We do this by integrating the equations, using a change of variables

due to Dubrovin, Gelfand and Dickey, which takes advantage of the fact that each

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stationary KdV equation is a finite-dimensional completely integrable Hamiltonian

system. We discuss applications to stability theory for KdV and other completely

integrable equations.

This is joint work with Nghiem Nguyen.

——————————————————————————————————–

(3) Jaime Angulo Pava, IME-USP, Brazil, [email protected]

Title: Instability of cnoidal-peak solutions for the NLS equation with a periodic

δ–interaction

Abstract: In this talk we establish a theory of existence and stability of space-

periodic standing waves for the cubic nonlinear Schrodinger equation (NLS-equation)

with a point defect determined by a space-periodic Dirac distribution at the ori-

gin. This equation admits a family of periodic-peak traveling waves solutions with

a profile determined by the cnoidal Jacobi elliptic function. Via a perturbation

method and continuation argument, we prove that in the case of a repulsive defect

the cnoidal profile is unstable in H1per([−π, π]) with respect to perturbations which

have the same space-periodic as the wave itself. Global well-posedness is verified

for the Cauchy problem in H1per([−π, π]).

——————————————————————————————————–

(4) Vanessa Barros de Oliveira, Univ. Federal da Bahia, Brazil, [email protected]

Title: Infinite energy solutions for Schrodinger-type equations with a nonlocal term

Abstract: We study the Cauchy problem associated with nonlinear Schrodinger-

type equations with a nonlocal term in Rn. Existence and uniqueness of local and

global solutions are established in spaces which allow singular initial data. Scatter-

ing, asymptotic stability, and decay rates are also proved.

Joint work with Ademir Pastor.

——————————————————————————————————–

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(5) Jerry Bona, UIC, Chicago, USA, [email protected]

Title: Long-crested wave propagation

Abstract: The lecture will discuss long-crested water waves which, while fully

three-dimensional, feature only gentle variation in the y-direction in an xyz-Cartesian

coordinate system, where gravity acts in the direction of decreasing values of z.

Such waves are common in wave channels and on near-shore zones of large bodies

of water.

Assuming the initial data has a limit as |y| becomes large, long-time well-posedness

theory for Boussinesq-type systems is developed. These systems can be shown to

provide accurate renditions of the full, three-dimensional Euler equations in the case

of long-crested waves and appear to be a superior approximation to the well-known

KP-system.

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(6) Thierry Cazenave, Universite Pierre et Marie Curie & CNRS, France, [email protected]

Title: Finite-time blowup and global existence for the complex Ginzburg-Landau

equation

Abstract: In this talk, I will discuss recent joint works with Flavio Dickstein

(UFRJ) and Fred B. Weissler (Paris-Nord) on the complex Ginzburg-Landau equa-

tion ∂tu = eiθ∆u+ eiγ|u|αu, set either on the whole space or on the torus (i.e., with

periodic boundary conditions). Here −π2≤ θ, γ ≤ π

2and α > 0. In the case θ = γ,

we prove finite-time blowup under a standard energy condition on the initial value

and study how the blowup time depends on θ. In the more general case γ 6= θ, we

prove the existence of standing waves on the torus, for sufficiently small α. I will

also comment several open questions that naturally arise.

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(7) Adan Corcho, IM-UFRJ, Brazil, [email protected]

Title: On the dynamics of symmetric solutions for the Schrodinger-Korteweg de

Vries system in the energy space

Abstract: In this work we study interactions between long and short waves that

arise in various physical situations. More precisely, we study the interactions mod-

eled by the Initial Value Problem (IVP) associated to the Schrodinger-Korteweg de

Vries system, that is,

iφτ + φξξ = αφη + β|φ|2φ, τ, ξ ∈ R,ητ + ηξξξ + ηηξ = γ(|φ|2)ξ,φ(ξ, 0) = φ0(ξ), η(ξ, 0) = η0(ξ),

(0.2)

where the short wave φ = φ(ξ, τ) is a complex valued function, the long wave

η = η(ξ, τ) is a real valued function and α, β and γ are real constants with αγ 6= 0.

Many works concerning local and global well-posedness for the IVP (0.2) have

been developed by several authors for initial data (u0, v0) in the classical Sobolev

spaces Hs ×Hκ.

All global well-posedness results known for the system (0.2) were obtained in

the case αγ > 0. Here, we present interesting results about the dynamics of the

solutions of the system (0.2) in the energy space H1 × H1, under symmetry as-

sumptions, in the case αγ < 0.

Joint work with X. Carvajal.

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(8) Gleb Doronin, Universidade Estadual de Maringa, Brazil, [email protected]

Title: Decay of solutions and critical size of spatial domains for the ZK equation

Abstract: Initial-boundary-value problems for the Zakharov-Kuznetsov equation

posed on bounded rectangles and on a strip are considered. The spectral properties

of a stationary operator are studied in order to show that the evolution problem

possesses restrictions on the size of a spatial domain. The exponential ecay of

regular solutions to problems with- and without above restrictions is established.

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(9) Amin Esfahani, Damghan University, Iran, [email protected]

Title: Scattering of solutions and stability of solitary waves for the generalized

BBM-ZK equation

Abstract: A two-dimensional version of the BBM equation will be considered.

The existence and scattering of global small amplitude solutions to this equation

will be studied. The orbital stability of solitary wave solutions of this equation will

be also investigated.

——————————————————————————————————–

(10) Luiz Gustavo Farah, UFMG, Brazil, [email protected]

Title: On the inhomogeneous nonlinear Schrodinger equation

Abstract: In this talk we consider the Inhomogeneous Nonlinear Schrodinger

Equation

i∂tu+ ∆u+ |x|−b|u|2σu = 0, x ∈ RN , t > 0,

where N ≥ 1, 0 < b < min{2, N} and2− bN

< σ <2− bN − 2

.

We prove a new Gagliardo-Nirenberg estimate and use it to obtain global well-

posedness in H1(RN) for small data. We also prove the existence of blow-up solu-

tions for certain initial data.

——————————————————————————————————–

(11) Lucas C. F. Ferreira, UNICAMP, Brazil, [email protected]

Title: On the Schrodinger equation with singular potentials

Abstract: We study the Cauchy problem for the non-linear Schrodinger equation

with singular potentials. For point-mass potential and nonperiodic case, we prove

existence and asymptotic stability of global solutions in weak-Lp spaces. Specific

interest is give to the point-like δ and δ′ impurity and for two δ-interactions in one

dimension. We also consider the periodic case which is analyzed in a functional

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space based on Fourier transform and local-in-time well-posedness is proved.

Joint work with Jaime Angulo Pava (IME-USP).

——————————————————————————————————–

(12) Nicolas Godet, Toulouse University, France, [email protected]

Title: Blow-up for the nonlinear Schrodinger equation on manifolds

Abstract: We study the quintic nonlinear Schrodinger equation posed on a mani-

fold M. If M is the real line, and near the ground state, the blow up theory for this

equation is well-known. In this setting, essentially only one regime is possible; it

is characterized by an almost self-similar blow-up rate and is known to be stable

by perturbation of the initial data. In this talk, we show a property of geometric

stability of this regime in the sense that we prove that it persists in noneuclidean

geometries but with a radial symmetry assumption.

——————————————————————————————————–

(13) Axel Grunrock, Heinrich-Heine-Universitat Dusseldorf, Germany, [email protected]

duesseldorf.de

Title: The Cauchy problem for the Zakharov-Kuznetsov equation in 2 D

Abstract: The Zakharov-Kuznetsov (ZK) equation

ut + ∂x∆u+ ∂x(u2) = 0, u = u(t, x, y), (x, y) ∈ R× Rn−1 (0.3)

is a generalization of the famous Korteweg-de Vries (KdV) equation to higher space

dimensions. If n = 2, the Cauchy-Problem for this equation is known to be globally

well-posed in Hs(R2), if s ≥ 1 (Faminskii, 1995) and locally well-posed, if s > 34

(Linares & Pastor, 2009). We report on a recent improvement of the local result

down to s > 12, which was obtained in collaboration with S. Herr (Grunrock & Herr,

2013). This progress is achieved by a symmetrisation of the differential equation

(using a linear transformation of the space variables) and by a bilinear Strichartz

type estimate for free solutions of the linear part of the transformed equation.

Joint work with S. Herr.

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(14) Cesar Adolfo Hernandez Melo, Universidade Estadual de Maringa, Brazil, cah-

[email protected]

Title: Stability of standing waves of a nonlinear Schrodinger equation with a Dirac

delta potential.

Abstract: The question of stability/instability of standing waves for nonlinear

Schrodinger equations with zero potential have been widely debated in the last

few decades. In contrast, this issue has received less attention when the potential

in the equation is nonzero. In our work, we address the stability/instability of

standing waves of a Schrodinger equation with a Dirac potential and a cubic-quintic

nonlinearity, namely

iut + uxx + Zδ(x)u+ u(|u|2 + |u|4) = 0, x, t ∈ R,

where δ is the Dirac distribution at the origin and Z is a real parameter. A main

difficulty in our approach is to compute the number of negative eigenvalues of the

linearized operator around the standing waves, and it is overcome by a perturbation

method and a continuation arguments. Among others, we show that the standing

wave solution is stable in H1(R) for Z > 0 and unstable in H1even(R) for Z <

−0.866025 approximately.

——————————————————————————————————–

(15) Rafael Jose Iorio, Jr., IMPA, Brazil, [email protected]

Title: On the Cauchy problem associated to the Birnkman flow in Rn

Abstract: In this work we deal with the Cauchy problem associated to the Brinkman

flow, which models fluid flow in certain types of porous media. We study local and

global well-posedness in Sobolev spaces Hs(Rn), s > n2

+ 1, using Kato’s theory for

quasilinear equations and parabolic regularization.

——————————————————————————————————–

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(16) Jose Manuel Jimenez, Universidad Nacional de Colombia, Colombia, [email protected]

Title: On the persistence properties of solutions of a fifth order KdV equation in

weighted Sobolev spaces

Abstract: We study persistence properties of solutions for a fifth order KdV equa-

tion in the weighted Sobolev spaces

Zs,r = Hs(Rn) ∩ L2(|x|2rdx)

and we establish that for a solution of a fifth order KdV equation to satisfy the

persistent property in L2(|x|2rdx) it is necessary to have a persistence property in

an appropriate Sobolev space Hs(Rn).

——————————————————————————————————–

(17) Felipe Linares, IMPA, Brazil, [email protected]

Title: Dispersive perturbations of Burgers and hyperbolic equations I: local theory

Abstract: The aim of this talk is to show how a weakly dispersive perturbation of

the inviscid Burgers equation improve (enlarge) the space of resolution of the local

Cauchy problem. More generally we will review several problems arising from weak

dispersive perturbations of nonlinear hyperbolic equations or systems.

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(18) Stefan C. Mancas, Embry -Riddle Aeronautical University, USA, [email protected]

Title: Fifth order BBM type equation: Weierstrass traveling wave solutions

Abstract: We use second order approximation to the higher-order Boussinesq type

system to derive a single Benjamin-Bona-Mahony (BBM) type equation.

Using a traveling wave reduction we show that the single 5BBM type equation

admits solutions that can be written of Weierstrass ℘ elliptic functions. For certain

relationships between constants, the solutions can be simplified to the well-known

Jacobi elliptic, hyperbolic functions or circular periodic functions.

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(19) Fabio Natali, Universidade Estadual de Maringa, Brazil, [email protected]

Title: Orbital stability of periodic waves for the Klein-Gordon type equations

Abstract: In this talk, we prove results of orbital stability and instability of

periodic waves related to the Klein-Gordon equation with general nonlinearities.

Among the equations studied here, we can cite, the logarithmic and quintic Klein-

Gordon as well as the Liouville equation. The main tool used in our approach is a

recent computational method in order to decide about the nonpositive spectrum of

the linearized operator combined with the classical stability theory.

——————————————————————————————————–

(20) Ricardo Pastran, Universidad Nacional de Colombia, Colombia, [email protected]

Title: On a perturbation of the Benjamin-Ono equation

Abstract: We recall some results about local and global well-posedness in the

Sobolev space Hs(R) of the initial value problem associated to a perturbation of

the Benjamin-Ono equation ut + uux + βHuxx + η(Hux − uxx) = 0, where x ∈ R,

t ≥ 0, η > 0 and H denotes the usual Hilbert transform. We will show some new

results about the unique continuation of the solutions to this equation.

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(21) Ademir Pazoto, Universidade Federal do Rio de Janeiro (UFRJ), Brazil, [email protected]

Title: Controllability of a 1-D tank containing a fluid modeled by a Boussinesq

system

Abstract: This paper is concerned with the exact controllability problem for a 1-D

tank containing an inviscid incompressible irrotational fluid. The tank is subject

to one-dimensional horizontal motion. We take as fluid model a Boussinesq system

of KdV-KdV type, and as control the acceleration of the tank. We derive for the

linearized system an exact controllability result in an appropriate space.

Joint work with Lionel Rosier and Dugan Nina.

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(22) Gustavo Ponce, University of California-Santa Barbara, USA, [email protected]

Title: Recent results on uniqueness and decay properties of solutions to the KdV

and BO equations

Abstract: In this talk we shall present some recent results concerning uniqueness

and decay properties of solutions to the Korteweg-de Vries (KdV) and Benjamin-

Ono (BO) equations. In the case of the KdV eq. our results are related with

sharp persistence properties and reflect the “parabolic” character of solutions of

this equation in exponential weighted spaces first observed by T. Kato. In the case

of the BO our results provide a unconditional uniqueness result involving only ”two

time observations”.

——————————————————————————————————–

(23) Javier Ramos, IMPA, Brazil, [email protected]

Title: On the existence of minimal blow-up solutions for the nonlinear H12 × H− 1

2

wave equation

Abstract: We prove the existence of minimal blow-up solutions for the nonlinear

wave equation with initial data in H12 ×H− 1

2 in dimensions d ≥ 2. The proof relays

on the nonlinear profile decomposition and one of the main difficulties comes from

the presence of the Lorentz symmetry of the equation.

——————————————————————————————————–

(24) Anne-Sophie de Suzzoni, Universite Paris 13, France, [email protected]

Title: Almost sure global well-posedness for the cubic wave equation

Abstract: We propose to use probabilities to solve the cubic wave equation on the

sphere S3 in spaces of low (sub critical) regularity. First, we use a result by Burq

and Lebeau to get a L2 basis consisting in spherical harmonics uniformly bounded

in Lp. Then, we use it to randomize an initial datum in L2 and deduce from its

probabilistic properties global well-posedness.

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(25) Nicola Visciglia, University of Pisa, Italy, [email protected]

Title: NLS in the partially periodic setting

Abstract: We study the long time behaviour of solutions to NLS in the partially

periodic case.

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Abstracts - Posters

——————————————————————————————————–

(1) Thiago Pinguello de Andrade, UNICAMP, Brazil, [email protected]

Title: Orbital stability of periodic travelling-wave solutions for the mKdV equation

Abstract: In this work we present the orbital stability of periodic travelling-wave

solutions for the mKdV equation

ut + uxxx + u2ux = 0. (0.4)

Travelling waves are special solutions having the form u(x, t) = φ(x − ct). This

leads us to look for periodic solutions of the ODE

−φ′′ + cφ− φ3 − A = 0, (0.5)

where A is an integration constant. In the case A = 0, periodic solutions for (0.5)

have been found and the orbital stability of the respective travelling waves estab-

lished. However, if we assume A 6= 0 we find a new family of travelling-waves

solutions which has not been studied yet. This is the main subject of our work.

Joint work with Ademir Pastor-UNICAMP.

——————————————————————————————————–

(2) Roberto Capistrano Filho, UFRJ, Brazil and Universite de Lorraine, France, capis-

[email protected]

Title: Internal controllability for the Korteweg-de Vries equations on a bounded

domain

Abstract: This work is devoted to the study of the internal controllability for the

Korteweg-de Vries equation posed on a bounded interval. The main part of the

work focus on the null controllability property of a linearized equation. Following a

classical duality approach (see e.g. J. -L. Lions, Exact controllability, stabilization

and perturbations for distributed system, SIAM 98’) the problem is reduced to the

study of an observability inequality which is proved by using a Carleman estimate.

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Then, making use of a cut-off argument and the duality approach, the exact con-

trollability is also investigated. In both cases, we return to the nonlinear system

by means of a fixed point argument.

Joint work with L. Rosier (UL) and A. F. Pazoto (UFRJ).

——————————————————————————————————–

(3) Alysson Cunha, UFG-CAJ, Brazil, [email protected]

Title: Unique continuation principles for the Benjamin-Ono-Zakharov-Kuznetsov

equation

Abstract: We consider the initial-value problem (IVP) associated with the Benjamin-

Ono-Zakharov-Kuznetsov equation

ut + H∂2xu+ uxyy + uux = 0, (x, y) ∈ R2, t > 0

and prove some unique continuation principles. To obtain our results, we use tools

from harmonic analysis such as the Stein derivative, the boundedness of the Hilbert

transform in weighted Sobolev spaces and the Calderon commutator estimate. In

particular, these continuation principles show that some persistence properties for

the IVP are sharp.

Joint work with Ademir Pastor-UNICAMP.

——————————————————————————————————–

(4) Pedro Gamboa Romero, UFRJ, Brazil, [email protected]

Title: Global well-posedness for the critical Schrodinger-Debye system

Abstract: We establish global well-posedness results for the initial value prob-

lem associated to the Schrodinger-Debye system in dimension two, for data in

Hs(R2)× L2(R2), 2/3 < s ≤ 1 and for data in H1(R2)×H1(R2).

Joint work with X. Carvajal from IM-UFRJ.

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(5) Nataliia Goloshchapova, IME–USP, Brazil, [email protected]

Title: Stability of standing waves for a nonlinear Schrodinger equation with δ-

potential. Extension theory approach

Abstract: The aim of our work is to demonstrate effectiveness of standard exten-

sion theory “tricks” in investigation of stability of standing waves for semi-linear

Schrodinger equation with a delta-potential. Our approach relies on the abstract

theory by Grillakis, Shatah and Strauss for Hamiltonian systems which are invariant

under a one-parameter group of operators.

——————————————————————————————————–

(6) Benson Muite, KAUST, Saudi Arabia, [email protected]

Title: Numerical simulation of rough solutions to dispersive wave equations

Abstract: We give numerical evidence that splitting methods can compute rough

solutions to nonlinear dispersive equations. Example computations are shown for

the Davey-Stewartson, 1,2 and 3 dimensional L2 critical nonlinear Schrodinger,

Maxwell’s equations and the p-system. Current analysis of splitting schemes seems

to require higher regularity than is shown in the numerical simulations. This is

work in progress with D. Acevedo-Feliz, A. Alghamdi, D. Ketcheson, N. Mauser,

M. Quezada De Luna, D. San Roman Alerigi, M. Srinivasan, and H-P. Stimming.

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Notes

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Notes

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nonlinear dispersive equations - ime.unicamp.brnde/abstracts.pdf · 6 stationary kdv equation is a...

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