Stability Of Solitary Wave Solutions For Equations Of Short And Long Dispersive Waves

In this paper, we consider the existence and stability of a novel set of solitary-wave solutions for two models of short and long dispersive waves in a two layer fluid. We prove the existence of solitary waves via the Concentration Compactness Method. We then introduce the sets of solitary waves obtained through our analysis for each model and we show that them are stable provided the associated action is strictly convex. We also establish the existence of intervals of convexity for each associated action. Our analysis does not depend of spectral conditions. © 2006 Texas State University.2006118Albert, J., Angulo, J., Existence and stability of ground-state solutions of a Schrödinger-KdV system (2003) Proc. Roy. Soc. Edinburgh, Sect. A, 133 (5), pp. 987-1029Angulo, J., Montenegro, J.F., Existence and evenness of solitary-wave solutions for an equation of short and long dispersive waves (2000) Nonlinearity, 13 (5), pp. 1595-1611Bekiranov, D., Ogawa, T., Ponce, G., Interaction equation for short and long dispersive waves (1998) J. Funct. Anal., 158, pp. 357-388Bekiranov, D., Ogawa, T., Ponce, G., Weak solvability and well-posedness of a coupled Schrödinger- Korteweg de Vries equation for capillary-gravity wave interactions (1997) Proc. Amer. Math. Soc., 125 (10), pp. 2907-2919Cazenave, T., Lions, P.-L., Orbital stability of standing waves for some nonlinear Schrödinger equations (1982) Comm. Math. Phys., 85, pp. 549-561Fernandez, A., Linares, F., (2004) Well-posedness for the Schrödinger-korteweg-de Vries Equation, , PreprintFunakoshi, M., Oikawa, M., The resonant interaction between a long internal gravity wave and a surface gravity wave packet (1983) J. Phys. Soc. Japan, 52, pp. 1982-1995Grillakis, M., Shatah, J., Strauss, W., Stability theory of solitary waves in the presence of symmetry II (1990) J. Funct. Anal., 94, pp. 308-348Kawahara, T., Sugimoto, N., Kakutani, T., Nonlinear interaction between short and long capillary-gravity waves (1975) J. Phys. Soc. Japan, 39, pp. 11379-11386Levandosky, S., Stability and instability of fourth-order solitary waves (1998) J. Dynam. Diff. Eqs., 10, pp. 151-188Lin, C., Orbital stability of solitary waves of the nonlinear Schrödinger-KDV equation (1999) J. Partial Diff. Eqs., 12, pp. 11-25Lions, P.L., The concentration-compactness principle in the calculus of variations. the locally compact case, part 1 (1984) Ann. Inst. H. Poincaré, Anal. Non Linéare, 1, pp. 109-145Lions, P.L., The concentration-compactness principle in the calculus of variations. the locally compact case, part 2 (1984) Ann. Inst. H. Poincaré, Anal. Non Lineare, 4, pp. 223-283Lopes, O., Variational systems defined by improper integrals (1998) International Conference on Differential Equations, pp. 137-151. , Magalhaes, L. et al, World ScientificLopes, O., Nonlocal Variational Problems Arising in Long Wave Propagation, , to appearShatah, J., Stable standing waves of nonlinear Klein Gordon equations (1983) Commun. Math. Phys., 91, pp. 313-327Tsutsumi, M., Well-posedness of the Cauchy problem for a coupled Schrödinger-KdV equation (1993) Math. Sci. Appl., 2, pp. 513-52

Periodic Pulses Of Coupled Nonlinear Schrödinger Equations In Optics

A system of coupled nonlinear Schrödinger equations arising in nonlinear optics is considered. The existence of periodic pulses as well as the stability and instability of such solutions are studied. It is shown the existence of a smooth curve of periodic pulses that are of cnoidal type. The Grillakis, Shatah and Strauss theory is set forward to prove the stability results. Regarding instability a general criteria introduced by Grillakis and Jones is used. The well-posedness of the periodic boundary value problem is also studied. Results in the same spirit of the ones obtained for single quadratic semilinear Schrödinger equation by Kenig, Ponce and Vega are established. Indiana University Mathematics Journal ©.562847877ALBERT, J.P., BONA, J.L., HENRY, D.B., Sufficient conditions for stability of solitary-wave solutions of model equations for long waves (1987) Phys. D, 24, pp. 343-366. , http://dx.doi.org/10.1016/0167-2789(87)90084-4. MR 887857, 89a:35166ALBERT, J.P., BONA, J.L., SAUT, J.-C., Model equations for waves in stratified fluids (1997) Proc. Roy. Soc. London Ser. A, 453, pp. 1233-1260. , MR 1455330 99g:76013ANGULO, J., Non-linear stability of periodic travelling-wave solutions to the Schrödinger and the modified Korteweg-de Vries (2007) J. Diff. Equations, 235, pp. 1-34BANG, O., CLAUSSEN, C.B., KIVSHAR, Y.S., Spatial solitons and induced Kerr effects in quase-phase-matched quadratic media (1995) Phys. Rev. Lett, 197, pp. 4749-4752BOURGAIN, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations (1993) Geom. Funct. Anal, 3, pp. 107-156. , http://dx.doi.org/10.1007/BF01896020. MR 1209299, 95d:35160aBOURGAIN, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation (1993) Geom. Funct. Anal, 3, pp. 209-262. , http://dx.doi.org/10.1007/ BF01895688. MR 1215780, 95d:35160bALEXANDER V. BURYAK and YURI S. KIVSHAR, Solitons due to second harmonic generation, Phys. Lett. A 197 (1995), 407-412, http://dx.doi.org/10.1016/0375-9601(94)00989-3. MR 1314169 (95j:35188)BYRD, P.F., FRIEDMAN, M.D., (1971) Handbook of Elliptic Integrals for Engineers and Scientists, 67. , Die Grundlehren der mathematischen Wissenschaften, Band, Springer-Verlag, New York, Second edition, revised. MR 0277773 43 #3506CRASOVAN, L.-C., LEDERER, F., MIHALACHE, D., Multiple-humped bright solitary waves in second-order nonlinear media (1996) Opt. Eng, 35, pp. 1616-1623DESALVO, R., HAGAN, D.J., SHEIK, M., AHAE, B., STEGEMAN, G., VANHERZEELE, H., VAN STRYLAND, E.W., Self-Focusing and Defocusing by Cascaded Second Order Nonlinearity in KTP (1992) Opt. Lett, 17, pp. 28-30FERRO, P., TRILLO, S., Periodical waves, domain walls, and modulations instability in dispersive quadratic nonlinear media (1995) Phys. Rev. E, 51, pp. 4994-4997. , http://dx.doi.org/10.1103/PhysRevE. 51.4994FRANKEN, P.A., HILL, A.E., PETERS, C.W., WEINREICH, G., Generation of optical harmonics (1961) Phys. Rev. Lett, 7, pp. 118-119. , http://dx.doi.org/10.1103/PhysRevLett.7.118MANOUSSOS, G., (1988) Linearized instability for nonlinear Schrödinger and Klein-Gordon equations, 41, pp. 747-774. , Comm. Pure Appl. Math, MR 948770 89m:35192MANOUSSOS, G., Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system (1990) Comm. Pure Appl. Math, 43, pp. 299-333. , MR 1040143 91d:58231MANOUSSOS GRILLAKIS, JALAL SHATAH, and WALTER STRAUSS, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), 160-197, http://dx.doi.org/10.1016/0022-1236(87)90044-9. MR 901236 (88g:35169)MANOUSSOS, G., Stability theory of solitary waves in the presence of symmetry. II (1990) J. Funct. Anal, 94, pp. 308-348. , http://dx.doi.org/10.1016/ 0022-1236(90)90016-E. MR 1081647, 92a:35135HE, H., WERNER, M.J., DRUMMOND, P.D., Simultaneous solitary-wave solutions in a nonlinear parametric waveguide (1996) Phys. Rev. E, 54, pp. 896-911. , http://dx.doi.org/10.1103/PhysRevE.54. 896INCE, E.L., The periodic Lame functions (1940) Proc. Roy. Soc. Edinburgh, 60, pp. 47-63. , MR 0002399 2,46cJONES, C.K.R.T., (1988) Instability of standing waves for nonlinear Schrödinger-type equations, 8, pp. 119-138. , Ergodic Theory Dynam. Systems * (, MR 967634 90d:35267KARAMZIN, Y.N., SUKHORUKOV, A.P., Nonlinear interaction of diffracted light beams in a medium with quadratic nolinearity: Mutual focusing of beams and limitation on the efficiency of optical frequency conveners (1974) JETP Lett, 20, pp. 339-342TOSIO, K., (1995) Perturbation Theory for Linear Operators, , Classics in Mathematics, Springer-Verlag, Berlin, ISBN 3-540-58661-X, Reprint of the 1980 edition. MR 1335452 96a:47025CARLOS E. KENIG, GUSTAVO PONCE, and LUIS VEGA, The Cauchy problem for the Koneweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1993), 1-21, http://dx.doi.org/10.1215/S0012-7094-93-07101-3. MR 1230283 (94g:35196)TOSIO, K., Quadratic forms for the 1-D semilinear Schrödinger equation (1996) Trans. Amer. Math. Soc, 348, pp. 3323-3353. , http://dx.doi.org/10.2307/2154688. MR 1357398, 96j:35233WILHELM, M., STANLEY, W., (1966) Interscience Tracts in Pure and Applied Mathematics, Hill's equation, 20. , Interscience Publishers John Wiley & Sons, New York-London-Sydney, MR 0197830 33 #5991MENYUK, C.R., SCHIEK, R., TORNER, L., Solitary waves due to Χ(2): Χ(2) cascading (1994) J. Opt. Soc. Amer. B, 11, pp. 2434-2443MICHAEL, R., BARRY, S., (1978) Methods of Modern Mathematical Physics. IV. Analysis of Operators, , Academic Press [Harcourt Brace Jovanovich Publishers, New York, ISBN 0-12-585004-2. MR 0493421 58 #12429cALICE C. YEW, Multipulses of nonlinearly coupled Schrödinger equations, J. Differential Equations 173 (2001), 92-137, http://dx.doi.org/10.1006/jdeq.2000.3922. MR 1836246 (2002h:35302)_, Stability analysis of multipulses in nonlinearly-coupled Schrödinger equations, Indiana Univ. Math. J. 49 (2000), 1079-1124, http://dx.doi.org/10.1512/iumj.2000.49.1826. MR 1803222 (2001m:35033)ALICE C. YEW, A.R. CHAMPNEYS, and P.J. MCKENNA, Multiple solitary waves due to second-harmonic generation in quadratic media, J. Nonlinear Sci. 9 (1999), 33-52, http://dx.doi.org/10.1007/s003329900063. MR 1656377 (99i:78025

Riociguat treatment in patients with chronic thromboembolic pulmonary hypertension: Final safety data from the EXPERT registry

Objective: The soluble guanylate cyclase stimulator riociguat is approved for the treatment of adult patients with pulmonary arterial hypertension (PAH) and inoperable or persistent/recurrent chronic thromboembolic pulmonary hypertension (CTEPH) following Phase

On The Instability Of Solitary-wave Solutions For Fifth-order Water Wave Models

This work presents new results about the instability of solitary-wave solutions to a generalized fifth-order Korteweg-deVries equation of the form ut + uxxxxx + buxxx = (G(u, ux, uxx))x, where G(q, r, s) = Fq(q, r) - rFqr(q, r) - sFrr (q, r) for some F(q, r) which is homogeneous of degree p + 1 for some p > 1. This model arises, for example, in the mathematical description of phenomena in water waves and magneto-sound propagation in plasma. The existence of a class of solitary-wave solutions is obtained by solving a constrained minimization problem in H2(ℝ) which is based in results obtained by Levandosky. The instability of this class of solitary-wave solutions is determined for b ≠ 0, and it is obtained by making use of the variational characterization of the solitary waves and a modification of the theories of instability established by Shatah & Strauss, Bona & Souganidis & Strauss and Gonçalves Ribeiro. Moreover, our approach shows that the trajectories used to exhibit instability will be uniformly bounded in H2(ℝ).2003118Amick, C.J., Toland, J.F., Homoclinic orbits in the dynamic phase-space analogy of an elastic strut (1992) European J. Appl. Math., 3 (2), pp. 97-114Angulo, J., On the instability of solitary waves solutions of the generalized Benjamin equation (2003) Advances in Differential Equations, , To appearBenjamin, T.B., A new kind of solitary waves (1992) J. Fluid Mechanics, 254, pp. 401-411Benjamin, T.B., Solitary and periodic waves of a new kind (1996) Phil. Trans. Roy. Soc. London Ser. A, 354, pp. 1775-1806Benney, D.J., A general theory for interactions between short and long waves (1977) Stud. Appl. Math., 56, pp. 81-94Bona, J.L., Souganidis, P.E., Strauss, W.A., Stability and instability of solitary waves of Korteweg- de Vries type (1987) Proc. Royal. Soc. London Ser. A, 411, pp. 395-412Bridges, T.J., Derks, G., Linear instability of solitary wave solutions of the Kawahara equation and its generalization (2002) SIAM J. Math. Anal., 33, pp. 1356-1378Champneys, A.R., Groves, M.D., A global investigation of solitary-wave solutions to a two-parameter model for water waves (1996) Fluid Mech., 342, pp. 199-229Craig, W., Groves, M.D., Hamiltonian long-wave approxiamtions to the water-wave problem (1994) Wave Motion, 19, pp. 367-389Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F., (1954) Tables of Integral Transform, 2. , McGraw-Hill, New YorkGonçalves Ribeiro, J.M., Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field (1992) Ann. Inst. H. PoincaréPhys. Théor., 54, pp. 403-433Gorshkov, K.A., Ostrovsky, L.A., Papko, V.V., Hamiltonian and non-Hamiltonian models for water waves (1984) Lecture Notes in Physics, 195, pp. 273-290. , Springer, BerlinGrillakis, M., Shatah, J., Strauss, W., Stability theory of solitary waves in the presence of symmetry I. (1987) J. Funct. Anal., 74, pp. 160-197Hunter, J., Scheurle, J., Existence of perturbed solitary wave solutions to a model equation for water waves (1988) Physica D, 32, pp. 253-268Kato, T., Quasilinear equations of evolution, with applications to Partial Differential Equations (1975) Lectures Notes in Math., 448, pp. 25-70Kawahara, T., Oscillatory solitary waves in dispersive media (1972) J. Phys. Soc. Jpn., 33, pp. 260-264Kenig, C., Ponce, G., Vega, L., Higher-order nonlinear dispersive equations (1994) Proc. Amer. Math. Soc., 122 (1), pp. 157-166Kichenassamy, S., Existence of solitary waves for water-wave models (1997) Nonlinearity, 10 (1), pp. 133-151Kichenassamy, S., Olver, P., Existence and nonexistence of solitary wave solutions to higher-order model evolution equations (1992) SIAM J. Math. Anal., 23, pp. 1141-1166Levandosky, S.P., A stability analysis of fifth-order water wave models (1999) Physica D, 125, pp. 222-240Lions, P.L., The concentration-compactness principle in the calculus of variations. The locally compact case, part 1 (1984) Ann. Inst. H. Poincaré, Anal. Non Linéare, 1, pp. 109-145Lions, P.L., The concentration-compactness principle in the calculus of variations. The locally compact case, part 2 (1984) Ann. Inst. H. Poincaré, Anal. Non Linéare, 4, pp. 223-283McKenna, P.J., Walter, W., Traveling waves in a suspension bridge (1990) SIAM, J. Appl. Math., 50, pp. 703-715Olver, P.J., Hamiltonian and non-Hamiltonian models for water waves (1984) Lecture Notes in Physics, 195, pp. 273-290. , Springer, BerlinPonce, G., Lax pairs and higher order models for water waves (1993) J. Differential Equations, 102 (2), pp. 360-381Saut, J.C., Quelques généralisations de l'équation de Korteweg-de Vries, II (1979) J. Differential Equations, 33, pp. 320-335Shatah, J., Strauss, W.A., Instability of nonlinear bound states (1985) Comm. Math. Phys., 100, pp. 173-190Souganidis, P.E., Strauss, W.A., Instability of a class of dispersive solitary waves (1990) Proc. Royal. Soc. Eding., 114 A, pp. 195-212Weinstein, M., Existence and dynamic stability of solitary wave solutions of equations in long wave propagation (1987) Comm. P.D.E., 12, pp. 1133-1173Zufiria, J., Symmetric breaking in periodic and solitary gravity-capillary waves on water of finite depth (1987) J. Fluid Mech., 184, pp. 183-20

On The Cauchy Problem For A Boussinesq-type System

The Cauchy problem for the following Boussinesq system, is considered. It is showed that this problem is locally well-posed in Hs(ℝ) × Hs-1(ℝ) for any s > 3/2. The proof involves parabolic regularization and techniques of Bona-Smith. It is also determined that the special solitary-wave solutions of this system are orbitally stable for the entire range of the wave speed. Combining these facts we can extend globally the local solution for data sufficiently close to the solitary wave.44457492Abdelouhad, L., Bona, J.L., Fellam, M., Saut, J.C., Non-local models for nonlinear dispersives waves (1989) Phys.D., 40, pp. 360-392Albert, J.P., Positivity properties, and stability solitary-wave solutions of model equations for long waves (1992) Comm. P.D.E., 17, pp. 1-26Albert, J.P., Bona, J.L., Total positivity, and the stability of internal waves in stratified fluids of finite depth (1991) IMA journal of App. Math., 46, pp. 1-19Angulo, J., Linares, F., Global existence of solutions of a nonlinear dispersive model (1995) J. Math. Analy., and Appl., 195, pp. 797-808Benjamin, T., The stability of solitary waves (1972) Proc. R. Soc. Lond. A., 328, pp. 153-183Beresticky, H., Lions, P., Nonlinear scalar field equation I (1983) Arch. Rat. Mech. Anal., 82, pp. 313-346Berezin, F.A., Shubin, M.A., (1991) The Schrödinger equation, , Mathematics, and its Application. Kluwer AcademicBergh, J., Lofstrom, J., (1980) Interpolation Spaces, , SpringerBona, J.L., Sachs, R., Global existence of smooth solutions, and stability of solitary waves for a generalized Boussinesq equation (1988) Commun. Math. Phys., 118, pp. 15-29Bona, J.L., Saut, J.-C., Toland, J.F., Boussinesq equations, and other systems for small-amplitude long waves in nonlinear dispersive media, , to appearBona, J.L., Smith, R., The initial-value problem for the Korteweg-de Vries equation (1975) Philos. Trans. Roy. Soc. London Ser., A 278, pp. 555-604Bona, J.L., Souganidis, P., Strauss, W., Stability, and instability of solitary waves of Korteweg-de Vries type (1987) Proc. R. Soc. Lond. A., A 411, pp. 395-412Brezis, H., Gallouet, T., Nonlinear Schrödinger evolution equations (1980) Nonlinear Anal. TMA., 4, pp. 677-681Burington, R.S., (1973) Handbook of Mathematical Tables, and Formulas, , McGraw-Hill,New YorkCoddington, E.A., Levinson, N., (1955) Theory of Ordinary Differential Equations, , McGraw-Hill, New YorkGlazman, I.M., (1965) Direct methods of qualitative spectral analysis of singular differential operators, , Jerusalem, IPSTGrillakis, M., Shatah, J., Strauss, W., Stability theory of solitary waves in the presence of symmetry I. (1987) J. Funct. Anal., 74, pp. 160-197Hirota, R., Classical Boussinesq equation is a reduction of the modified KP equation (1985) J. Phys. Soc. Jpn., 54, pp. 2409-2415Hirota, R., Nakamura, A., A new example of explode-decay solitary waves in onedimension (1985) J. Phys. Soc. Jpn., 54, pp. 491-499Iorio, R.S., On the Cauchy problem for the Benjamin-Ono equation (1986) Comm. P.D.E, 11, pp. 1031-1081Iorio, R.J., (1990) Functional-Analytic Methods for P.D.E. Lectures Notes in Math., 145, pp. 104-121. , KdV B.O. and friends in weighted Sobolev spacesIorio, R.J., Nunes, W.L., (1991) Introdução às Equações de Evolução Não lineares, , 18 Colóquio Brasileiro de MatemáticaKato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equation (1983) Studies in Appl. Math. Adv. in Math. Suppl. Studies, 8, pp. 93-128Kato, T., Non-stationary flows of viscous, and ideal fluids (1972) J. Funct. Anal., 9 (3), pp. 29-36Kato, T., On the Korteweg-de Vries equation (1979) Manuscripta Math., 28, pp. 89-99Kato, T., Quasi-linear equations of evolution with application to partial differential equations (1975) Spectral theory, and Differential equations, Lectures Notes in Mathematics, Springer-Verlag, 448, pp. 25-70Kato, T., Fujita, H., On the non-stationary Navier-Stokes system (1962) Red. Sem. Mat. Uni. Padova., 32, pp. 243-260Kato, T., Ponce, G., Commutator estimates, and Euler, and Navier-Stokes equation (1988) Comm. Pure. Appl. Math., 41, pp. 891-907Kaup, D.J., A higher-order water-wave equation, and the method for solving it (1975) Prog. Theor. Phys., 54, pp. 396-408Matveev, V.B., Yavor, M.I., Solutions presque périodiques et a N-solitons de l'équation hydrodynamique non linéaire de Kaup (1979) Ann. Inst. Henri Poincaré-Physique théorique, 31 (1), pp. 25-41Pazy, A., (1983) Semigroups of linear operators, and applications to P.D.E., , Springer-VerlagPego, R.L., Weinstein, M.I., Asymptotic stability of solitary waves (1994) Comm. Math. Phys., 164, pp. 305-349Ponce, G., Smoothing properties of solutions to Benjamin-Ono equation (1990) Lectures Notes in Pure, and App. Math., 122, pp. 667-679. , (Ed.C.Sadosky) Marcel DekkerReed, S., Simon, B., (1975) Methods of Modern Mathematical Physics: Analysis of Operator AP, 5 (4)Sachs, R.L., On the integrable variant of the Boussinesq system (1988) Physica D, 30, pp. 1-28Saut, J.C., Sur quelques généralisations de l'équation de Korteweg-de Vries (1979) J. Math. Pure Appl., 58, pp. 21-61Saut, J.C., Teman, R., Remarks on the Korteweg-de Vries equation (1976) Israel J. of Math., 24, pp. 78-87Souganidis, P., Strauss, W., Instability of a class of dispersive solitary waves (1990) Proc. Roy. Soc. of Edinburgh, 114 A, pp. 195-212Tom, M., Existence of global solutions for nonlinear dispersive equations (1993) Nonlinear Analysis, TMA, 20, pp. 175-189Weinstein, M., Existence, and dynamic stability of solitary-wave solutions of equations arising in long waves propagation (1987) Comm. P.D.E., 12, pp. 1133-1173Whitham, G.B., (1974) Linear, and nonlinear waves, , Wiley-Interscience, New YorkYosida, K., (1966) Functional Analysis, , Springer-Verla

A Dispersive System Of Long Waves In Weighted Sobolev Spaces

In this article we treat the Cauchy problem for the dispersive system of long waves, in weighted Sobolev spaces. It is shown that this problem is locally well-posed in Hs(ℝ)×Hs-1(ℝ) ∩ L2 γ (ℝ) × H-1 γ (ℝ) for > 3/2 and 0 ≦ γ ≦ s.The proof involves parabolic regularization and techniques of Bona-Smith. It is also determined, using the orbital stability of the special solitary-wave solutions of this system, that we can extend globally the local solution for data sufficiently close to the solitary wave in the norm H1 (ℝ) × L2 (ℝ).32227248Abdelouhad, L., Nonlocal dispersive equations in weighted Sobolev spaces (1992) Differential and Integral Equations, 5 (2), pp. 307-338Angulo, J., On the Cauchy problem for a Boussinesq-type system to appearBergh, J., Lofstrom, J., Interpolation Spaces (1980), SpringerBona, J.L., Smith, R., The initial-value problem for the Korteweg-de Vries equation (1975) Philos. Trans. Roy. Soc. London, Ser. A, 278, pp. 555-604Grillakis, M., Shatah, J., Strauss, W., Stability theory of solitary waves in the presence of symmetry I (1987) J. Funct. Anal., 74, pp. 160-197Iorio, R.J., The Benjamin-Ono equation in weighted Sobolev spaces (1991) Math. Anal. and Appl., 157 (2), pp. 557-590Iorio, R.J., KdV, BO and friends in weighted Sobolev spaces, Functional-Analytic Methods for P.D.E. (1990) Lectures Notes in Math., 1450, pp. 104-121Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equation (1983) Studies in Appl. Math. Adv. in Math. Suppl. Studies, 8, pp. 93-128Kato, T., Non-stationary flows of viscous and ideal fluids (1972) J. Funct. Anal., 9 (3), pp. 29-36Kato, T., Ponce, G., Commutator estimates and Euler and Navier-Stokes equation (1988) Comm. Pure. Appl. Math., 41, pp. 891-907Ponce, G., Smoothing properties of solutions to Benjamin-Ono equation (1990) Lectures Notes in Pure and Appl. Math., 122, pp. 667-679. , (ed. C. Sadosky), Marcel DekkerPonce, G., Regularity of solutions to nonlinear dispersive equations (1989) J. Diff. Equations, 78 (1), pp. 122-135Procópio de Borba, M., The intermediate long-wave equation in weighted Sobolev spaces (1992) Matemática Contemporânea, 3, pp. 9-20Stein, E.M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton University PressTriebel, H., Interpolation Theory, Function Spaces Differential Operators (1972), North-HollandTsutsumi, M., Weighted Sobolev spaces and rapidly decreasing solutions of some, nonlinear dispersive wave equations (1981) J. Diff. Equations, 42, pp. 260-28

On The Instability Of Solitary Waves Solutions Of The Generalized Benjamin Equation

This work is concerned with instability properties of solutions u(x, t) = φ(x -ct) of the equation ut + (up)x + lHuxx + uxxx = 0 in R, where p ε ℕ, p ≥ 2, and H is the Hilbert transform. Here, π will be a solution of the pseudo-differential equation π'' + lHπ'-cπ =-πp solving a certain variational problem. We prove that the set ωφ = {π(· + y): y ε ℝ } is unstable by the flow of the evolution equation above provided l is small, c > 1/4l2 and p ≥ 5. Moreover, the trajectories used to exhibit instability are global and uniformly bounded. Finally, we extend these results for a natural generalization of the evolution equation above with general forms of competing dispersion, in particular, we obtain instability results for some Korteweg-de Vries type equations without requiring spectral conditions.815582Abdelouhab, L., Bona, J.L., Felland, M., Saut, J.-C., Nonlocal models for nonlinear dispersive waves (1989) NPhysica D, 40, pp. 360-392Albert, J.P., Positive properties and stability of Solitary-wave solutions of model equations for long waves (1992) Comm. PDE, 17, pp. 1-22Albert, J.P., Bona, J.L., Restrepo, J.M., Solitary-wave solutions of the Benjamin equation (1999) SIAM J. Appl. Math., 59, pp. 2139-2161Albert, J.P., Linares, F., Stability of solitary-wave solutions to long-wave equations with general dispersion. Fifth Workshop on Partial Differential Equations (Rio de Janeiro 1997) (1998) Mat. Contemp., 15, pp. 1-19Angulo, J., Existence and stability of solitary wave solutions of the Benjamin equation (1999) J. Diff. Equations, 152, pp. 136-159Angulo, J., Linares, F., Global existence of solutions of a nonlinear dispersive model (1995) J. Math. Anal. Appl., 195, pp. 797-808Benjamin, T.B., A new kind of solitary waves (1992) J. Fluid Mechanics, 254, pp. 401-411Benjamin, T.B., Solitary and periodic waves of a new kind (1996) Phil. Trans. Roy. Soc. London Ser. A, 354, pp. 1775-1806Benjamin, T.B., The stability of solitary waves (1972) Proc. Roy. Soc. London A, 338, pp. 153-183Bona, J.L., On the stability theory of solitary waves (1975) Proc. Roy. Soc. London A, 344, pp. 363-374Bona, J.L., Chen, H., Existence and asymptotic properties of solitary-wave solutions of Benjamin-type equations (1998) Adv. Diff. Eq., 3, pp. 51-84Bona, J.L., Souganidis, P.E., Strauss, W.A., Stability and instability of solitary waves of Korteweg-de Vries type (1987) Proc. Royal. Soc. London Ser. A, 411, pp. 395-412Gonçalves Ribeiro, J.M., Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field (1992) Ann. Inst. H. PoincaréPhys. Théor., 54, pp. 403-433Grillakis, M., Shatah, J., Strauss, W., Stability theory of solitary waves in the presence of symmetry I (1987) J. Funct. Anal., 74, pp. 160-197Iorio, R.J., On the Cauchy problem for the Benjamin-Ono equation (1986) Comm. Partial Differential Equations, 111, pp. 1031-1081Kato, T., Quasilinear equations of evolution with applications to Partial Differential Equations. (1975) Lectures Notes in Math., 448, pp. 25-70Kenig, C.E., Ponce, G., Vega, L., Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle (1993) Comm. Pure Appl. Math., 46, pp. 527-620Li, Y.A., Bona, J.L., Analicyticity of solitary-wave solutions of model equations for long waves (1996) SIAM J. Math. Anal., 27, pp. 725-737Linares, F., L2(R) global well-posedness of the initial value problem associated to the Benjamin equation (1999) J. Differential Equations, 152, pp. 377-393Linares, F., Scialom, M., (2001) On generalized Benjamin type equations, , PreprintLions, P.L., The concentration-compactness principle in the calculus of variations. The locally compact case, part 1 (1984) Ann. Inst. H. Poincaré, Anal. Non linéare, 1, pp. 109-145Lions, P.L., The concentration-compactness principle in the calculus of variations. The locally compact case, part 2 (1984) Ann. Inst. H. Poincaré, Anal. Non linéare, 4, pp. 223-283Martel, Y., Merle, F., (2000) Instability of solitons for the critical generalized Korteweg-de Vries equation, , preprintShatah, J., Strauss, W.A., Instability of nonlinear bound states (1985) Comm. Math. Phys., 100, pp. 173-190Souganidis, P.E., Strauss, W.A., Instability of a class of dispersive solitary waves (1990) Proc. Royal. Soc. Eding., 114 A, pp. 195-21

On The Schrödinger Equation With Singular Potentials

We study the Cauchy problem for the non-linear Schrödinger equation with singular potentials. For the point-mass potential and nonperiodic case, we prove existence and asymptotic stability of global solutions in weak-L p spaces. Specific interest is given to the point-like δ and δ′ impurity and to two δ-interactions in one dimension. We also consider the periodic case, which is analyzed in a functional space based on Fourier transform and local-in-time well-posedness is proved.2707/08/15767800Adami, R., Noja, D., Existence of dynamics for a 1D NLS equation perturbed with a generalized point defect (2009) J. Phys. A, 42 (49), p. 495302. , 19 ppAdami, R., Noja, D., Stability and symmetry-breaking bifurcation for the ground states of a NLS with a 6'-interaction (2013) Comm. Math. Phys., 318, pp. 247-289Adami, R., Noja, D., Visciglia, N., Constrained energy minimization and ground states for NLS with point defects (2013) Discrete Contin. Dyn. Syst. Ser. B, 18, pp. 1155-1188Albeverio, S., Brzezniak, Z., Dabrowski, L., Fundamental solution of the heat and schrödinger equations with point interaction (1995) J. Funct. Anal., 130, pp. 220-254Albeverio, S., Gestezy, F., Koegh-Krohn, R., Holden, H., (1988) Solvable Models in Quantum Mechanics, , Spreing-Verlag, New York Russian Transi MIR, MoscowAlbeverio, S., Kurasov, P., Singular perturbations of differential operators (2000) London Mathematical Society Lecture Note Series 271, , Cambridge University Press, CambridgeBennett, C., Sharpley, R., Interpolation of operators (1988) Pure and Applied Mathematics, 129. , Academic PressBergh, J., Lofstrom, J., (1976) Interpolation Spaces, , Springer-Verlag, Berlin-New YorkBraz E Silva, P., Ferreira, L.C.F., Villamizar-Roa, E.J., On the existence of infinite energy solutions for nonlinear schrödinger equations (2009) Proc. Amer. Math. Soc, 137, pp. 1977-1987Caudrelier, V., Mintchev, M., Ragoucy, E., The quantum non-linear schrödinger equation with point-like defect (2004) J. Physics A: Mathematical and General, 37, pp. L367-L375Caudrelier, V., Mintchev, M., Ragoucy, E., Solving the quantum non-linear schrödinger equation with δ-type impurity (2005) J. Math. Phys., 46, 24pCazenave, T., Semilinear schrödinger equations (2003) Courant Lecture Notes in Mathematics, 10. , Providence, RI: American Mathematical SocietyCazenave, T., Vega, L., Vilela, M.C., A note on the nonlinear schrödinger equation in weak Lp spaces (2001) Commun. Contemp. Math., 3, pp. 153-162Cazenave, T., Weissler, F.B., Asymptotically self-similar global solutions of the nonlinear schrödinger and heat equations (1998) Math. Z., 228, pp. 83-120Duchêne, V., Marzuola, J., Weinstein, M., Wave operator bounds for one-dimensional schrödinger operators with singular potentials and applications (2011) J. Math. 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Existence And Stability Of Ground-state Solutions Of A Schrödinger-kdv System

We consider the coupled Schrödinger-Korteweg-de Vries system i(u t + c1ux) + δ1uxx = αuv, vt + c2vx + δ 2vxxx + γ(v2)x = β(|u|2)x, which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Ground states of the system are, by definition, minimizers of the energy functional subject to constraints on conserved functionals associated with symmetries of the system. In particular, ground states have a simple time dependence because they propagate via those symmetries. For a range of values of the parameters α, β, γ, δi, ci, we prove the existence and stability of a two-parameter family of ground states associated with a two-parameter family of symmetries.13359871029Albert, J., Concentration compactness and the stability of solitary-wave solutions to nonlocal equations (1999) Applied Analysis, pp. 1-29. , (ed. J. Goldstein et al.) 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Psychosocial functioning during the treatment of major depressive disorder with fluoxetine.

BACKGROUND: Major depressive disorder (MDD) is associated with significant disability, having a profound impact on psychosocial functioning. Therefore, studying the impact of treatment on psychosocial functioning in MDD could help further improve the standard of care. METHODS: Two hundred twenty-two MDD outpatients were treated openly with 20 mg fluoxetine for 8 weeks. The self-report version of the Social Adjustment Scale was administered at baseline and during the final visit. We then tested for the relationships between (1) self-report version of the Social Adjustment Scale scores at baseline and clinical response, (2) nonresponse, response and remission status and overall psychosocial adjustment at end point, (3) the number/severity of residual depressive symptoms and overall psychosocial adjustment at end point in responders, and (4) the time to onset of response and overall psychosocial adjustment at end point. RESULTS: An earlier onset of clinical response predicted better overall psychosocial functioning at end point (P = 0.0440). Responders (n = 128) demonstrated better overall psychosocial adjustment at end point than nonresponders (P = 0.0003), while remitters (n = 64) demonstrated better overall psychosocial adjustment at end point than nonremitted responders (P = 0.0031). In fact, a greater number/severity of residual symptoms predicted poorer overall psychosocial adjustment at end point in responders (P = 0.0011). Psychosocial functioning at baseline did not predict response. CONCLUSIONS: While MDD patients appear equally likely to respond to treatment with fluoxetine, regardless of their level of functioning immediately before treatment, the above results stress the importance of achieving early symptom improvement then followed by full remission of depressive symptoms with respect to restoring psychosocial functioning in MDD