On the Stability of Periodic Solutions of the Generalized Benjamin-Bona-Mahony Equation

We study the stability of a four parameter family of spatially periodic traveling wave solutions of the generalized Benjamin-Bona-Mahony equation to two classes of perturbations: periodic perturbations with the same periodic structure as the underlying wave, and long-wavelength localized perturbations. In particular, we derive necessary conditions for spectral instability to perturbations to both classes of perturbations by deriving appropriate asymptotic expansions of the periodic Evans function, and we outline a nonlinear stability theory to periodic perturbations based on variational methods which effectively extends our periodic spectral stability results.Comment: 27 pages, 3 figure

Speleological objects of the south and southeast Velebit

Velebit je najprostranija planina u Hrvatskoj i najveće zaštićeno područje. Dijeli se na četiri dijela. Ovaj rad bavi se analizom fizičko-geografskih obilježja južnog i jugoistočnog Velebita, a prvenstveno speleoloških objekata te kratke povijesti dosadašnjih istraživanja nekih važnih objekata (špiljski sustav Kita Gaćešina – Draženova Puhaljka i Cerovačke špilje). Južni i jugoistočni Velebit skriva speleološke objekte koji su od velikog znanstvenog značaja. Rad se temelji na korištenju dostupne literature, speleoloških nacrta i baza podataka, internetskih izvora, te terenskog rada.Velebit is the most widespread mountain in Croatia and the largest protected area. It is divided into four parts. This paper deals with the analysis of the physical and geographical features of the south and southeastern Velebit, primarily speleological objects and a short history of previous research of some important objects (Kita Gaćešina - Draženova Puhaljka and Cerovačka caves). Southern and Southeastern Velebit hides speleological objects that are of great scientific significance. The work is based on the use of available literature, speleological layouts and databases, internet sources, and field work

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

The Cuban Conundrum: Proposing an International Trademark Registry for Well-Known Foreign Marks

The analysis of the U.S.-Cuba trademark disputes can provide guidance for the future in weighing how a country should balance its international politics with its obligations under intellectual property law. This Comment argues that countries set a dangerous precedent when they rely on temporary political considerations to block well-known foreign trademarks or interfere with internationally agreed-upon intellectual property rights. This Comment searches for a feasible solution that would prevent a country from violating international intellectual property treaties by registering its own version of a well-known foreign mark\u27thus leading to consumer confusion about the source and origin of the mark\u27simply because a political relationship has deteriorated between the country of registration and the country of the foreign mark\u27s origin. Because avoidance of consumer confusion is the ultimate purpose of trademark law, this Comment advocates for an international, centralized registration system for well-known foreign trademarks