On the Unique Continuation Property for the Higher Order Nonlinear Schrödinger Equation With Constant Coefficients

We solve the unique continuation property: If u is a solution of the higher order nonlinear Schrödinger equation with constant coefficients with t1 < t2 which is sufficiently smooth and such that supp u( . , tj) \subset (a, b), -\infty < a < b < \infty, j = 1, 2, then u \equiv 0.

On the Unique Continuation Property for the Higher Order Nonlinear Schrödinger Equation With Constant Coefficients

We solve the unique continuation property: If u is a solution of the higher order nonlinear Schrödinger equation with constant coefficients with t1 < t2 which is sufficiently smooth and such that supp u( . , tj) \subset (a, b), -\infty < a < b < \infty, j = 1, 2, then u \equiv 0.

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  • Bona, J. and Scott, R.: Solutions of the Korteweg - de Vries equation in fractional order Sobolev space. Duke Math. J. 87-99. 43(1976).
  • Bona, J. and Smith, R.: The initial value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. London. 555-604. A278(1975).
  • Bourgain J.: On the compactness of the support of solutions of dispersive equations. Inter- nat. Math. Res. Notices. 437-447. 9(1997).
  • Carleman, T.: Sur les systemes lin´eaires aux d´eriv´ees partielles du premier ordre a deux variables. C. R. Acad. Sci. Paris. 471-474. 197(1933).
  • Carvajal, X. and Linares, F.: Ahigher order nonlinear Schr¨odinger equation with variable coefficients. Differential and Integral Equations. 1111-1130. 16(2003).
  • Carvajal, X.: Local well-posedness for a higher order nonlinear Schr¨odinger equation in Sobolev space of negative indices. EJDE. 1-10. 13(2004).
  • Cohen, A.: Solutions of the Korteweg - de Vries equations from irregular data. Duke Math., J. 149-181. 45(1991).
  • Escauriaza, L., Kenig C. E., Ponce G. and Vega L.: On unique continuation of solutions of Schr¨odinger equations. Preprint. Ginibre, J. and Tsutsumi, Y.: Uniqueness of solutions for the generalized Korteweg-de Vries equation. SIAM J. Math. Anal. 1388-1425. 20(1989).
  • Hasegawa, A. and Kodama, Y.: Nonlinear pulse propagation in a monomode dielectric guide. IEEE. J. Quant. Elect. 510-524. 23(1987).
  • Hayashi, N., Nakamitsu, K. and Tsutsumi, M.: On solutions on the initial value problem for the nonlinear Schr¨odinger equations in One Space Dimension. Math. Z. 637-650. 192(1986).
  • Hayashi, N., Nakamitsu, K. and Tsutsumi, M.: On solutions of the initial value problem for nonlinear Schr¨odinger equations. J. of Funct. Anal. 218-245. 71(1987). Hormander, L.: Linear Partial Differential Operators. Springer.Verlag.
  • Berlin/Heidelberg/New York. 1969.
  • Kato, T.: On the cauchy problem for the (generalized) Korteweg-de Vries equation. Ad- vancesin MathematicsSupplementary Studiesin Applied Math. 93-128. 8(1983).
  • Kenig, C. E., Ponce, G. and Vega, L.: Oscillatory integrals and regularity of dispersive equations. Indiana University Math. J. 33-69. 40(1991).
  • Kenig, C. E., Ponce, G. and Vega, L.: Well-posedness and scattering results for the gener- alized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. 620. 46(1993).
  • Kenig, C. E., Ponce, G. and Vega, L.: Higher-order nonlinear dispersive equations. Proc. Amer. Math. Soc. 157-166. 122(1994).
  • Kenig, C. E., Ponce, G. and Vega, L.: On the support of solutions to the generalized KdV equation. Analise non linare. 191-208. 19(1992).
  • Kenig, C. E., Ponce, G. and Vega, L.: On unique continuation for nonlinear Schr¨odinger equations. Comm. on Pure and Appl. Math. 1247-1262. 55(2002).
  • Kenig, C. E., Ruiz A. and Sogge, C.: Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J. 329-347. 55(1987).
  • Kenig, C. E. and Sogge, C.: Anote on unique continuation for Schrodinger’s operator. Proc. Amer. Math. Soc. 543-546. 103(1988).
  • Kodama, Y.: Optical solitons in a monomode Şber. J. Phys. Stat. 596-614. 39(1985).
  • Kozakevicius, A. and Vera, O.: On the unique continuation property for a nonlinear dis- persive system. EJQTDE. 1-23. 14(2005).
  • Laurey, C.: Le problme de Cauchy pour une quation de Schr¨odinger non-linaire de ordre 3. C. R. Acad. Sci. Paris. 165-168. 315(1992).
  • Mizohata, S.: Unicit´e du prolongement des solutions pour quelques op´erateurs diff´erentiels paraboliques. Mem. Coll. Sci. Univ. Kyoto. 219-239. A31(1958).
  • Nirenberg, L.:Uniqueness of Cauchy problems for differential equations with constant leading coefficient. Comm. Pure Appl. Math. 89-105. 10(1957).
  • Peetre, J.: Espaces d’interpolation et th´eor´eme de Sobolev. Ann. Inst. Fourier. 279-317. (1966).
  • Lions, J. L.: Quelques mthodes de rsolution des problmes aux limites non linaires. Gauthiers- Villars. Paris. 1969.
  • Robbiano, L.: Th´eor´eme d’unicit´e adapt´e au contr´ole des solutions des problemes hyper- boliques. Comm. PDE. 789-800. 16(1991).
  • Saut, J. C. and Scheurer, B.: Unique continuation for some evolution equations. J. Diff. Eqs. 118.139. 66(1987).
  • Saut, J. C. and Temam, R.: Remark on the Korteweg-de Vries equation. Israel J. Math. 87. 24(1976).
  • Schechter, M. and Simon, B.: Unique continuation for Schrodinger o perators with un- bounded potentials. J. Math. Anal. Appl. 482-492. 77(1980).
  • Staffilani, G.: On the generalized Korteweg-de Vries type equation. Vol. 10. 777-796. 4(1997).
  • Temam R.: Sur un probleme non lin´eaire. J. Math. PuresAppl. 157-172. 48(1969).
  • Vera, O.: Gain in regularity for the higher order nonlinear Schr¨odinger equation with constant coefficients. Submitted. Zhang, B. Y.: Hardy function and unique continuation for evolution equations. J. Math. Anal. Appl. 381-403. 178(1993).
  • Zhang, B. Y.: Unique continuation for the Korteweg-de Vries equation. SIAM, J. Math. Anal. 55-71. 23. Vanilde BISOGNIN ´ Area de CiˆenciasNaturaise Tecnol´ogicasda UNIFRA Centro Universit´ario Franciscano Santa Maria, RS. BRAZIL e-mail: vanilde@unifra.br Octavio Paulo Vera VILLAGR ´AN Departamento de Matem´atica, Universidad del B´ıo-B´ıo, Collao 1202, Casilla 5-C, Concepci´on, Chile e-mail: overa@ubiobio.cl