Scattering of the Radial Focusing Mass-Supercritical and Energy-Subcritical Nonlinear Schrödinger Equation in 3D

Affiliation(s)

Department of Mathematics, Faculty of Education, Kassala University, Kassala, Sudan.

Department of Mathematics, Faculty of Pure and Applied Sciences, Fourah Bay College, University of Sierra Leone, Freetown, Sierra Leone.

Department of Mathematics, Faculty of Education, Kassala University, Kassala, Sudan.

Department of Mathematics, Faculty of Pure and Applied Sciences, Fourah Bay College, University of Sierra Leone, Freetown, Sierra Leone.

ABSTRACT

This paper studies the global behavior to 3*D* focusing nonlinear Schrodinger equation (NLS), the scaling index here is (*0**＜**s _{c}*

Cite this paper

M. M-Ali and A. Kamara, "Scattering of the Radial Focusing Mass-Supercritical and Energy-Subcritical Nonlinear Schrödinger Equation in 3D,"*Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 164-171. doi: 10.4236/apm.2013.31A023.

M. M-Ali and A. Kamara, "Scattering of the Radial Focusing Mass-Supercritical and Energy-Subcritical Nonlinear Schrödinger Equation in 3D,"

References

[1] T T. Tao, “On the Asymptotic Behavior of Large Radial Data for a Focusing Non-Linear Schrodinger Equation,” Dynamics of Partial Differential Equations, Vol. 1, No. 1, 2004, pp. 1-48.

[2] T. Tao, “A (Concentration-) Compact Attractor for High-Dimensional Non-Linear Schrodinger Equation,” Dynamics of Partial Differential Equations, Vol. 4, No. 1, 2007, pp. 1-53.

[3] T. Cazenave, “Semilinear Schrodinger Equations. Courant Lecture Notes in Mathematics,” New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, 2003.

[4] T. Tao, “Nonlinear Dispersive Equations: Local and Global Analysis,” CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington DC, American Mathematical Society, Providence, 2006.

[5] M. Weinstein, “Nonlinear Schrodinger Equations and Sharp Interpolation Estimates,” Communications in Mathematical Physics, Vol. 87, No. 4, 1982, pp. 567-576.

[6] J. Holmer and S. Roudenko, “A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrodinger Equations,” Communications in Mathematical Physics, Vol. 282, No. 2, 2008, pp. 435-467.

[7] C. E. Kenig and F. Merle, “Global Well-Posedness, Scattering, and Blow-Up for the Energy-Critical Focusing Nonlinear Schrodinger Equation in the Radial Case,” Inventiones Mathematicae, Vo. 166, No. 3, 2006, pp. 645- 675.

[8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, “Global Existence and Scattering for Rough Solutions of a Nonlinear Schrodinger Equation on R3,” Communications on Pure and Applied Mathematics, Vol. 57, No. 8, 2004, pp. 987-1014.

[9] T. Hmidi and S. Keraani, “Blowup Theory for the Critical Nonlinear Schrodinger Equations Revisited,” International Mathematics Research Notices, Vol. 2005, No. 46, 2005, pp. 2815-2828. doi:10.1155/IMRN.2005.2815

[10] F. Merle and Y. Tsutsumi, “L2 Concentration of Blow-Up Solutions for the Nonlinear Schrodinger Equation with Critical Power Nonlinearity,” Journal of Differential Equations, Vol. 84, No. 2, 1990, pp. 205-214.

[11] W. A. Strauss, “Existence of Solitary Waves in Higher Dimensions,” Communications in Mathematical Physics, Vol. 55, No. 2, 1977, pp. 149-162.

[12] D. Foschi, “Inhomogeneous Strichartz Estimates,” J. Hyper. Diff. Eq., Vol. 2, No. 1, 2005, pp. 1-24.

[13] C. Kenig, G. Ponce and L. Vega, “Well-Posedness and Scattering Results for the Generalized Korteweg-De Vries Equation via the Contraction Principle,” Communications on Pure and Applied Mathematics, Vol. 46, No. 4, 1993, pp. 527-620.

[14] S. Keraani, “On the Defect of Compactness for the Strichartz Estimates of the Schrodinger Equation,” Journal of Differential Equations, Vol. 175, No. 2, 2001, pp. 353- 392. doi:10.1006/jdeq.2000.3951

[15] E. Donley, N. Claussen, S. Cornish, J. Roberts, E. Cornell and C. Wieman, “Dynamics of Collapsing and Exploding Bose-Einstein Condensates,” Nature, Vol. 412, No. 6844, 2001, pp. 295-299. doi:10.1038/35085500

[16] C. Sulem and P.-L. Sulem, “The Nonlinear Schrodinger Equation. Self-Focusing and Wave Collapse,” Applied Mathematical Sciences, Vol. 139, 1999.

[17] M. Vilela, “Regularity of Solutions to the Free Schrodinger Equation with Radial Initial Data,” Illinois Journal of Mathematics, Vol. 45, No. 2, 2001, pp. 361-370.

[18] L. Bergé, T. Alexander and Y. Kivshar, “Stability Criterion for Attractive Bose-Einstein Condensates,” Physical Review A, Vol. 62, No. 2, 2000, 6 p.

[19] P. Bégout, “Necessary Conditions and Sufficient Conditions for Global Existence in the Nonlinear Schrodinger Equation,” Advances in Applied Mathematics and Mechanics, Vol. 12, No. 2, 2002, pp. 817-827.

[1] T T. Tao, “On the Asymptotic Behavior of Large Radial Data for a Focusing Non-Linear Schrodinger Equation,” Dynamics of Partial Differential Equations, Vol. 1, No. 1, 2004, pp. 1-48.

[2] T. Tao, “A (Concentration-) Compact Attractor for High-Dimensional Non-Linear Schrodinger Equation,” Dynamics of Partial Differential Equations, Vol. 4, No. 1, 2007, pp. 1-53.

[3] T. Cazenave, “Semilinear Schrodinger Equations. Courant Lecture Notes in Mathematics,” New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, 2003.

[4] T. Tao, “Nonlinear Dispersive Equations: Local and Global Analysis,” CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington DC, American Mathematical Society, Providence, 2006.

[5] M. Weinstein, “Nonlinear Schrodinger Equations and Sharp Interpolation Estimates,” Communications in Mathematical Physics, Vol. 87, No. 4, 1982, pp. 567-576.

[6] J. Holmer and S. Roudenko, “A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrodinger Equations,” Communications in Mathematical Physics, Vol. 282, No. 2, 2008, pp. 435-467.

[7] C. E. Kenig and F. Merle, “Global Well-Posedness, Scattering, and Blow-Up for the Energy-Critical Focusing Nonlinear Schrodinger Equation in the Radial Case,” Inventiones Mathematicae, Vo. 166, No. 3, 2006, pp. 645- 675.

[8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, “Global Existence and Scattering for Rough Solutions of a Nonlinear Schrodinger Equation on R3,” Communications on Pure and Applied Mathematics, Vol. 57, No. 8, 2004, pp. 987-1014.

[9] T. Hmidi and S. Keraani, “Blowup Theory for the Critical Nonlinear Schrodinger Equations Revisited,” International Mathematics Research Notices, Vol. 2005, No. 46, 2005, pp. 2815-2828. doi:10.1155/IMRN.2005.2815

[10] F. Merle and Y. Tsutsumi, “L2 Concentration of Blow-Up Solutions for the Nonlinear Schrodinger Equation with Critical Power Nonlinearity,” Journal of Differential Equations, Vol. 84, No. 2, 1990, pp. 205-214.

[11] W. A. Strauss, “Existence of Solitary Waves in Higher Dimensions,” Communications in Mathematical Physics, Vol. 55, No. 2, 1977, pp. 149-162.

[12] D. Foschi, “Inhomogeneous Strichartz Estimates,” J. Hyper. Diff. Eq., Vol. 2, No. 1, 2005, pp. 1-24.

[13] C. Kenig, G. Ponce and L. Vega, “Well-Posedness and Scattering Results for the Generalized Korteweg-De Vries Equation via the Contraction Principle,” Communications on Pure and Applied Mathematics, Vol. 46, No. 4, 1993, pp. 527-620.

[14] S. Keraani, “On the Defect of Compactness for the Strichartz Estimates of the Schrodinger Equation,” Journal of Differential Equations, Vol. 175, No. 2, 2001, pp. 353- 392. doi:10.1006/jdeq.2000.3951

[15] E. Donley, N. Claussen, S. Cornish, J. Roberts, E. Cornell and C. Wieman, “Dynamics of Collapsing and Exploding Bose-Einstein Condensates,” Nature, Vol. 412, No. 6844, 2001, pp. 295-299. doi:10.1038/35085500

[16] C. Sulem and P.-L. Sulem, “The Nonlinear Schrodinger Equation. Self-Focusing and Wave Collapse,” Applied Mathematical Sciences, Vol. 139, 1999.

[17] M. Vilela, “Regularity of Solutions to the Free Schrodinger Equation with Radial Initial Data,” Illinois Journal of Mathematics, Vol. 45, No. 2, 2001, pp. 361-370.

[18] L. Bergé, T. Alexander and Y. Kivshar, “Stability Criterion for Attractive Bose-Einstein Condensates,” Physical Review A, Vol. 62, No. 2, 2000, 6 p.

[19] P. Bégout, “Necessary Conditions and Sufficient Conditions for Global Existence in the Nonlinear Schrodinger Equation,” Advances in Applied Mathematics and Mechanics, Vol. 12, No. 2, 2002, pp. 817-827.