Agrawal, G.P.: Nonlinear Fiber Optics. Academic Press Inc, Cambridge (1995)

Google Scholar 

Adhikari, S.K.: Mean-field description of collapsing and exploding Bose-Einstein condensates. Phys. Rev. A 66, 13611–13619 (2002)

Article  ADS  Google Scholar 

Ahn, J., Kim, J., Seo, I.: On the radius of spatial analyticity for defocusing nonlinear Schrödinger equations. Discrete Contin. Dyn. Syst. 40, 423–439 (2020)

Article  MathSciNet  Google Scholar 

Barab, J.: Nonexistence of asymptotically free solutions of a nonlinear Schrödinger equation. J. Math. Phys. 25, 3270–3273 (1984)

Article  ADS  MathSciNet  Google Scholar 

Bao, W., Jaksch, D., Markowich, P.A.: Three-dimensional simulation of jet formation in collapsing condensates. J. Phys. B At. Mol. Opt. Phys. 37, 329 (2004)

Article  ADS  Google Scholar 

Biswas, A.: Optical soliton perturbation with nonlinear damping and saturable amplifiers. Math. Comput. Simul. 56, 521–37 (2001)

Article  MathSciNet  Google Scholar 

Cazenave, T.: Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2003)

Cazenave, T., Han, Z.: Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation. Descrete Contin. Dyn. Syst. 40, 4801–4819 (2020)

Article  Google Scholar 

Cazenave, T., Han, Z., Naumkin, I.: Asymptotic behavior for a dissipative nonlinear Schrödinger equation. Nonlinear Anal. 205, 112243 (2021)

Article  Google Scholar 

Cazenave, T., Naumkin, I.: Local existence, global existence, and scattering for the nonlinear Schrödinger equation. Commun. Contemp. Math. 19, 1650038 (2017)

Article  MathSciNet  Google Scholar 

Cazenave, T., Naumkin, I.: Modified scattering for the critical nonlinear Schrödinger equation. J. Funct. Anal. 274, 402–432 (2018)

Article  MathSciNet  Google Scholar 

Cazenave, T., Weissler, F.: The Cauchy problem for the nonlinear Schrödinger equation in \(H^1\). Manuscr. Math. 61, 477–494 (1988)

Article  Google Scholar 

Chihara, H.: Gain of analyticity for semilinear Schrödinger equations. In: Proceedings of Sapporo Guest House Mini Symposium on Nonlinear Wave Equations, pp. 28–29 (1999)

Doi, S.: On the Cauchy problem for Schrödinger type equations and regularity of solutions. J. Math. Kyoto Univ. 79, 319–328 (1994)

Google Scholar 

Fibich, G.: The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse. Springer, Berlin (2015)

Book  Google Scholar 

Ferlaino, F., Knoop, S., Berninger, M., Harm, W., D’Incao, J.P., Nagerl, H.C., Grimm, R.: Evidence for universal four-body states tied to an efimov trimer. Phys. Rev. Lett. 102, 140401 (2009)

Article  ADS  Google Scholar 

Ginible, J., Velo, G.: On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. J. Funct. Anal. 32, 1–31 (1979)

Article  Google Scholar 

Hayashi, N., Miao, C., Naumkin, P.I.: Global existence of small solutions to the generalized derivative nonlinear Schrödinger equahon. Asymptot. Anal. 21, 133–147 (1999)

MathSciNet  Google Scholar 

Hayashi, N., Naumkin, P.I.: Asymptotics for large time of solutions to nonlinear Schrödinger and Hartree equations. Am. J. Math. 120, 369–389 (1998)

Article  Google Scholar 

Hayashi, N., Li, C., Naumkin, P.I.: Time decay for nonlinear dissipative Schrödinger equations in optical fields. Adv. Math. Phys. 3702738 (2016)

Hoshino, G.: Global well-posedness and analytic smoothing effect for the dissipative nonlinear Schrödinger equations. J. Dyn. Differ. Equ. 31, 2339–2351 (2019)

Article  Google Scholar 

Hoshino, G.: Asymptotic behavior for solutions to the dissipative nonlinear Schrödinger equations with the fractional Sobolev space. J. Math. Phys. 60, 111504, 11 (2019)

Article  Google Scholar 

Hoshino, G., Ozawa, T.: Analytic smoothing effect for nonlinear Schrödingier equations with quintic nonlinearity. J. Math. Anal. Appl. 419, 285–297 (2014)

Article  MathSciNet  Google Scholar 

Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Theor. 46, 113–129 (1987)

MathSciNet  Google Scholar 

Kagan, Y., Muryshev, A.E., Shlyapnikov, G.V.: Collapse and Bose–Einstein condensation in a trapped Bose gas with negative scattering length. Phys. Rev. Lett. 81, 933 (1998)

Article  ADS  Google Scholar 

Keel, M., Tao, T.: Endpoint Strichartz estimates. Am. J. Math. 120, 955–980 (1998)

Article  MathSciNet  Google Scholar 

Kim, D., Sunagawa, H.: Remarks on decay of small solutions to systems of Klein-Gordon equations with dissipative nonlinearities. Nonlinaer. Anal. 97, 94–105 (2014)

Article  MathSciNet  Google Scholar 

Kita, N., Shimomura, A.: Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity. J. Differ. Equ. 242, 192–210 (2007)

Article  ADS  Google Scholar 

Kita, N., Shimomura, A.: Large time behavior of solutions to Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data. J. Math. Soc. Jpn. 61, 39–64 (2009)

Article  Google Scholar 

Kita, N., Sato, T.: Optimal \(L^2\)-decay of solutions to a cubic dissipative nonlinear Schrödinger equation. Asymptot. Anal. 129, 505–517 (2022)

MathSciNet  Google Scholar 

Kobayashi, M., Tsubota, M.: Kolmogorov spectrum of superfluid turbulence: numerical analysis of the Gross–Pitaevskii equation with a small-scale dissipation. Phys. Rev. Lett. 94, 065302 (2005)

Article  ADS  Google Scholar 

Kobayashi, M., Tsubota, M.: Thermal dissipation in quantum turbulence. Phys. Rev. Lett. 97, 145301 (2006)

Article  ADS  Google Scholar 

Kobayashi, M., Tsubota, M.: Quantum turbulence in a trapped Bose–Einstein condensate. Phys. Rev. A 76, 045603 (2007)

Article  ADS  Google Scholar 

Linares, F., Miyazaki, H., Ponce, G.: On a class of solutions to the generalized KdV type equation. Commun. Contemp. Math. 21, 1850056, 21 (2019)

Article  MathSciNet  Google Scholar 

Linares, F., Ponce, G., Santos, G.N.: On a class of solutions to the generalized derivative Schrödinger equations. Acta Math. Sin. (Engl. Ser.) 35, 1057–1073 (2019)

Article  MathSciNet  Google Scholar 

Linares, F., Ponce, G., Santos, G.N.: On a class of solutions to the generalized derivative Schrödinger equations II. J. Differ. Equ. 267, 97–118 (2019)

Article  ADS  Google Scholar 

Li, C., Nishii, Y., Sagawa, Y., Sunagawa, H.: Upper and lower \(L^2\)-decay bounds for a class of derivative nonlinear Schrödinger equations. Discrete Contin. Dyn. Syst. 42(12), 5893–5908 (2022)

Article  MathSciNet  Google Scholar 

Li, C., Nishii, Y., Sagawa, Y., Sunagawa, H.: Recent advances on Schrödinger equations with dissipative nonlinearities, preprint (2023). arXiv:2204.07320

Li, C., Sunagawa, H.: On Schrödinger systems with cubic dissipative nonlinearities of derivative type. Nonlinearity 29, 1537–1563 (2016)

Article  ADS  MathSciNet  Google Scholar 

Masaki, S., Sugiyama, K.: Optimal decay rate of solutions for nonlinear Klein–Gordon systems of critical type. Differ. Integral Equ. 33, 247–256 (2020)

MathSciNet  Google Scholar 

Moll, K.D., Gaeta, A.L., Fibich, G.: Self-similar optical wave collapse: observation of the townes profile. Phys. Rev. Lett. 90(20) (2003)

Nishii, Y., Sunagawa, H.: On Agemi-type structural conditions for a system of semilinear wave equations. J. Hypabolic Differ. Equ. 17, 459–473 (2020)

Article  MathSciNet  Google Scholar 

Ogawa, T., Sato, T.: \(L^2\)-decay rate for the critical nonlinear Schrödinger equation with a small smooth data. Nonlinear Differ. Equ. Appl. 27, 18 (2020)

Article  Google Scholar 

Ozawa, T.: Long range scattering for nonlinear Schrödinger equations in one space dimension. Commun. Math. Phys. 139, 479–493 (1991)

Article  ADS  Google Scholar 

Schonbek, M.E.: \(L^2\) decay for weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 88, 209–222 (1985)

Article  Google Scholar 

Sato, T.: \(L^2\)-decay estimate for the dissipative nonlinear Schrödinger equation in the Gevrey class. Arch. Math. 115, 575–588 (2020)

Article  MathSciNet  Google Scholar 

Sato, T.: Lower bound estimate for the dissipative nonlinear Schrödinger equation. SN Partial Differ. Equ. Appl. 2, 66 (2021)

Article  Google Scholar 

Sunagawa, H.: Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms. J. Math. Soc. Jpn. 58, 379–400 (2006)

Article  MathSciNet  Google Scholar 

Shimomura, A.: Asymtotic behavior of solutions for Schrödinger equations with dissipative nonlinearities. Commun. Partial Differ. Equ. 31, 1407–1423 (2006)

Article  Google Scholar 

Stein, E., Weiss, G.: Introduction to Fourier Analysis on Euclidian spaces, Princeton Landmarks in Mathematics, Princeton University Press, Princeton NJ (1971). Reprint of the second (1956) edition; Princeton Paperbacks

Tesfahun, A.: On the radius of spatial analyticity for cubic nonlinear Schrödinger equations. J. Differ. Equ. 263, 7496–7512 (2017)

Article  ADS  Google Scholar 

Tsutsumi, Y.: \(L^2\) solution for nonlinear Schrödinger equation and nonlinear groups. Funk. Ekva. 30, 115–125 (1987)

Google Scholar 

Tsutsumi, Y., Yajima, K.: The asymptotic behavior of nonlinear Schrödinger equations. Bull. Am. Math. Soc. 11, 186–188 (1984)

Article  Google Scholar 

Uchida, H.: Analicity of Solutions to Nonlinear Schrödinger Equahons, \(\rm SUT \). J. Math. 37, 105–135 (2001)