Stochastic Extended Korteweg-De Vries Equation

Yıl 2019, Cilt: 2 Sayı: 2, 74 - 81, 30.08.2019

Anna Karczewska Maciej Szczecinski

https://doi.org/10.33187/jmsm.459461

Öz

In the paper, we consider stochastic Korteweg-de Vries - type equation. We give sufficient conditions for the existence and uniqueness of the local mild solution to the equation with additive noise.  We discuss the possibility of the globalization of mild solution, as well.

Anahtar Kelimeler

Extended KdV equation, Extended KdV, Mild solution, Near identity transformation

Kaynakça

• [1] D.J. Korteweg, H. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philosophical Magazine, 39 (1895), 422-443.
• [2] P.G. Drazin, R.S. Johnson, . Solitons: An introduction, Cambridge University Press, 1989.
• [3] E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos, 2nd ed., Cambridge University Press, 2000.
• [4] A. Jeffrey, Role of the Korteweg-de Vries equation in plasma physics, Q. Jl R. Astr. Soc., 14 (1973), 183-189.
• [5] M. Remoissenet, Waves called solitons, Springer, 1994.
• [6] T.R. Marchant, N.F. Smyth, The extended Korteweg-de Vries equation and the resonant flow of a fluid over topography. J. Fluid Mech.,S 221 (1990), 263-288.
• [7] E. Infeld, E., Karczewska, A., Rowlands, G. and Rozmej, P.: Exact cnoidal solutions of the extended KdV equation, Acta Phys. Pol. A, 133 (2018), 1191-1199. DOI: 10.12693/APhysPolA.133.1191
• [8] P. Rozmej, A. Karczewska, New Exact Superposition Solutions to KdV2 Equation, Advances in Mathematical Physics, 2018 Article ID 5095482, 1-9. DOI: 10.1155/2018/5095482
• [9] P. Rozmej, A. Karczewska, E. Infeld, Superposition solutions to the extended KdV equation for water surface waves, Nonlinear Dynamics 91 (2018), 1085-1093. DOI: 10.1007/s11071-017-3931-1
• [10] F. Linares, G. Ponce, Introduction to Nonlinear Dispersive Equations. Universitext, Springer, 2009.
• [11] T. Tao, Nonlinear Dispersive Equations, Local and Global Analysis, CBMS Regional Conference Series, 106, American Mathematical Society, 2006.
• [12] A. de Bouard, A. Debussche, On the stochastic Korteweg-de Vries equation, J. Funct. Anal. 154 (1998), 215-251.
• [13] C.E. Kenig, G. Ponce, L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.
• [14] C.E. Kenig, G. Ponce, L. Vega, Well-posedness for the generalized Korteweg-de Vries equation via contraction principle Comm. Pure Appl. Math., 46 (1993), 527-620.
• [15] A. Karczewska, P. Rozmej, E. Infeld, G. Rowlands, Adiabatic invariants of the extended KdV equation, Phys. Lett. A, 381 (2017), 270-275.
• [16] Y. Kodama, On integrable systems with higher order corrections, Phys. Lett. A., 107 (1985), 245-249.
• [17] H.R. Dullin, G.A. Gottwald, D.D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett. 87 (2001), 194501.
• [18] R.A. Adams, Sobolev Spaces, Academic Press, 1975.
• [19] R. Grimshaw, Internal Solitary Waves, Presented at the international conference ”Progress in Nonlinear Science”, held in Nizhni Novgorod in July 2001, and dedicated to the 100-th Anniversary of Alexander A. Andronov.
• [20] G. Grimshaw, G. El, K. Khusnutdinova, Nonlinear Waves, Lecture 12: Higher-order KdV equations 2010.