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An Initial-Boundary Value Problem for a Modified Transitional Korteweg-de Vries Equation

DOI: 10.4236/jamp.2025.131005, PP. 138-147

Keywords: Modified Transitional KdV Equation, Initial-Boundary Value Problem, Semi-Group, Local and Global Existence

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Abstract:

We study the following modified transitional Korteweg-de Vries equation u t +f( t ) u p u x + u xxx =0 , ( x,t ) R + × R + , ( p2 is an even integer) with initial value u( x,0 )=g( x ) H 4 ( R + ) and inhomogeneous boundary value u( 0,t )=Q( t ) C 2 ( [ 0, ) ) . Under the conditions either (i) f( t )0 , f ( t )0 or (ii) f( t )α where α>0 , we prove the

References

[1]  Bona, J. and Smith, R. (1975) The Initial Value Problem for the Korteweg-de Vries Equation. Philosophical Transactions of the Royal Society of London Series A, 278, 555-601.
[2]  Miura, R. (1974) The Korteweg-de Vries Equation: A Model for Nonlinear Dispersive Waves. In: Leibovich, S. and Seebass, R., Eds., Nonlinear Waves, Cornell Univ. Press.
[3]  Benjamin, T., Bona, J. and Mahony, J. (1972) Model Equations for Long Waves in Nonlinear Dispersive Systems. Philosophical Transactions of the Royal Society of London Series A, 227, 47-48.
[4]  Bona, J.L. and Bryant, P.J. (1973) A Mathematical Model for Long Waves Generated by Wavemakers in Non-Linear Dispersive Systems. Mathematical Proceedings of the Cambridge Philosophical Society, 73, 391-405.
https://doi.org/10.1017/s0305004100076945
[5]  Hammack, J. and Segur, H. (1973) The Korteweg-de Vries Equation and Water Waves. Journal of Fluid Mechanics, 60, 769-799.
[6]  Newell, A.C. (1985) Solitons in Mathematics and Physics. Society for Industrial and Applied Mathematics.
https://doi.org/10.1137/1.9781611970227
[7]  Carroll, R. (1992) Topics in Soliton Theory. North-Holland Mathematics Studies, 167.
[8]  Miura, R.M., Gardner, C.S. and Kruskal, M.D. (1968) Korteweg-de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion. Journal of Mathematical Physics, 9, 1204-1209.
https://doi.org/10.1063/1.1664701
[9]  Bona, J. and Smith, R. (1974) Existence of Solutions to the Korteweg-de Vries Initial Value Problem. Nonlinear Wave Motion (Lecture Notes in Appl. Math.), 15.
[10]  Lax, P.D. (1976) Almost Periodic Solutions of the Kdv Equation. SIAM Review, 18, 351-375.
https://doi.org/10.1137/1018074
[11]  Kato, T. (1979) On the Korteweg-De Vries Equation. Manuscripta Mathematica, 28, 89-99.
https://doi.org/10.1007/bf01647967
[12]  Sjöberg, A. (1970) On the Korteweg-de Vries Equation: Existence and Uniqueness. Journal of Mathematical Analysis and Applications, 29, 569-579.
https://doi.org/10.1016/0022-247x(70)90068-5
[13]  Saut, J.C. and Temam, R. (1976) Remarks on the Korteweg-De Vries Equation. Israel Journal of Mathematics, 24, 78-87.
https://doi.org/10.1007/bf02761431
[14]  Ablowitz, M.J. and Cornille, H. (1979) On Solutions of the Korteweg-De Vries Equation. Physics Letters A, 72, 277-280.
https://doi.org/10.1016/0375-9601(79)90467-5
[15]  Gardner, C.S., Greene, J.M., Kruskal, M.D. and Miura, R.M. (1974) Korteweg‐Devries Equation and Generalizations. VI. Methods for Exact Solution. Communications on Pure and Applied Mathematics, 27, 97-133.
https://doi.org/10.1002/cpa.3160270108
[16]  Ginibre, J. and Tsutsumi, Y. (1989) Uniqueness of Solutions for the Generalized Korteweg-de Vries Equation. SIAM Journal on Mathematical Analysis, 20, 1388-1425.
https://doi.org/10.1137/0520091
[17]  Tsutsumi, M. (1971) On Global Solutions of the Generalized Korteweg-De Vrles Equation. Publications of the Research Institute for Mathematical Sciences, 7, 329-344.
https://doi.org/10.2977/prims/1195193545
[18]  Bu, C. (1995) Modified Korteweg-de Vries Equation with Generalized Functions as Initial Values. Journal of Mathematical Physics, 36, 3454-3460.
https://doi.org/10.1063/1.530972
[19]  Dushane, T. (1973) Generalization of the Korteweg-de Vries Equation. Proceedings of Symposia in Pure Mathematics, Vo. 23, 303-307.
[20]  Ginibre, J., Tsutsumi, Y. and Velo, G. (1990) Existence and Uniqueness of Solutions for the Generalized Korteweg De Vries Equation. Mathematische Zeitschrift, 203, 9-36.
https://doi.org/10.1007/bf02570720
[21]  Kenig, C.E., Ponce, G. and Vega, L. (1989) On the (Generalized) Korteweg-De Vries Equation. Duke Mathematical Journal, 59, 585-610.
https://doi.org/10.1215/s0012-7094-89-05927-9
[22]  Saut, J. (1979) Sur quelques generalisation de l’equation de Korteweg-de Vries. Journal de Mathématiques Pures et Appliquées, 58, 21-61.
[23]  Calogero, F. and Degasperis, A. (1985) A Modified Korteweg-de Vries Equation. Inverse Problems, 1, 57-66.
https://doi.org/10.1088/0266-5611/1/1/006
[24]  Gesztesy, F., Schweiger, W. and Simon, B. (1991) Commutation Methods Applied to the mKdV-Equation. Transactions of the American Mathematical Society, 324, 465-525.
https://doi.org/10.1090/s0002-9947-1991-1029000-7
[25]  Kato, T. (1983) On the Cauchy Problem for the (Generalized) Korteweg-de Vries Equation. Studies in Applied Mathematics (Advances in Mathematics Supplementary Studies), 8, 93-128.
[26]  Calogero, F. and Degasperis, A. (1985) Spectral Transform and Solitons II. North-Holland.
[27]  Bona, J.L. and Smith, R. (1976) A Model for the Two-Way Propagation of Water Waves in a Channel. Mathematical Proceedings of the Cambridge Philosophical Society, 79, 167-182.
https://doi.org/10.1017/s030500410005218x
[28]  Bona, J. and Winther, R. (1983) The Korteweg-de Vries Equation, Posed in a Quarter-plane. SIAM Journal on Mathematical Analysis, 14, 1056-1106.
https://doi.org/10.1137/0514085
[29]  Bona, J.L. and Winther, R. (1989) The Korteweg-de Vries Equation in a Quarter Plane, Continuous Dependence Results. Differential and Integral Equations, 2, 228-250.
https://doi.org/10.57262/die/1371648746
[30]  Strauss, W. and Bu, C. (2001) An Inhomogeneous Boundary Value Problem for Nonlinear Schrödinger Equations. Journal of Differential Equations, 173, 79-91.
https://doi.org/10.1006/jdeq.2000.3871
[31]  Gao, H. and Bu, C. (2004) Dirichlet Inhomogeneous Boundary Value Problem for the N+1 Complex Ginzburg-landau Equation. Journal of Differential Equations, 198, 176-195.
https://doi.org/10.1016/j.jde.2003.09.006
[32]  Bu, C. (1997) On a Forced Modified Kdv Equation. Physics Letters A, 229, 221-227.
https://doi.org/10.1016/s0375-9601(97)00148-5
[33]  Knickerbocker, C.J. and Newell, A.C. (1980) Internal Solitary Waves near a Turning Point. Physics Letters A, 75, 326-330.
https://doi.org/10.1016/0375-9601(80)90830-0
[34]  Nunes, W.V.L. (1998) Global Well-Posedness for the Transitional Korteweg-De Vries Equation. Applied Mathematics Letters, 11, 15-20.
https://doi.org/10.1016/s0893-9659(98)00072-x
[35]  Bu, C. (2024) A Modified Transitional Korteweg-De Vries Equation: Posed in the Quarter Plane. Journal of Applied Mathematics and Physics, 12, 2691-2701.
https://doi.org/10.4236/jamp.2024.127160
[36]  Kato, T. (1975) Quasi-Linear Equations of Evolution, with Applications to Partial Differential Equations. In: Everitt, W.N., Ed., Spectral Theory and Differential Equations, Springer, 25-70.
https://doi.org/10.1007/bfb0067080
[37]  Adams (1975) Sobolev Spaces. Acad. Press.
[38]  Nirenberg, L. (1959) On Elliptic Partial Differential Equations. Annali della Scuola Normale Superiore di Pisa, 13, 115-162.
[39]  Pazy, A. (1983) Semigroups of Linear Operators and Applications to PDE. Springer.

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