July 2019, 39(7): 3635-3670. doi: 10.3934/dcds.2019149

A uniqueness result for 2-soliton solutions of the Korteweg-de Vries equation

Department of Mathematics, University of Oklahoma, Norman OK 73019, USA

Received  October 2018 Revised  January 2019 Published  April 2019

Multisoliton solutions of the KdV equation satisfy nonlinear ordinary differential equations which are known as stationary equations for the KdV hierarchy, or sometimes as Lax-Novikov equations. An interesting feature of these equations, known since the 1970's, is that they can be explicitly integrated, by virtue of being finite-dimensional completely integrable Hamiltonian systems. Here we use the integration theory to investigate the question of whether the multisoliton solutions are the only nonsingular solutions of these ordinary differential equations which vanish at infinity. In particular we prove that this is indeed the case for $ 2 $-soliton solutions of the fourth-order stationary equation.

Citation: John P. Albert. A uniqueness result for 2-soliton solutions of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3635-3670. doi: 10.3934/dcds.2019149
References:
[1]

M. Adler, On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-Devries type equations, Inventiones Mathematicae, 50 (1978/79), 219-248. doi: 10.1007/BF01410079.

[2]

E. D. Belokolos, V. Enol'skii, A. R. Its and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, 1994.

[3]

M. Bochner, Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence, Trans. Amer. Math. Soc., 2 (1901), 139-149. doi: 10.2307/1986214.

[4]

J. Boussinesq, Essai sur la théorie des eaux courantes, Mémoires présentés par divers savants à l'Académie des Sciences Inst. France (séries 2), 23 (1877), 1-680.

[5]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955.

[6]

P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math., 32 (1979), 121-251. doi: 10.1002/cpa.3160320202.

[7]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[8]

L. A. Dickey, Soliton Equations and Hamiltonian Systems (1st edition), World Scientific, 1991. doi: 10.1142/1109.

[9]

L. A. Dickey, Soliton Equations and Hamiltonian Systems (2nd edition), World Scientific, 2003. doi: 10.1142/5108.

[10]

J. Drach, Sur l'intégration par quadratures de l'équation différentiel $y'' = [\theta(x)+h]y$, Compt. Rend. Acad. Sci., 168 (1919), 337-340.

[11]

B. A. Dubrovin, Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials, Functional Analysis and its Applications, 9 (1975), 215-223.

[12]

B. A. DubrovinV. B. Matveev and S. P. Novikov, Nonlinear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian varieties, Uspehi Mat. Nauk, 31 (1976), 55-136.

[13]

B. A. Dubrovin and S. P. Novikov, Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg-de Vries equation, Soviet Physics JETP, 40 (1974), 1058-1063.

[14]

L. Faddeev and V. E. Zakharov, Korteweg-de Vries equation: A completely integrable Hamiltonian system, Functional Anal. Appl., 5 (1971), 280-287.

[15]

C. S. Gardner, Korteweg-de Vries equation and generalizations. Ⅳ. The Korteweg-de Vries equation as a Hamiltonian system, J. Mathematical Phys., 12 (1971), 1548-1551. doi: 10.1063/1.1665772.

[16]

C. S. GardnerJ. M. GreeneM. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), 1095-1097.

[17]

I. M. Gel'fand and L. A. Dikii, Integrable nonlinear equations and the Liouville theorem, Funktsional. Anal. i Prilozhen., 13 (1979), 8-20.

[18]

F. Gesztesy and H. Holden, Soliton Equations and their Algebro-Geometric Solutions: Volume 1, (1+1)-Dimensional Continuous Models, Cambridge, 2003. doi: 10.1017/CBO9780511546723.

[19]

R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27 (1971), 1192-1194. doi: 10.1103/PhysRevLett.27.1192.

[20]

R. Hirota, The Direct Method in Soliton Theory, Cambridge, 2004. doi: 10.1017/CBO9780511543043.

[21]

A. R. Its and V. B. Matveev, Schrödinger operators with finite-gap spectrum and N-soliton solutions of the Korteweg-de Vries equation, Theoret. and Math. Phys., 23 (1975), 51-68.

[22]

T. Kato, Perturbation Theory for Linear Operators, Springer, 1995.

[23]

C. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-560. doi: 10.1002/cpa.3160460405.

[24]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary wave, Phil. Mag., 39 (1895), 422-443. doi: 10.1080/14786449508620739.

[25]

P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490. doi: 10.1002/cpa.3160210503.

[26]

N. Macon and A. Spitzbart, Inverses of Vandermonde matrices, Amer. Math. Monthly, 65 (1958), 95-100. doi: 10.1080/00029890.1958.11989147.

[27]

J. Maddocks and R. Sachs, On the stability of KdV multi-solitons, Comm. Pure Appl. Math., 46 (1993), 867-901. doi: 10.1002/cpa.3160460604.

[28]

M. Mariş, Profile decomposition for sequences of Borel measures, arXiv: 1410.6125.

[29]

V. B. Matveev, 30 years of finite-gap integration theory, Phil. Trans. R. Soc. A, 366 (2008), 837-875. doi: 10.1098/rsta.2007.2055.

[30]

V. B. Matveev, Positons: Slowly decreasing analogues of solitons, Theoret. and Math. Phys., 131 (2002), 483-497. doi: 10.1023/A:1015149618529.

[31]

A. C. Newell, Solitons in Mathematics and Physics, SIAM, 1985. doi: 10.1137/1.9781611970227.

[32]

S. P. Novikov, The periodic problem for the Korteweg-de Vries equation, Functional Analysis and its Applications, 8 (1974), 236-246.

[33]

G. Pólya and G. Szegö, Aufgaben und Lehrsätze auf der Analysis, Springer-Verlag, 1954.

[34]

I. Schur, Über vertauschbare lineare Differentialausdrücke, Sitzungsber. Berliner Math. Ges., 4 (1905) 2–8; Gesammelte Abhandlungen, Band I, Springer, 1973,170–176.

[35]

V. E. Zakharov and L. D. Faddeev, Korteweg-de Vries equation: A completely integrable Hamiltonian system, Functional Analysis and Its Applications, 5 (1971), 280-287.

[36]

N. J. Zabusky and M. D. Kruskal, Interaction of solutions in a collisionless plasma and the recurrence of initial states, Phys. Rev. Letters, 15 (1965), 240-243.

show all references

References:
[1]

M. Adler, On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-Devries type equations, Inventiones Mathematicae, 50 (1978/79), 219-248. doi: 10.1007/BF01410079.

[2]

E. D. Belokolos, V. Enol'skii, A. R. Its and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, 1994.

[3]

M. Bochner, Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence, Trans. Amer. Math. Soc., 2 (1901), 139-149. doi: 10.2307/1986214.

[4]

J. Boussinesq, Essai sur la théorie des eaux courantes, Mémoires présentés par divers savants à l'Académie des Sciences Inst. France (séries 2), 23 (1877), 1-680.

[5]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955.

[6]

P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math., 32 (1979), 121-251. doi: 10.1002/cpa.3160320202.

[7]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[8]

L. A. Dickey, Soliton Equations and Hamiltonian Systems (1st edition), World Scientific, 1991. doi: 10.1142/1109.

[9]

L. A. Dickey, Soliton Equations and Hamiltonian Systems (2nd edition), World Scientific, 2003. doi: 10.1142/5108.

[10]

J. Drach, Sur l'intégration par quadratures de l'équation différentiel $y'' = [\theta(x)+h]y$, Compt. Rend. Acad. Sci., 168 (1919), 337-340.

[11]

B. A. Dubrovin, Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials, Functional Analysis and its Applications, 9 (1975), 215-223.

[12]

B. A. DubrovinV. B. Matveev and S. P. Novikov, Nonlinear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian varieties, Uspehi Mat. Nauk, 31 (1976), 55-136.

[13]

B. A. Dubrovin and S. P. Novikov, Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg-de Vries equation, Soviet Physics JETP, 40 (1974), 1058-1063.

[14]

L. Faddeev and V. E. Zakharov, Korteweg-de Vries equation: A completely integrable Hamiltonian system, Functional Anal. Appl., 5 (1971), 280-287.

[15]

C. S. Gardner, Korteweg-de Vries equation and generalizations. Ⅳ. The Korteweg-de Vries equation as a Hamiltonian system, J. Mathematical Phys., 12 (1971), 1548-1551. doi: 10.1063/1.1665772.

[16]

C. S. GardnerJ. M. GreeneM. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), 1095-1097.

[17]

I. M. Gel'fand and L. A. Dikii, Integrable nonlinear equations and the Liouville theorem, Funktsional. Anal. i Prilozhen., 13 (1979), 8-20.

[18]

F. Gesztesy and H. Holden, Soliton Equations and their Algebro-Geometric Solutions: Volume 1, (1+1)-Dimensional Continuous Models, Cambridge, 2003. doi: 10.1017/CBO9780511546723.

[19]

R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27 (1971), 1192-1194. doi: 10.1103/PhysRevLett.27.1192.

[20]

R. Hirota, The Direct Method in Soliton Theory, Cambridge, 2004. doi: 10.1017/CBO9780511543043.

[21]

A. R. Its and V. B. Matveev, Schrödinger operators with finite-gap spectrum and N-soliton solutions of the Korteweg-de Vries equation, Theoret. and Math. Phys., 23 (1975), 51-68.

[22]

T. Kato, Perturbation Theory for Linear Operators, Springer, 1995.

[23]

C. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-560. doi: 10.1002/cpa.3160460405.

[24]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary wave, Phil. Mag., 39 (1895), 422-443. doi: 10.1080/14786449508620739.

[25]

P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490. doi: 10.1002/cpa.3160210503.

[26]

N. Macon and A. Spitzbart, Inverses of Vandermonde matrices, Amer. Math. Monthly, 65 (1958), 95-100. doi: 10.1080/00029890.1958.11989147.

[27]

J. Maddocks and R. Sachs, On the stability of KdV multi-solitons, Comm. Pure Appl. Math., 46 (1993), 867-901. doi: 10.1002/cpa.3160460604.

[28]

M. Mariş, Profile decomposition for sequences of Borel measures, arXiv: 1410.6125.

[29]

V. B. Matveev, 30 years of finite-gap integration theory, Phil. Trans. R. Soc. A, 366 (2008), 837-875. doi: 10.1098/rsta.2007.2055.

[30]

V. B. Matveev, Positons: Slowly decreasing analogues of solitons, Theoret. and Math. Phys., 131 (2002), 483-497. doi: 10.1023/A:1015149618529.

[31]

A. C. Newell, Solitons in Mathematics and Physics, SIAM, 1985. doi: 10.1137/1.9781611970227.

[32]

S. P. Novikov, The periodic problem for the Korteweg-de Vries equation, Functional Analysis and its Applications, 8 (1974), 236-246.

[33]

G. Pólya and G. Szegö, Aufgaben und Lehrsätze auf der Analysis, Springer-Verlag, 1954.

[34]

I. Schur, Über vertauschbare lineare Differentialausdrücke, Sitzungsber. Berliner Math. Ges., 4 (1905) 2–8; Gesammelte Abhandlungen, Band I, Springer, 1973,170–176.

[35]

V. E. Zakharov and L. D. Faddeev, Korteweg-de Vries equation: A completely integrable Hamiltonian system, Functional Analysis and Its Applications, 5 (1971), 280-287.

[36]

N. J. Zabusky and M. D. Kruskal, Interaction of solutions in a collisionless plasma and the recurrence of initial states, Phys. Rev. Letters, 15 (1965), 240-243.

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