Abstract
The periodic soliton solutions are obtained from two-soliton solution with complex conjugate soliton wave numbers and frequencies. The propagation properties depend on the ratio of imaginary part to real part of the wave number. The periodic soliton with the high ratio is propagated with large velocity even if the wave number is small. This is in contrast with the propagation property of line soliton. The exact solutions to the Kadomtsev-Petviashvili (KP) equation with positive dispersion and Davey-Stewartson I equation are analyzed to investigate the natures of the interactions between two periodic solitons and between periodic soliton and another kind of soliton. The interactions are classified into several types according to the combinations of parameters which are related to the phase shifts. There are two types of singular interactions: one is the resonant interaction where two solitons interact so as to make a new soliton, the other is the extremely long-range interaction where two solitons interact infinitely apart from each other. Here we call these singular interactions”periodic soliton resonances”. The periodic soliton resonances are irrelevant to the divergence of the soliton solutions. This is crucially different from the the solution of the resonant interaction between two line solitons to the KP equation with negative dispersion which is on the borderline between regular and singular regimes in the parameter space. It is also shown that the periodic soliton resonances are related to the instability of solitons including their decay and merger.
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References
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia (1981).
M. J. Ablowitz and J. Satsuma, J. Math. Phys. 19, 2180 (1978).
D. Anker and N. C. Freeman, Proc. R. Soc. London Ser. A 360, 529 (1978).
T. B. Benjamin and J. E. Feir, J. Fluid Mech. 27, 417 (1967)
T. B. Benjamin, Proc. R. Soc. London Ser. A 299, 59 (1967).
A. Davey and K. Stewartson, Proc. R. Soc. London Ser. A 338, 101 (1974).
J. E. Feir, Proc. R. Soc. London Ser. A 299, 54 (1967).
N. C. Freeman and A. Davey, Proc. Roy. Soc. London A344, 427 (1975).
A. C. Freeman, Adv. Appl. Mech. 20, 1 (1980).
K. A. Gorshkov, D. E. Pelinovskii and Yu. A. Stepanyants, JETP 77, 237 (1993).
H. Hasimoto and H. Ono, J. Phys. Soc. Jpn. 33, 805 (1972).
R. S. Johnson and S. Thompson, Phy. Lett. 66A, 279 (1978).
B. B. Kadomtsev and V. I. Petviashvili, Sov. Phys. Dokl. 15, 539 (1970).
B. M. Lake and H. C. Yuen and H. Rungaldier and W. E. Ferguson, J. Fluid Mech. 83, 49 (1977).
S. V. Manakov, V. E. Zakharov, L. A. Bordag, A. R. Its and V. B. Matveev, Phys. Lett. A 63, 205 (1977).
J. W. Miles, J. Fluid Mech. 79, 157, 171 (1977).
Y. Murakami and M. Tajiri, J. Phys. Soc. Jpn. 58, 2207 (1989).
Y. Murakami and M. Tajiri, Wave Motion 14, 169 (1991).
Y. Murakami and M. Tajiri, J. Phys. Soc. Jpn. 61, 791 (1992).
D. E. Pelinovsky and Yu. A. Stepanyants, JETP 77, 602 (1993).
D. E. Pelinovskii and Yu. A. Stepanyants, JETP Lett. 57, 24 (1993).
D. Pelinovsky, Phys. Lett. A 196, 181 (1994).
V. I. Petviashvili, Plasma Phys. 1, 364 (1975).
V. I. Petviashvili and V. V. Yan’kov, ”Soliton and Turbulence”, in Reviews of Plasma Physics 14 ed. by B. B. Kadomtsev, Consultants Bureau (1989).
J. Satsuma, J. Phys. Soc. Jpn. 40, 286 (1976).
J. Satsuma and M. J. Ablowitz, J. Math. Phys. 20, 1496 (1979).
M. Tajiri and Y. Murakami, J. Phys. Soc. Jpn. 58, 3029 (1989).
M. Tajiri and Y. Murakami, Phys. Lett. A 143, 217 (1990).
M. Tajiri and Y. Fujimura and Y. Murakami, J. Phys. Soc. Jpn. 61, 783 (1992).
M. Tajiri and Y. Murakami and K. Kimura, Bull. Univ. Osaka Pref. A 43, 55 (1994).
M. Tajiri and T. Arai and Y. Watanabe, J. Phys. Soc. Jpn. 67, 4051 (1998).
M. Tajiri and Y. Watanabe, Phys. Rev. E 57, 3510 (1998).
M. Tajiri and T. Arai, Phys. Rev. E 60, 2297 (1999).
M. Tajiri and K. Takeuchi and T. Arai, Phys. Rev. E 64, 056622–1 (2001).
M. Tajiri and K. Takeuchi and T. Arai, J. Phys. Soc. Jpn. 70, 1505 (2001).
M. Tajiri and H. Miura and T. Arai, Phys. Rev. E 66, 067601–1 (2002).
T. Taniuti, wave Motion 12, 373 (1990).
T. Taniuti and A. Hasagawa, Wave Motion 13, 133 (1991).
Y. Watanabe and M. Tajiri, J. Phys. Soc. Jpn. 67, 705 (1998).
N. Yajima, Prog. Theor. Phys. 69, 678 (1983).
A. A. Zaitsev, Sov. Phys. Dokl., 28, 720 (1983).
V. E. Zakharov, J. App. Mech. Tech. Phys. 4, 190 (1968).
V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl. 8, 226 (1974).
V. E. Zakharov and E. A. Kuzunetsov, Sov. Phys. JETP 39, 285 (1974).
S. K. Zhdanov, Sov. Phys. Dokl. 30, 769 (1985).
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Tajiri, M. (2003). Periodic Soliton Resonances. In: Gao, D.Y., Ogden, R.W. (eds) Advances in Mechanics and Mathematics. Advances in Mechanics and Mathematics, vol 4. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0247-6_4
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DOI: https://doi.org/10.1007/978-1-4613-0247-6_4
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