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The Cauchy problem and stability of solitary-wave solutions for RLW–KP-type equations. (English) Zbl 1023.35085
Summary: The Kadomtsev Petviashvilli (KP) equation, $(u_t+u_x+ uu_x+u_{xxx})_x +\varepsilon u_{yy}=0, \tag{1}$ arises in various contexts where nonlinear dispersive waves propagate principally along the $$x$$-axis, but with weak dispersive effects being felt in the direction parallel to the $$y$$-axis perpendicular to the main direction of propagation. We propose and analyze here a class of evolution equations of the form $(u_t+u_x+ u^pu_x+Lu_t)_x+ \varepsilon u_{yy}=0, \tag{2}$ which provides an alternative to (1) in the same way the regularized long-wave equation is related to the classical Korteweg-de Vries (KdV) equation. The operator $$L$$ is a pseudo-differential operator in the $$x$$-variable, $$p\geq 1$$ is an integer and $$\varepsilon=\pm 1$$.
After discussing the underlying motivation for the class (2), a local well-posedness theory for the initial-value problem is developed. With assumptions on $$L$$ and $$p$$ that include conditions pertaining to models of interesting physical phenomenon, the solutions defined locally in time $$t$$ are shown to be smoothly extendable to the entire time-axis. In the particularly interesting case where $$L=-\partial^2_x$$ and $$\varepsilon=-1$$, (1) possesses travelling-wave solutions $$u(x,y,t)= \varphi_c(x-ct,y)$$ provided $$c<1$$ and $$0<p<4$$. It is shown here that these solitary waves are stable for $$0<p< {4\over 3}$$ and $$c>1$$ and for $${4\over 3}<p< 4$$ if $$c>(4p)/(4+p)$$.
The paper concludes with commentary on extensions of the present theory to more than two space dimensions.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems 76B25 Solitary waves for incompressible inviscid fluids
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##### References:
 [1] Abdelouhab, L.; Bona, J.L.; Felland, M.; Saut, J.-C., Nonlocal models for nonlinear dispersive waves, Physica D, 40, 360-392, (1989) · Zbl 0699.35227 [2] Abramyan, L.A.; Yu., A.Stepanyants, The structure of two-dimensional solitons in media with anomalously small dispersion, Sov. phys. JETP, 61, 963-966, (1985) [3] Albert, J.P.; Bona, J.L., Comparisons between model equations for long waves, J. nonlinear sci., 1, 345-374, (1991) · Zbl 0791.35123 [4] Benjamin, T.B., The stability of solitary waves, Proc. roy. soc. London, ser. A, 328, 153-183, (1972) [5] Benjamin, T.B.; Bona, J.L.; Bose, D.K., Solitary wave solutions of nonlinear problems, Philos. trans. royal. soc. London, ser. A, 331, 195-244, (1990) · Zbl 0707.35131 [6] Benjamin, T.B.; Bona, J.L.; Mahony, J.J., Model equations for long waves in nonlinear dispersive systems, Philos. trans. roy. soc. London, ser. A, 272, 47-78, (1972) · Zbl 0229.35013 [7] Besov, O.V.; Il’in, V.P.; Nikolskii, S.M., Integral representations of functions-imbedding theorems, (1978), J. Wiley New York [8] Bona, J.L., On the stability theory of solitary waves, Proc. roy. soc. London, ser. A, 344, 363-374, (1975) · Zbl 0328.76016 [9] Bona, J.L.; Chen, H., Comparison of model equations for small-amplitude long waves, Nonlinear anal. TMA, ser. B, 38, 625-664, (1999) · Zbl 0948.35108 [10] Bona, J.L.; Pritchard, W.G.; Scott, L.R., A comparison of solutions of two model equations for long waves, Lectures appl. math., 20, 235-265, (1983) · Zbl 0534.76024 [11] Bona, J.L.; Scialom, M., The effect of change in the nonlinearity and dispersion relation of model equations for long waves, Canad. appl. math. quart., 3, 1-41, (1995) · Zbl 0832.35123 [12] Bona, J.L.; Smith, R., A model for the two-way propagation of water waves in a channel, Math. proc. Cambridge philos. soc., 79, 167-182, (1976) · Zbl 0332.76007 [13] Bona, J.L.; Souganidis, P.E.; Strauss, W.A., Stability and instability of solitary waves of KDV-type, Proc. roy. soc. London, ser. A, 411, 395-412, (1987) · Zbl 0648.76005 [14] de Bouard, A.; Saut, J.-C., Remarks on the stability of generalized KP solitary waves, Contemp. math., 200, 75-84, (1996) · Zbl 0863.35095 [15] de Bouard, A.; Saut, J.-C., Solitary waves of generalized kadomtsev – petviashvili equations, Ann. inst. H. Poincaré, anal. non linéaire, 15, 211-236, (1997) · Zbl 0883.35103 [16] de Bouard, A.; Saut, J.-C., Symmetries and decay of the generalized KP solitary waves, SIAM J. math. anal., 28, 1064-1085, (1997) · Zbl 0889.35090 [17] Bourgain, J., On the Cauchy problem for the kadomtsev – petviashvili equations, Geom. funct. anal., 3, 315-341, (1993) · Zbl 0787.35086 [18] Craig, W., An existence theory for water waves and the Boussinesq and korteweg – de Vries scaling limits, Comm. partial differential equations, 10, 787-1003, (1985) · Zbl 0577.76030 [19] A. V. Faminskii, On the Cauchy problem for certain multidimensional generalizations of the KDV equation, Memorias Escueia de Verano, Universidad Nacional de Colombia, 1994, pp. 15-26. [20] Fokas, A.S.; Sung, L.Y., On the solvability of the N-wave, davey – stewartson and kadomtsev – petviashvili equations, Inverse problems, 8, 673-708, (1992) · Zbl 0768.35069 [21] Hammack, J.L., A note on tsunamis: their generation and propagation in an Ocean of uniform depth, J. fluid mech., 60, 769-799, (1973) · Zbl 0273.76010 [22] Hammack, J.L.; Segur, H., The korteweg – de Vries equation and water waves, part 2. comparison with experiments, J. fluid mech., 65, 289-324, (1974) · Zbl 0373.76010 [23] Iorio Jr., R.J.; Nunes, W.V.L., On equations of KP-type, Proc. roy. soc. Edinburgh A, 128, 725-743, (1998) · Zbl 0911.35103 [24] Isaza, P.; Mejia, J.; Stallhohm, V., Local solutions for the kadomtsev – petviashvili equations in $$R$$2, J. math. anal. appl., 196, 566-587, (1995) · Zbl 0844.35107 [25] Karpman, V.I.; Yu. Belashov, V., Dynamics of two-dimensional solitons in weakly dispersive media, Phys. lett. A, 154, 131-139, (1991) [26] Karpman, V.I.; Belashov, V.Yu., Evolution of three-dimensional nonlinear pulses in weakly dispersive media, Phys. lett. A, 154, 140-144, (1991) [27] Kato, T., On the Cauchy problem for the (generalized) korteweg – de Vries equation, advances in mathematics, supplementary studies, Stud. appl. math., 8, 93-128, (1983) [28] Lamb, G.L., Elements of soliton theory, (1980), Wiley New York · Zbl 0445.35001 [29] Lions, J.-L.; Magenes, E., Probléme aux limites non homogénes et applications, (1968), Dunod Paris · Zbl 0165.10801 [30] Liu, Y., Blow-up and instability of solitary-wave solutions to a generalized kadomtsev – petviashvili equation, Trans. amer. math. soc., 353, 191-208, (2001) · Zbl 0949.35120 [31] Liu, Y.; Wang, X.P., Nonlinear stability of solitary waves of the generalized kadomtsev – petviashvili equation, Comm. math. phys., 183, 253-266, (1997) · Zbl 0879.35136 [32] Saut, J.-C., Sur quelques généralisations de l’équation de korteweg – de Vries, J. math. pures appl., 58, 21-61, (1979) · Zbl 0449.35083 [33] Saut, J.-C., Remarks on the generalized kadomtsev – petviashvili equations, Indiana univ. math. J., 42, 1011-1026, (1993) · Zbl 0814.35119 [34] Saut, J.-C., Recent results on the generalized kadomtsev – petviashvili equations, Acta appl. math., 39, 477-487, (1995) · Zbl 0839.35121 [35] Schwarz, M., Periodic solutions of kadomtsev – petviashvili, Adv. math., 66, 217-233, (1987) · Zbl 0659.35084 [36] Tom, M.M., On the generalized kadomtsev – petviashvili equation, Contemp. math., 200, 193-210, (1996) · Zbl 0861.35103 [37] M. M. Tom, Generalization of the KP-equations in several space dimension, preprint, 1997. [38] Tom, M.M., Remarks on global solutions of some generalizations of the KP equations, (), 220-224 · Zbl 1051.35509 [39] Tom, M.M., Some generalizations of the kadomtsev – petviashvili equations, J. math. anal. appl., 243, 64-84, (2000) · Zbl 0947.35144 [40] Ukai, S., Local solutions of the kadomtsev – petviashvili equations, J. fac. sci. univ. Tokyo sect. 1A math., 36, 193-209, (1989) · Zbl 0703.35155 [41] Wang, X.P.; Ablowitz, M.J.; Segur, H., Wave collapse and instability of solitary waves of a generalized kadomtsev – petviashvili equation, Physica D, 78, 241-265, (1994) · Zbl 0824.35116 [42] Weinstein, M., Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. partial differential equations, 12, 1133-1173, (1987) · Zbl 0657.73040 [43] Whitham, G.B., Linear and nonlinear waves, (1974), Wiley New York · Zbl 0373.76001 [44] Zabusky, N.J.; Galvin, C.J., Shallow-water waves, the korteweg – de Vries equation and solitons, J. fluid mech., 47, 811-824, (1971) [45] Zhou, X., Inverse scattering transform for the time dependent Schrödinger equations with application to the KP equations, Comm. math. phys., 128, 551-564, (1990) · Zbl 0702.35241
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