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Traveling waves to a Burgers-Korteweg-de Vries-type equation with higher-order nonlinearities. (English) Zbl 1119.35075
The main goal of this paper is to focus on travelling wave solutions of equations $$u_t+\alpha uu_x+\beta u_{xx}+ su_{xxx}= 0\tag1$$ and $$u_t+\alpha u^p u_x+\beta u^{2p} u_x+\gamma u_{xx}+\mu u_{xxx}= 0,\tag2$$ where $\alpha$, $\beta$, $\gamma$, $\mu$ and $s$ are real constants, and $p$ is a positive number. The authors transform the so-called Burgers-KdV-type equation (2) to a two-dimensional autonomous system and apply the qualitative theory of planar dynamical systems to analyze the resultant system for its solitary waves. A qualitative analysis to the equation (2) is presented, which indicates that under given parametric conditions, the equation (2) has neither nontrivial bell-profile solitary waves, nor periodic waves. The authors show that a solitary wave solution is obtained by using the first-integral method.

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
37K40Soliton theory, asymptotic behavior of solutions
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References:
[1] Benney, D. J.: Long waves on liquid films. J. math. Phys. 45, 150-155 (1966) · Zbl 0148.23003
[2] Johnson, R. S.: Shallow water waves on a viscous fluid --- the undular bore. Phys. fluids 15, 1693-1699 (1972) · Zbl 0264.76014
[3] Van Wijngaarden, L.: On the motion of gas bubbles in a perfect fluid. Ann. rev. Fluid mech. 4, 369-373 (1972)
[4] Johnson, R. S.: A nonlinear equation incorporating damping and dispersion. J. fluid mech. 42, 49-60 (1970) · Zbl 0213.54904
[5] Grad, H.; Hu, P. W.: Unified shock profile in a plasma. Phys. fluids 10, 2596-2602 (1967)
[6] Hu, P. N.: Collisional theory of shock and nonlinear waves in a plasma. Phys. fluids 15, 854-864 (1972)
[7] Karahara, T.: Weak nonlinear magneto-acoustic waves in a cold plasma in the presence of effective electron -- ion collisions. J. phys. Soc. Japan 27, 1321-1329 (1970)
[8] Gao, G.: A theory of interaction between dissipation and dispersion of turbulence. Sci. sinica (Ser. A) 28, 616-627 (1985) · Zbl 0589.76067
[9] Liu, S. D.; Liu, S. K.: KdV -- Burgers equation modelling of turbulence. Sci. sinica (Ser. A) 35, 576-586 (1992) · Zbl 0759.76036
[10] Wadati, M.: Wave propagation in nonlinear lattice. J. phys. Soc. Japan 38, 673-680 (1975)
[11] Feudel, F.; Steudel, H.: Nonexistence of prolongation structure for the Korteweg -- de Vries -- Burgers equation. Phys. lett. A 107, 5-8 (1985) · Zbl 1177.37056
[12] Dodd, R.; Fordy, A.: The prolongation structures of quasipolynomial flows. Proc. roy. Soc. London A 385, 389-429 (1983) · Zbl 0542.35069
[13] Burgers, J. M.: Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Trans. roy. Neth. acad. Sci. Amsterdam 17, 1-53 (1939) · Zbl 0061.45709
[14] Korteweg, D. J.; De Vries, G.: On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves. Phil. mag. 39, 422-443 (1895) · Zbl 26.0881.02
[15] Dey, B.: Domain wall solution of KdV like equation higher-order nonlinearity. J. phys. A 19, L9-L12 (1986)
[16] Coffey, M. W.: On series expansions giving closed form of Korteweg -- de Vries-like equations. SIAM J. Appl. math. 50, 1580-1592 (1990) · Zbl 0712.76025
[17] Dey, B.: KdV like equations with domain wall solutions and their Hamiltonians, solitons. Springer series on nonlinear dynamics (1988) · Zbl 0694.35191
[18] Liu, Z. R.; Li, J. B.: Bifurcations of solitary waves and domain wall waves for KdV-like equation with higher-order nonlinearity. Internat. J. Bifur. chaos 12, 397-407 (2002) · Zbl 1042.35067
[19] Miura, R. M.: The Korteweg -- de Vries equation: A survey of results. SIAM rev. 8, 412-459 (1976) · Zbl 0333.35021
[20] Dodd, R. K.; Eilbeck, J. C.; Gibbon, J. D.; Morris, H. C.: Solitons and nonlinear wave equations. (1982) · Zbl 0496.35001
[21] Bona, J. L.; Schonbek, M. E.: Traveling wave solutions to Korteweg -- de Vries -- Burgers equation. Proc. roy. Soc. Edinburgh sect. A 101, 207-226 (1985) · Zbl 0594.76015
[22] Bona, J. L.; Souganidis, P. E.; Strauss, W. A.: Stability and instability of solitary waves. Proc. R. Soc. lond. Ser. A 411, 395-412 (1987) · Zbl 0648.76005
[23] Laedke, E. W.; Spatschek, K. H.: Stability theorem for KdV type equations. J. plasma phys. 32, 263-272 (1984) · Zbl 0542.76028
[24] Weinstein, M. I.: On the solitary traveling wave of the generalized Korteweg -- de Vries equation. Lectures in applied mathematics 23 (1986) · Zbl 0593.35074
[25] Weinstein, M. I.: 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
[26] Pego, R. L.; Smereka, P.; Weinstein, M. I.: Oscillatory instability of traveling waves for a KdV -- Burgers equation. Phys. D 67, 45-65 (1993) · Zbl 0787.76031
[27] Pego, R. L.; Weinstein, M. I.: Eigenvalues, and solitary wave instabilities. Philos. trans. R. soc. Lond. ser. A 34, 47-94 (1992) · Zbl 0776.35065
[28] Zhang, W. G.; Chang, Q. S.; Jiang, B. G.: Explicit exact solitary-wave solutions for compound KdV-type and compound KdV -- Burgers-type equations with nonlinear term of any order. Chaos solitons fractals 13, 311-319 (2002) · Zbl 1028.35133
[29] Feng, Z.: Traveling solitary wave solutions to evolution equations with nonlinear terms of any order. Chaos solitons fractals 17, 861-868 (2003) · Zbl 1030.35137
[30] Li, B.; Chen, Y.; Qing, Z. H.: Explicit exact solutions for new general two-dimensional KdV-type and two-dimensional KdV -- Burgers-type equations with nonlinear term of any order. J. phys. A 35, 8253-8265 (2002) · Zbl 1040.35100
[31] Jeffrey, A.: Some aspects of the mathematical modelling of long nonlinear waves. Arch. mech. 31, 559-574 (1979) · Zbl 0437.73010
[32] Canosa, J.; Gazdag, J.: The Korteweg -- de Vries -- Burgers equation. J. comput. Phys. 23, 393-403 (1977) · Zbl 0356.65107
[33] Dauletiyarov, K. Z.: Investigation of the difference method for the bona -- Smith and Burgers -- Korteweg -- de Vries equations. Zh. vychisl. Mat. mat. Fiz. 24, 383-402 (1984) · Zbl 0556.65090
[34] Avilov, V. V.; Krichever, I. M.; Novikov, S. P.: Evolution of the whiteham zone in the Korteweg -- de Vries theory. Soviet phys. Dokl. 32, 345-349 (1987) · Zbl 0655.65132
[35] Guan, K. Y.; Gao, G.: The qualitative theory of the mixed Korteweg -- de Vries -- Burgers equation. Sci. sinica (Ser. A) 30, 64-73 (1987)
[36] Guan, K. Y.; Lei, J. Z.: Intergrability of second-order antonomous system. Ann. differential equations 10, 117-135 (2002) · Zbl 1014.34002
[37] Gao, J. X.; Lei, J. Z.; Guan, K. Y.: Integrable condition on traveling wave solutions of Burgers -- KdV equation. J. northern jiaotong university 27, 38-42 (2003)
[38] Zhu, G. W.; Qin, B. Q.; Gao, G.: Direct evidence of phosphorus outbreak release from sediment to overlying water in a large shallow lake caused by strong wind wave disturbance. Chinese sci. Bull. 50, 577-582 (2005)
[39] Shu, J. J.: The proper analytical solution of the Korteweg -- de Vries equation. J. phys. A 20, L49-L56 (1987) · Zbl 0663.35091
[40] Gibbon, J. D.; Radmore, P.; Tabor, M.; Wood, D.: The Painlevé and Hirota’s method. Stud. appl. Math. 72, 39-63 (1985) · Zbl 0581.35074
[41] Feng, Z.: Qualitative analysis and exact solutions to the Burgers -- KdV equation. Dyn. contin. Discrete impuls. Syst. ser. A 9, 563-580 (2002) · Zbl 1017.35100
[42] Xiong, S. L.: An analytic solution of Burgers -- KdV equation. Chinese sci. Bull. 34, 1158-1162 (1989) · Zbl 0704.35126
[43] Mcintosh, I.: Single phase averaging and traveling wave solutions of the modified Burgers -- Korteweg -- de Vries equation. Phys. lett. A 143, 57-61 (1990)
[44] Zhang, W. G.: Exact solutions of the Burgers-combined KdV mixed type equation. Acta. math. Sci. 16, 241-248 (1996) · Zbl 0958.35125
[45] Liu, S. D.; Liu, S. K.; Ye, Q. X.: Explicit traveling wave solutions of nonlinear evolution equations. Math. practice theory 28, 289-301 (1998)
[46] Jeffrey, A.; Xu, S.: Exact solutions to the Korteweg -- de Vries -- Burgers equation. Wave motion 11, 559-564 (1989) · Zbl 0698.35139
[47] Jeffrey, A.; Mohamad, M. N. B.: Exact solutions to the KdV -- Burgers equation. Wave motion 14, 369-375 (1991) · Zbl 0833.35124
[48] Parkes, E. J.; Duffy, B. R.: An automated tanh-fucntion method for finding solitary wave solutions to nonlinear evolution equations. Comput. phys. Comm. 98, 288-300 (1996) · Zbl 0948.76595
[49] Parkes, E. J.; Duffy, B. R.: Traveling solitary wave solutions to a compound KdV -- Burgers equation. Phys. lett. A 229, 217-220 (1997) · Zbl 1043.35521
[50] Wang, M.: Exact solutions for a compound KdV -- Burgers equation. Phys. lett. A 213, 279-287 (1996) · Zbl 0972.35526
[51] Demiray, H.: A note on the exact traveling wave solution to the KdV -- Burgers equation. Wave motion 38, 367-369 (2003) · Zbl 1163.74335
[52] Halford, W. D.; Vlieg-Hulstman, M.: The Korteweg -- de Vries -- Burgers equation: A reconstruction of exact solutions. Wave motion 14, 267-271 (1991) · Zbl 0832.35129
[53] Ince, E. I.: Ordinary differential equations. (1956) · Zbl 0063.02971
[54] Weiss, J.; Tabor, M.; Carnevale, G.: The Painlevé property for partial differential equations. J. math. Phys. 24, 522-526 (1983) · Zbl 0514.35083
[55] Halford, W. D.; Vlieg-Hulstman, M.: The Korteweg -- de Vries -- Burgers equation and Painlevé property. J. phys. A 25, 2375-2379 (1992) · Zbl 0754.35138
[56] Feng, Z.: The first-integral method to the Burgers -- Korteweg -- de Vries equation. J. phys. A 35, 343-350 (2002) · Zbl 1040.35096
[57] Feng, Z.: On explicit exact solutions to the compound Burgers -- Korteweg -- de Vries equation. Phys. lett. A 293, 57-66 (2002) · Zbl 0984.35138
[58] Feng, Z.: Exact solutions in terms of elliptic functions for the Burgers -- Korteweg -- de Vries equation. Wave motion 38, 109-115 (2003) · Zbl 1163.74349
[59] Feng, Z.: Traveling solitary wave solutions to several nonlinear differential equations. Amer. math. Soc. contemp. Math. 357, 269-286 (2004)
[60] Ablowitz, M. J.; Zeppetella, A.: Explicit solution of Fisher’s equation for a special wave speed. Bull. math. Biol. 41, 835-840 (1979) · Zbl 0423.35079
[61] Lawden, D. F.: Elliptic functions and applications. (1989) · Zbl 0689.33001
[62] Abramowitz, M.; Stegun, I.: Handbook of mathematical functions. (1965) · Zbl 0171.38503
[63] Feng, Z.: Algebraic curve solution for second-order polynomial autonomous systems. Electron J. Linear algebra 8, 14-25 (2001) · Zbl 0989.34004
[64] Bourbaki, N.: Commutative algebra. (1972) · Zbl 0279.13001