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1.M. Affouf and R.E. Caflsch, “A numerical study of Riemann problem solutions and stability for a system of viscous conservation laws of mixed type,” SIAM J. Appl. Math. 51, 605-634 (1991).
2.M. Slemrod and J.E. Flaherty, Numerical integration of a Riemann problem for a van der Waals fluid (Elsevier, Amsterdam, 1986).
3.H. Fan, “A vanishing viscosity approach on the dydamics of phase transitions in Van Der Waals fluids,” J. Differential Equations. 103, 179-204 (1993).
4.H. Hattori, “The Riemann problem and the existence of weak solution to a system of Mixed-type in dynamic phase transition,” J. Differential Equations. 146, 287-319 (1998).
5.H. Hattori, “The existence and large time behavior of solutions to a system related to a phase transition problem,” SIAM J. Math. Anal. 34(4), 774-804 (2003).
6.M. Shearer and Y. Yang, “The Riemann problem for a system of conservation laws of mixed type with a cubic nonlinearity,” P. Roy. Soc. Edinb. A. 125, 675-699 (1995).
7.J. Mercier and B. Piccoli, “Admissible Riemann Solvers for Genuinely Nonlinear p-Systems of Mixed Type,” J. Differential Equations. 180, 395-426 (2002).
8.D.J. Korteweg, “Sur la forme que prennent lea quation des mouvements des fluids si l’on tient compte des forces capillarires par des variations de densit,” Arch. Neerl. Sci. Exactes Nat. Ser. II. 6, 1-24 (1901).
9.B.U. Felderhof, “The gas-liquid interface near a critical point,” Phys. 48, 541-560 (1970).
10.V. Bongiorno, L.E. Scriven, and H.T. Davis, “Molecular theory of fluid interfaces,” J. Colloid. Interface. Sci. 57, 462-475 (1976).
11.D.Q. Lu and A.T. Chwang, “Unsteady free-surface waves due to a submerged body moving in a viscous fluid,” Phys. Rev. E. 71, 066303 (2005).
12.V. Bongiorno, L.E. Scriven, and H.T. Davis, “Molecular theory of fluid interfaces,” Int. J. Nonlin. Mech. 72, 80-83 (2015).
13.W.X. Ma and J.H. Lee, “A transformed rational function method and exact solutions to the 3 + 1 dimensional Jimbo-Miwa equation,” Chaos. Solitons. Fract. 42, 1356-1363 (2009).
14.W.X. Ma, H. Wu, and J. He, “Partial differential equations possessing Frobenius integrable decompositions,” Phys. Lett. A. 364, 29-32 (2007).
15.W.X. Ma and B. Fuchssteiner, “Explicit and Exact Solutions to a Kolmogorov-Petrovskii-Piskunov Equation,” Int. J. Nonlin. Mech. 31, 329-338 (1996).
16.M. Grillakis, J. Shatah, and W. Strauss, “Stability theory of solitary waves in the presence of symmetry I,” J. Funct. Anal. 74, 160-197 (1987).
17.M. Grillakis, J. Shatah, and W. Strauss, “Stability theory of solitary waves in the presence of symmetry II,” J. Funct. Anal. 94, 308-348 (1990).
18.B. Guo and Y. Wu, “Orbital stability of solitary waves for the nonlinear derivative Schrodinger equation,” J. Differential Equations. 123, 35-55 (1995).
19.W. Zhang, Y. Qin, Y. Zhao, and B. Guo, “Orbital stability of solitary waves for Kundu equation,” J. Differential Equations. 247, 1591-1615 (2009).
20.S. Hakkaev and K. Kirchev, “Local well-posedness and orbital stability of solitary wave solutions for a generalized Camassa-Holm Equation,” Commun. Part. Diff. Eq. 30, 761-781 (2005).
21.X. Liu, Y. Liu, P. Olver, and C. Qu, “Orbital stability of peakons for a generalization of the modified Camassa-Holm equation,” Nonlinearity. 27, 2297-2319 (2014).
22.C. Qu, Y. Zhang, X. Liu, and Y. Liu, “Orbital stability of periodic peakons to a generalized -Camassa-Holm equation,” Arch. Rational Mech. Anal. 211, 593-617 (2014).
23.V. Nemytskii and V. Stepanov, Qualitative theory of differential equations (Dover Publicaton, Dover, 1989).
24.Z. Zhang, T. Ding, W. Huang, and Z. Dong, Qualitative theory of differential equations, Translations of Mathematical Monographs (American Mathematical Society, Providence, 1992).
25.J. Li and L. Zhang, “Bifurcations of traveling wave solutions in generalized Pochhammer-Chree equation,” Chaos. Solitons. Fract. 14, 581-593 (2002).
26.W. Zhang, Q. Chang, and B. Jiang, “Explicit exact solitary-wave solutions for compound KdV-type and compound KdV Burgers-type equations with nonlinear term of any order,” Chaos. Solitons. Fract. 13, 311-319 (2002).
27.M.E. Taylor, Partial Differential Equations I: Basic Theory, 2nd ed. (Springer, Beijing, 2014).
28.Q. Ye and Z. Li, Introduction of reaction-diffusion equations (Science Press, Beijing, 1990) (in Chinese).

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In this paper, we discuss the existence of traveling wave solutions for compressible fluid equations by applying the theory and method of planar dynamical system, and obtain explicit expressions for all bounded traveling wave solutions by undetermined coefficient method, including kink and bell profile traveling wave solutions, as well as periodic wave solutions. We prove the kink profile solitary wave solution, both sides of which asymptotic values are not zero, is orbitally stable by the theory of Grillakis-Shatah-Strauss orbital stability.


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