Publications (peer-reviewed papers + preprints). More information can be found in Google Scholar
Papers (published or accpeted):
- J. Carrillo, G. Hong and Z.A. Wang
Convergence of boundary layers of chemotaxis models with physical boundary conditions~I: degenerate initial data,
SIAM J. Math. Anal, 2024 (supported by the Hong Kong RGC GRF grant No. 15306121 and an internal grant ZZPY). - D. Tang and Z.A. Wang
Coexistence of heterogenous predator-prey systems with prey-dependent dispersal,
J. Differential Equations, 409:461-497, 2024 (supported by the grant from the NSFC/RGC Joint Research Scheme No. N_PolyU509/22). - W.R. Tao, Z.A. Wang and W. Yang
Global dynamics of a two species clustering model with Lotka-Volterra competition,
Nonlinear Differential Equations and Applications NoDEA, 31, Article No 47, 42 pages, 2024 (supported by Hong Kong RGC GRF grant No. PolyU 15307222 and Postdoc Matching Fund Scheme W15F). - Q. Liu, H. Peng and Z.A. Wang
The relaxation limit of a quasi-linear hyperbolic-parabolic chemotaxis system modeling vasculogenesis,
Commun. Math. Anal. Appl., 3(1):1-18, 2024 (supported by the Hong Kong RGC GRF grant No. 15304720). - L. Mu, W. Tao and Z.A. Wang
Global dynamics and spatiotemporal heterogeneity of a preytaxis model with prey-induced acceleration,
European J. Appl. Math., 35:601-633,2024 (Supported by GRF 15307222 and an internal grant W15F). - H. Tang and Z.A. Wang
Strong solutions to nonlinear aggregation-diffusion equations with random birth-death dynamics ,
Comm. Contemp. Math., 26(2), 2250073, 39 pages, 2024. (supported by the Hong Kong RGC GRF grant No. PolyU153055/18P). - R. Hou, Z.A. Wang, W.-B. Xu, Z. Zhang
The uniform spreading speed in cooperative systems with non-uniform initial data,
Discrete Contin. Dyn. Syst.-S, 17(2): 585-601, 2024 (special issue for Professor Yihong Du's 60th birthday), (supported by HK RGC GRF grant no. PolyU 15307222 and an internal grant no. 1-WZ03). - R. Peng, Z.A. Wang,G. Zhang and M. Zhou
Novel spatial profiles of some diffusive SIS epidemic models,
J. Math. Biol., 82, Paper No. 81, 36 pages, 2023 (Supported by GRF 15307222 and an internal grant ZZRC). - X. Deng, Q. Huang and Z.A. Wang
Global dynamics and pattern formation in a diffusive population-toxicant model with negative toxicant-taxis,
SIAM J. Appl. Math., 83(6): 2212-2236, 2023 (supported by GRF 15306121 and an internal grant W18M). - Z.A. Wang, A. Yang and Kun Zhao
Wave propagation and stabilization in the Boussinesq-Burgers system,
Phys. D, 447, 133687, 13 pp, 2023. (supported by HK RGC GRF grant No. PolyU 15304720). - H.Y. Jin, Z.A. Wang and L. Wu
Global solvability and stability of an alarm-taxis system,
SIAM J. Math. Anal., 55(4): 2838-2876, 2023. (supported by the HK RGC GRF grant No. PolyU 15306121 and 2020 Hong Kong Scholars Program). - D. Tang and Z.A. Wang
Population dynamics with resource-dependent dispersal: single- and two-species models,
J. Math. Biol., 86, no. 2, Paper No. 23, 42 pp, 2023. (supported by HK RGC GRF grant No. PolyU 15303019 and an internal grant no. UAH0). - L. Wu and Z.A. Wang
Lotka-Volterra diffusion-advection competition system with dynamical resources,
Discrete Contin. Dyn. Syst. - B, 28(6): 3322-3348, 2023. (supported by the Hong Kong Scholars Program Grant No. YZ3Z). - W.B. Lyu and Z.A. Wang
Global boundedness and asymptotics of a class of prey-taxis models with singular response,
Math. Meth. Appl. Sci., 46:6705-6721, 2023. (supported by HK RGC GRFgrant No. PolyU 15304720 and an internal grant No. UAH0). - Z.A. Wang and W.-B. Xu
Acceleration of propagation in a chemotaxis-growth system with slowly decaying initial data,
Bull. London Math. Soc., 55:447-469, 2023. (supported by the Hong Kong RGC GRF grant No. PolyU153055/18P and an internal grant No. UAH0). - W. Lyu and Z.A. Wang
Logistic damping effect in chemotaxis models with density-suppressed motility,
Adv. Nonlinear Anal., 12: 336-355, 2023. (supported by HK RGC GRF grant No. 15303019 and an internal grant No. UAH0). - L. Battaglia, A. Jevnikar, Z.A. Wang, and W. Yang
Prescribing Gaussian curvature on surfaces with conical singularities and geodesic boundary,
Annali di Matematica Pura ed Applicata, 202:1173-1185, 2023. (supported by the Hong Kong RGC GRF grant No. PolyU15306121). - H.Y. Jin, Z.A. Wang and L. Wu
Global dynamics of a three-species spatial food chain model,
J. Differential Equations, 333:144-183, 2022. (supported by the GRF grant No. PolyU 15306121 and the 2020 Hong Kong Scholars Program (Project ID P0031250)) . - T. Li and Z.A. Wang
Traveling wave solutions to the singular Keller-Segel system with logistic source,
Math. Biosci. Eng., 19(8): 8107-8131, 2022. (supported by the Hong Kong RGC GRF grant No. PolyU 153055/18P and PolyU 15304720). - W. Tao and Z.A. Wang
On a new type of chemotaxis model with acceleration,
Commun. Math. Anal. Appl., 1(2): 319-344, 2022. (supported by GRF grant No. PolyU 15303019 and Postdoc Matching Fund Scheme W15F). - H.Y. Peng, Z.A. Wang and C.J. Zhu
Global weak solutions and asymptotics of a singular PDE-ODE chemotaxis system with discontinuous data,
Sci. China Math., 65:269-290, 2022. (supported by GRF PolyU 153031/17P and internal grant 4-ZZHY). - Q.Q. Liu, H.Y. Peng and Z.A. Wang
Convergence to nonlinear diffusion waves for a hyperbolic-parabolic chemotaxis system modelling vasculogenesis,
J. Differential Equations, 413:251-286, 2022. (supported by GRF PolyU 15304720) . - W. Lyu and Z.A. Wang
Global classical solutions for a class of reaction-diffusion system with density-suppressed motility,
Electronic Research Archive, 30(3): 995-1015, 2022. (supported by RGC GRF grant No. 15303019 and an internal grant No. UAH0) . - Z.A. Wang and X. Xu
Radial spiky steady states of a flux-limited Keller-Segel model: existence, asymptotics and stability,
Stud. Appl. Math.,148: 1251-1273, 2022. DOI: 10.1111/sapm.12474 (supported by GRF grant No. PolyU 153055/18P (P0005472)). - Z.A. Wang
A kinetic chemotaxis model with internal states and temporal sensing,
Kinet. Relat. Models, 15(1): 27-48, 2022. Doi:10.3934/krm.2021043(supported by GRF grant No. PolyU 153055/18P (P0005472)). - Q.Q. Liu, H.Y. Peng and Z.A. Wang
Asymptotic stability of diffusion waves of a quasi-linear hyperbolic-parabolic model for vasculogenesis,
SIAM J. Math. Anal, 54(1): 1313-1346, 2022. (supported by RGC GRF grant no. PolyU 15304720). - Z.A. Wang and L. Wu
Global solvability of a class of reaction-diffusion systems with cross-diffusion,
Appl. Math. Lett., 124, Paper No. 107699, 8 pp, 2022. (supported by RGC GRF PolyU 15303019 and the 2020 Hong Kong Scholars Program (Project ID P0031250)). - Y. Cai, Q. Cao and Z.A. Wang
Asymptotic dynamics and spatial patterns of a ratio-dependent predator-prey system with prey-taxis,
Applicable Analysis, 101:81-99, 2022. https://doi.org/10.1080/00036811.2020.1728259, 2020(supported by GRF grant PolyU 153298/16P (Project ID P0005162)). - J. Li and Z.A. Wang
Traveling wave solutions to the density-suppressed motility model ,
J. Differential Equations, 301:1-36,2021. (supported by RGC GRF grant No. PolyU 15303019 (Project ID P0030816). - S. Ji, Z.A. Wang, T. Xu and J. Yin
A reducing mechanism on wave speed for chemotaxis systems with degenerate diffusion,
Calc. Var. Partial Differential Equations, Vol. 60, Paper No. 178, 19 pp, 2021 (supported by GRF PolyU 153055/18P - Q65K). - G. Hong and Z.A. Wang
Asymptotic stability of exogenous chemotaxis systems with physical boundary conditions,
Quart. Appl. Math., 79:717-743,2021. https://doi.org/10.1090/qam/1599. (supported by GRF PolyU 153031/17P - Q62H and ZZHY from HKPU). - Z.A. Wang
On the parabolic-elliptic Keller-Segel system with signal-dependent motilities: a paradigm for global boundedness and steady states,
Math. Methods Appl. Sci., 44:10881-10898, 2021. https://doi.org/10.1002/mma.7455 (supported by GRF grant No. 15303019 - Q75G and UAH0). - Z.A. Wang and X. Xu
Steady states and pattern formation of the density-suppressed motility model,
IMA J. Appl. Math., 86:577-603, 2021(supported by RGC grant no. PolyU 15303019 - Q75G). - Z.A. Wang and J. Zheng
Global boundedness of the fully parabolic Keller-Segel system with signal-dependent motilities,
Acta Appl Math., Vol. 171, no. 25, Paper No. 25, 19 pp, 2021. https://doi.org/10.1007/s10440-021-00392-8. (supported by GRF grant No. 15303019 and an internal grant no. UAH0 (P0005472)). - Z.A. Wang and J. Xu
On the Lotka-Volterra competition system with dynamical resources and density-dependent diffusion,
J. Math. Biol., Vol. 82, no. 1-2, Paper No. 7, 37 pp, 2021. https://doi.org/10.1007/s00285-021-01562-w (supported by PolyU 153298/16P). - G. Hong, H. Peng, Z.A. Wang and C. Zhu
Nonlinear stability of phase transition steady states to a hyperbolic-parabolic system modelling vascular networks,
J. London Math. Soc., 103:1480-1514, 2021 (supported by PolyU 153055/18P (P0005472) and an internal grant No. ZZKN from HKPU (P0031013)). - D. Wang, Z.A. Wang and K. Zhao
Cauchy problem of a system of parabolic conservation laws arising from a Keller-Segel type chemotaxis model in multi-dimensions,
Indiana Univ. Math. J., 70(1):1-47, 2021 (supported by PolyU 153031/17P). - T. Li, D. Wang, F. Wang, Z.A. Wang and K. Zhao
Large time behavior and diffusion limit for a system of balance laws from chemotaxis in multi-dimensions,
Comm. Math. Sci., 19:229-272, 2021 (supported by PolyU 153031/17P (Project ID:P0005368) and internal grant ZZHY (Project ID:P0001905)). - J.A. Carrillo, J. Li and Z.A. Wang
Boundary spike-layer solutions of the singular Keller-Segel system: existence and stability,
Proc. London Math. Soc., 122:42-68, 2021. doi:10.1112/plms.12319 (supported by PolyU 153031/17P (Project ID P0005368)). - H.Y. Jin and Z.A. Wang
Global stability and spatio-temporal patterns of predator-prey systems with density-dependent motion,
Euro. Jnl of Applied Mathematics, 32:652-682, 2021 (supported by PolyU 153298/16P). - H.Y. Jin and Z.A. Wang
The Keller-Segel system with logistic growth and signal-dependent motility,
Disc. Cont. Dyn. Syst.-B, 26(6): 3023-3041, 2021. doi: 10.3934/dcdsb.2020218 (supported by PolyU No. 15303019 (Project Q75G)). - Q. Hou, T.-C. Lin and Z.A. Wang
On a singularly perturbed semi-linear problem with Robin boundary conditions ,
Disc. Cont. Dyn. Syst.-B, 26(1): 401-414, 2021(supported by PolyU 153031/17P (Project ID P0005368) and an internal grant P0001905). - H.Y. Jin, S. Shi and Z.A. Wang
Boundedness and asymptotics of a reaction-diffusion system with density-dependent motility,
J. Differential Equations, 269:6758-6793, 2020. (supported by PolyU No. 15303019 (Project Q75G)). - C.C. Lee, Z.A. Wang and W. Yang
Boundary-layer profile of a singularly perturbed non-local semi-linear problem arising in chemotaxis ,
Nonlinearity, 33:5111-5141, 2020(supported by PolyU 153031/11P (Project ID P0005368)). - H.Y. Jin and Z.A. Wang
Critical mass on the Keller-Segel system with signal-dependent motility,
Proc. Amer. Math. Soc., 148:4855-4873,2020(supported by an internal grant ZZHY). - B. Perthame, N. Vauchelet and Z.A. Wang
The flux limited Keller-Segel system: properties and derivation from kinetic equations,
Rev. Mat. Iberoam., 36: 357-386, 2020. Doi 10.4171/rmi/1132 (supported by PolyU 153031/17P and an internal grant No. ZZHY). - J.Y. Li and Z.A. Wang
Convergence to traveling waves of a singular PDE-ODE hybrid chemotaxis system in the half space,
J. Differential Equations, 268:6940-6970, 2020 (supported by PolyU153032/15P). - M. Ma, R. Peng and Z. Wang
Stationary and non-stationary patterns of the density-suppressed motility model,
Phys. D, 402, 132259, 2020 (supported by PolyU153298/16P (Q56F)). - H.Y. Peng and Z. Wang
On a parabolic-hyperbolic chemotaxis system with discontinuous data: well-posedness, stability and regularity,
J. Differential Equations, 268: 4374-4415, 2020. (supported by PolyU 153032/15P and ZZHY). - J.A. Carrillo, X. Chen, Q. Wang, Z. Wang and L. Zhang
Phase transitions and bump solutions of the Keller-Segel model with volume exclusion ,
SIAM J. Appl. Math., 80:232-261, 2020 (supported by PolyU 153031/17P). - H.Y. Jin and Z. Wang
Global stabilization of the full attraction-repulsion Keller-Segel system,
Disc. Cont. Dyn. Syst., 40:3509-3527,2020. (special volume for W-.M. Ni's 70th birthday)(supported by PolyU 5091/13P). - J. Wang, Z. Wang and W. Yang
Uniqueness and convergence on equilibria of the Keller-Segel system with subcritical mass,
Comm. Partial Differential Equations, 44:545-572, 2019 (supported by internal grant PolyU 153041/15P). - L.G. Rebholz, D. Wang. Z. Wang, K. Zhao and C. Zerfas
Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions,
Disc. Cont. Dyn. Syst., 39:3789-3838, 2019 (supported by PolyU 153031/17P). - Q. Hou and Z.A. Wang
Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane,
J. Math. Pures Appl., 130:251-287, 2019 (supported by PolyU 153031/17P). - C. Li, R. Peng and Z.A. Wang
On a diffusive SIS epidemic model with mass action mechanism and birth-death effect: analysis, simulations and comparison with other mechanisms,
SIAM J. Appl. Math., 78:2129-2153, 2018 (supported by PolyU153298/16P). - H.Y. Jin, Y.-J. Kim and Z.A. Wang
Boundedness, stabilization and pattern formation driven by density-suppressed motility,
SIAM J. Appl. Math., 78:1632-1657, 2018. (published on June 2018, supported by PolyU 153032/15). - Q.Q. Hou, C.J. Liu, Y.G. Wang and Z.A. Wang
Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis: one dimensional case,
SIAM J. Math. Anal., 50:3058-3091, 2018. (published on June 2018, supported by PolyU 153032/15P) - V. Martinez, Z.A. Wang and K. Zhao
Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology,
Indiana Univ. Math. J., 67:1383-1424, 2018 (supported by PolyU 153032/15P). - H.Y. Peng and Z.A. Wang
Nonlinear stability of strong traveling waves for the singular Keller-Segel system with large perturbations,
J. Differential Equations, 265: 2577-2613, 2018. (published on September 2018, supported by PolyU 153032/15P) - H.Y. Jin and Z.A. Wang
A dual-gradient chemotaxis system modeling the spontaneous aggregation of microglia in Alzheimer's disease,
Analysis and Applications, 16:307-338, 2018. (published on May 2018, supported by PolyU 5091/13P) - H. Peng, Z.A. Wang, K. Zhao and C.J. Zhu
Boundary layers and stabilization of the singular Keller-Segel system,
Kinetic and Related Models, 11: 1085-1123, 2018. (published on October 2018, supported by PolyU 153032/15P) - H.Y. Jin and Z.A. Wang
Global stability of prey-taxis systems,
J. Differential Equations, 262:1257-1290, 2017. (published on October 2016, supported by PolyU 153298/16P) - M. Ma and Z.A. Wang
Patterns in a generalized volume-filling chemotaxis model with cell proliferation,
Analysis and Applications, 15:83-106, 2017. (published on September 2015, supported by G-YBCS, A-PL15 and PolyU 153032/15P) . - Q.Q. Hou, Z.A. Wang and K. Zhao
Boundary layer problem on a hyperbolic system arising from chemotaxis,
J. Differential Equations, 261:5035-5070, 2016. - Z.A. Wang, Z. Xiang and P. Yu
Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis,
J. Differential Equations, 260:2225-2258, 2016. - H. Jin and Z.A. Wang
Boundedness, blowup and critical mass phenomenon in competing chemotaxis,
J. Differential Equations, 260:162-196, 2016. - M. Ma and Z.A. Wang
Global bifurcation and stability of steady states for a reaction-diffusion-chemotaxis model with volume-filling effect,
Nonlinearity, 28: 2639-2660, 2015. - M. Mei, H. Peng and Z.A. Wang
Asymptotic profile of a parabolic-hyperbolic system with boundary effect arising from tumor angiogenesis,
J. Differential Equations, 259: 5168-5191, 2015. - W. Ding and Z.A. Wang
Global existence and asymptotic behavior of the Boussinesq-Burgers system,
J. Math. Anal. Appl., 424: 584-597, 2015. - S.B. Ai and Z.A. Wang
Traveling bands for the Keller-Segel model with population growth,
Mathematical Biosciences and Engineering, 12:717-737, 2015. - S.B. Ai, W.Z. Huang and Z.A. Wang
Reaction, diffusion and chemotaxis in wave propagation,
Discrete Contin. Dyn. Syst.-Series B, 21(1):1-21, 2015. - H.Y. Jin, Z.A. Wang and L. Xiong
Cauchy problem of the Magnetohydrodynamic Burgers (MHD-Burgers) system,
Commu. Math. Sci., 13(1): 127-151, 2015. - H.Y. Jin and Z.A. Wang
Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model,
Math. Methods Appl. Sci., 38:444-457, 2015 . - J. Li. T. Li and Z.A. Wang
Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity,
Math. Models Methods Appl. Sci., 24(14): 2819-2849, 2014. - T.C. Lin and Z.A. Wang
Development of traveling waves in an interacting two-species chemotaxis model,
Discrete Contin. Dyn. Syst., 34(7):2907-2927, 2014. - P. Liu, J.P. Shi and Z.A. Wang
Pattern formation of the attraction-repulsion Keller-Segel system,
Discrete Contin. Dyn. Syst-Series B, 18(10): 2597-2625, 2013. - H.Y. Jin, J. Li and Z.A. Wang
Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity,
J. Differential Equations, 255:193-219, 2013 . - Z.A. Wang and K. Zhao
Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model,
Comm. Pure Appl. Anal., 12(6): 3027-3046, 2013. - Z.A. Wang
Mathematics of traveling waves in chemotaxis,
Discrete Contin. Dyn. Syst-Series B.,18(3): 601-641, 2013. - Y.S. Tao, L.H. Wang and Z.A. Wang
Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension,
Discrete Contin. Dyn. Syst-Series B., 18(3): 821-845, 2013. - Y.S. Tao and Z.A. Wang
Competing effects of attraction vs. repulsion in chemotaxis,
Math. Models Methods Appl. Sci., 23: 1-36, 2013. - Z.A. Wang, M. Winkler and D. Wrzosek
Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume filling effect and degenerate diffusion,
SIAM J. Math. Anal., 44: 3502-3525, 2012. - T. Li and Z.A. Wang
Steadily propagating waves of a chemotaxis model,
Mathematical Biosciences, 240: 161-168, 2012 - M.J. Ma, C.H. Ou and Z.A. Wang
Stationary solutions of a volume filling chemotaxis model with logistic growth and their stability,
SIAM J. Appl. Math., 72: 740-766, 2012. - Z.A. Wang
Wavefront of an angiogenesis model,
Discrete Contin. Dyn. Syst-Series B., 17(8): 2849-2860, 2012. - Z.A. Wang, M. Winkler and D. Wrzosek
Singularity formation in chemotaxis systems with volume-filling effect,
Nonlinearity, 24: 3279-3297, 2011. - J. Liu and Z.A. Wang
Classical solutions and steady states of an attraction-repulsion chemotaxis model in one dimension,
J. Biol. Dyn, 6: 31-41, 2012 . - T. Li and Z.A. Wang
Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis,
J. Differential Equations, 250: 1310-1333, 2011. - T. Li and Z.A. Wang
Nonlinear stability of large amplitude viscous shock waves of a hyperbolic-parabolic system arising in chemotaxis,
Math. Models Methods. Appl. Sci., 20: 1967-1998, 2010. - Z.A. Wang
On chemotaxis models with cell population interactions,
Math. Model Nat. Phenom.,5:173-190, 2010 - R. Lui and Z.A. Wang
Traveling wave solutions from microscopic to macroscopic chemotaxis models,
J. Math. Biol, 61: 739-761, 2010. - T. Hillen, P. Hinow and Z.A. Wang
Mathematical analysis of a kinetic model for cell movement in network tissues,
Discrete Contin. Dyn. Syst-Series B, 14:1055-1080, 2010. - Y.S. Choi and Z.A. Wang
Prevention of blow up in chemotaxis by fast diffusion,
J. Math. Anal. Appl., 362: 553-564, 2010. - T. Li and Z.A. Wang
Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,
SIAM J. Appl. Math., 70: 1522-1541, 2009. - Z.A. Wang, T. Hillen and M. Li
Mesenchymal motion models in one dimension,
SIAM J. Appl. Math., 69: 375-397, 2008. - Z.H. Guo, M.N. Jiang, Z.A. Wang and G.F. Zheng
Existence of global weak solutions to the Camassa-Holm equation,
Discrete Contin. Dyn. Syst., 21: 883-906, 2008. - Z.A. Wang and T. Hillen
Shock formation in a chemotaxis model,
Math. Methods. Appl. Sci., 31: 45-70, 2008. - Z.A. Wang and T. Hillen
Classical solutions and pattern formation for a volume filling chemotaxis model,
Chaos 17, 037108, 2007. - Z.A. Wang and H. Sang
Asymptotic profile to the nonlinear dissipative evolution equations with conservation form ,
Math. Methods. Appl. Sci., 20): 977-994, 2007. - Z.A. Wang
Optimal convergence rates toward diffusion wave of solutions to non-linear evolution equations with conservational form,
J. Math. Anal. Appl., 319: 740-763, 2006. - Z.A. Wang
Optimal decay rates of solutions to dissipative nonlinear evolution equations with ellipticity,
Z. Angew. Math. Phys., 57: 399-418, 2006. - Z.A. Wang
Large time behaviors of solutions for a dissipative nonlinear evolution system with conservation form,
J. Phys. A: Math. Gen., 38: 10955-10969, 2005. - C.J. Zhu and Z.A. Wang
Decay rates of solutions to dissipative nonlinear evolution equations with ellipticity,
Z. Angew. Math. Phys., 55: 994-1014, 2004. - Z.A. Wang, C.J. Zhu
Stability of the rarefaction wave for the generalized KdV-Burgers equation,
Acta Math Scientia., 22B(3): 319-328, 2002.