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The hyperbolic nature of the zero dispersion KdV limit. (English) Zbl 0678.35081

The authors study the weak limit of solutions \(u^{\epsilon}(t,x)\) of \(u_ t^{\epsilon}-6u^{\epsilon}u_ x^{\epsilon}+\epsilon^ 2u^{\epsilon}_{xxx}=0\) whose initial values \(u^{\epsilon}(0,x)=u(x)\) are independent of \(\epsilon\). It is known that in the neighborhood of almost all (t,x) there exists an integer N such that the weak limit is determined by \((2N+1)\)-dimensional real vector-valued function \(\lambda =(\lambda_ 0,\lambda_ 1,...,\lambda_{2N})\) with the condition \(\lambda_ 0<\lambda_ 1<...<\lambda_{2N},\) and satisfying the hyperbolic system in Riemann invariant form \(\partial_ t\lambda_ k+s_ k(\lambda)\partial_ x\lambda_ k=0,\) \(k=0,...,2N\); where characteristic speeds, \(s_ k(\lambda)\), are real and have a functional form depending on N. The authors prove that this system is strictly hyperbolic, with the characteristic speeds ordered as \(s_ 0({\underline \lambda})>...>s_{2N}(\lambda)\) and genuinely nonlinear, with \(\partial_{\lambda_ k}s_ k({\underline \lambda})<0\), \(k=0,...,2N\). The authors also prove that the evolution of \(\partial_ x\lambda_ k\) along kth characteristic satisfies some Riccati equation.
Reviewer: A.M.Šermenev

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35B40 Asymptotic behavior of solutions to PDEs
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References:

[1] Ercolani N.M., in preparation
[2] DOI: 10.1002/cpa.3160330605 · Zbl 0454.35080
[3] Lax P.D., Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (1973) · Zbl 0268.35062
[4] DOI: 10.1073/pnas.76.8.3602 · Zbl 0411.35081
[5] DOI: 10.1137/0126036 · Zbl 0273.35055
[6] DOI: 10.1002/cpa.3160380202 · Zbl 0571.35095
[7] Whitham G.B., Linear and Nonlinear waves (1974) · Zbl 0373.76001
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