Symmetry and decay of traveling wave solutions to the Whitham equation

Abstract

This paper is concerned with decay and symmetry properties of solitary wave solutions to a nonlocal shallow water wave model. It is shown that all supercritical solitary wave solutions are symmetric and monotone on either side of the crest. The proof is based on a priori decay estimates and the method of moving planes. Furthermore, a close relation between symmetric and traveling wave solutions is established

On Whitham and Related Equations

“…The existence of solitary wave solutions to the Whitham equation decaying to zero at infinity and close to the KdV soliton has been proven in by variational methods. Symmetry and decay properties of Whitham solitary waves are established in . The analysis in is made on with under the scaling …”

“…Similarly, a solitary wave of the Whitham equation satisfies the equation Remark Following the method used in , one can prove that solutions of that tend to 0 as decay to 0 exponentially in the sense that for some , .…”

“…Remark 10. Following the method used in [46], one can prove that solutions of (60) that tend to 0 as |x| → ∞ decay to 0 exponentially in the sense that for some ν > 0, e ν|·| φ ∈ L 1 (R) ∩ L ∞ (R).…”

Symmetric waves are traveling waves for a shallow water equation modeling surface waves of moderate amplitude

“…An important feature of equation (1) is that it captures the non-linear phenomenon of wave breaking, in the sense that the surface profile remains bounded but its slope may form singularities in finite time [5,8]. The Cauchy problem associated to (1) is locally well-posed in H s , for s > 3/2, on the real line as well as on the circle [11,9], and the data-to-solution map was shown to be non-uniformly continuous [11]. For a wellposedness result in the context of Besov spaces we refer the reader to [25], whereas a result on low regularity solutions with 1 < s ≤ 3/2 may be found in [22], and for global conservative solutions and continuation of solutions beyond wave breaking we point out [29].…”

Families of Rational and Semirational Solutions of the Partial Reverse Space-Time Nonlocal Mel′nikov Equation

“…Most of the NLEEs studied in the literature are local equations where the evolution depends only on the local value of the dependent variable and its local space and time derivatives. Nonlocal evolution equations arise in many fields of applications too, where the nonlocal nature frequently comes from a dependence on the global properties, e.g., an integral of the dependent variable [21][22][23]. Physical scenarios include modulation of nonlinear waves (Whitham's equation) [21] and reaction diffusion systems [22,23].…”