Higher dimensional generalization of the Benjamin‐Ono equation: 2D case

Abstract

We consider a higher‐dimensional version of the Benjamin‐Ono (HBO) equation in the 2D setting: ut−R1normalΔu+12false(u2false)x=0,(x,y)∈R2$u_t- \mathcal {R}_1 \Delta u + \frac{1}{2}(u^2)_x=0, (x,y) \in \mathbb {R}^2$, which is L2$L^2$‐critical, and investigate properties of solutions both analytically and numerically. For a generalized equation (fractional 2D gKdV) after deriving the Pohozaev identities, we obtain nonexistence conditions for solitary wave solutions, then prove uniform bounds in the energy space or conditional global existence, and investigate the radiation region, a specific wedge in the negative x$x$‐direction. We then introduce our numerical approach in a general context, and apply it to obtain the ground state solution in the 2D critical HBO equation, then show that its mass is a threshold for global versus finite time existing solutions, which is typical in the focusing (mass‐)critical dispersive equations. We also observe that globally existing solutions tend to disperse completely into the radiation in this nonlocal equation. The blow‐up solutions travel in the positive x$x$‐direction with the rescaled ground state profile while also radiating dispersive oscillations into the radiative wedge. We conclude with examples of different interactions of two solitary wave solutions, including weak and strong interactions.

Stability and instability of solitary waves in fractional generalized KdV equation in all dimensions

“…These identities can be obtained by similar arguments as in the proof of [75, Lemma 1]. We remark that to adapt the proof from [75], we require the following identities…”

“…admits non-trivial solutions in H α 2 (R d ), for example, via Pohozhaev identities, see Lemma 2.9 and further details in [75] (here, we are only interested in real-valued and more precisely positive solutions). 6 We recall the following fractional Gagliardo-Nirenberg inequality ([14, p.168], [19, Theorem 1.3.7], [33]), which shows that any f ∈ H α 2 (R d ) also belongs to L m+1 (R d ) by interpolation:…”

“…In general, the fKdV equation in dimention d = 1 has been used in a variety of wave phenomena, and the lower dimensional case 0 < α < 1 has been investigated as a model to measure the influence of dispersive effects on the dynamics of Burger's equation. For other studies on the 1d fKdV equation, for example, see [7,45,64,74,65,77], for the higher dimensional fKdV, see [76,38,79,75], and reference therein.…”

On decay properties for solutions of the Zakharov-Kuznetsov equation

“…The condition (1.10) also appears in the wellposedness result for the exponential initial condition in the work of Deléage and Linares [11]. In general, it seems that similar conditions to (1.10) are also needed in some studies of dispersive generalizations of the ZK equation, see the numerical investigations in [59] and the propagation of regularity results in [48].…”

Well-posedness and dynamics of solutions to the generalized KdV with low power nonlinearity

“…(We could consider higher polynomial decay as well, but this is the slowest rate of decay we are able to handle by numerics by using rational basis functions, see Appendix or [56], [55].) We take the computational domain [−L, L] large enough (L = 1600π) to approximate the actual solution on R. We point out that taking the larger values of L leads to similar numerical results.…”