Mixed Rational Lump-Solitary Wave Solutions to an Extended (2+1)-Dimensional KdV Equation

Based on the bilinear method, rational lump and mixed lump-solitary wave solutions to an extended (2+1)-dimensional KdV equation are constructed through the different assumptions of the auxiliary function in the trilinear form. It is found that the rational lump decays algebraically in all directions in the space plane and its amplitude possesses one maximum and two minima. One kind of the mixed solution describes the interaction between one lump and one line solitary wave, which exhibits fission and fusion phenomena under the different parameters. The other kind of the mixed solution shows one lump interacting with two paralleled line solitary waves, in which the evolution of the lump gives rise to a two-dimensional rogue wave. This shows that these three interesting phenomena exist in the corresponding physical model.

New Periodic Wave, Cross-Kink Wave, Breather, and the Interaction Phenomenon for the (2 + 1)-Dimensional Sharmo–Tasso–Olver Equation

In this paper, with the aid of symbolic computation, several kinds of exact solutions including periodic waves, cross-kink waves, and breather are proposed by using a trilinear form for the (2 + 1)-dimensional Sharmo–Tasso–Olver equation. Then, by combing the different forms, the interactions between a lump and one-kink soliton and between a lump and periodic waves are generated. Moreover, the dynamic characteristics of interaction solutions are analyzed graphically by selecting suitable parameters with the help of Maple.

The Stokes Complex for Virtual Elements with Application to Navier–Stokes Flows

In the present paper, we investigate the underlying Stokes complex structure of the Virtual Element Method for Stokes and Navier–Stokes introduced in previous papers by the same authors, restricting our attention to the two dimensional case. We introduce a Virtual Element space $${\varPhi }_h \subset H^2({\varOmega })$$ Φ h ⊂ H 2 ( Ω ) and prove that the triad $$\{{\varPhi }_h, {\varvec{V}}_h, Q_h\}$$ { Φ h , V h , Q h } (with $${\varvec{V}}_h$$ V h and $$Q_h$$ Q h denoting the discrete velocity and pressure spaces) is an exact Stokes complex. Furthermore, we show the computability of the associated differential operators in terms of the adopted degrees of freedom and explore also a different discretization of the convective trilinear form. The theoretical findings are supported by numerical tests.

Trilinear forms with double Kloosterman sums

We obtain several estimates for trilinear form with double Kloosterman sums. In particular, these bounds show the existence of nontrivial cancellations between such sums.

Invariant trilinear forms on spherical principal series of real rank one semisimple Lie groups

Let G be a connected semisimple real-rank one Lie group with finite center. We consider intertwining operators on tensor products of spherical principal series representations of G. This allows us to construct an invariant trilinear form [Formula: see text] indexed by a complex multiparameter [Formula: see text] and defined on the space of smooth functions on the product of three spheres in 𝔽n, where 𝔽 is either ℝ, ℂ, ℍ, or 𝕆 with n = 2. We then study the analytic continuation of the trilinear form with respect to (ν1, ν2, ν3), where we locate the hyperplanes containing the poles. Using a result due to Johnson and Wallach on the so-called "partial intertwining operator", we obtain an expression for the generalized Bernstein–Reznikov integral [Formula: see text] in terms of hypergeometric functions.

Random trilinear forms and the Schur multiplication of tensors

We obtain estimates for the distribution of the norm of the random trilinear formA:ℓrM×ℓpN×ℓqK→ℂ, defined byA(ei,ej,ek)=aijk, where theaijk's are uniformly bounded, independent, mean zero random variables. As an application, we make progress on the problem whenℓr⊗⌣ℓp⊗⌣ℓqis a Banach algebra under the Schur multiplication.