Nonlocal solitons has been a hot topic during the past decades [1], [2], [3], [4]. Nonlocal nonlinearity [5] is an inherent characteristics of many nonlinear media [2], [3], [4], [6]. Nonlocality sustain a series of novel solitons states, such as elliptic solitons and vortex solitons [6], two-color vector solitons [7], surface-wave solitons [8], and incoherent solitons [9], etc. Nonlocality also have deep impacts on soliton self-bending [10], soliton mobility [11], refraction and reflection of solitons [12], incoherent soliton turbulence [13], optical shock waves [14], as well as solitons interactions [15], [16].

In recent years, much attention has been focused on competing nonlocal nonlinearity with multiple different physical processes in the media. For instance, with simultaneous thermal and re-orientational effect [17], [18], [19], [20], [21], nematic liquid crystal exhibits competing nonlocal cubic-nonlocal cubic nonlinearities [22], [23], [24], [25], [26]. Furthermore, solitons have also been studied with competing local cubic-nonlocal cubic nonlinearities [27], [28], [29]. This kind of nonlinearity can be addressed in semiconductor material, such as AlGaAs, with both Kerr and thermal nonlinearities [27]. In Tonks-Girardeau gas [30] or Bose–Einstein condensate [31], the nonlinearity can be represented as competing nonlocal cubic-local quintic nonlinearities [32], [33], [34].

Another competing nonlocal media is competing CQ nonlocal nonlinear media which was firstly introduced by D. Mihalache et al. [35]. The CQ nonlinearity is regarded as expansion of saturable nonlinearity [36], competing CQ nonlocal nonlinear media can be treated as nonlocal media with a saturation of the nonlinear response [36], such as in atomic vapors [37], [38], [39], [40]. Competing CQ nonlocal nonlinear media can support solitons of even–odd parities [35], dark solitons [40], vortex solitons [41], elliptic solitons [42], [43], super-Gaussian solitons [44], as well as high dimensional dipole and quadrupole solitons [45].

HG [46], [47], [48] and LG solitons [49], [50], classical higher-order modal solitons, were studied extensively in nonlocal media [51], [52] during the past years. It has been shown that nontrivial model transformations between HG and LG are induced with the help of nonlocality [51], [52], [53]. However, HG and LG solitons clusters have not been investigated in nonlinear media with competing CQ nonlocal nonlinearities, in particular, with arbitrary degree of nonlocality.

In this paper, using linear-stability analysis, we study two- dimensional MI in competing CQ nonlocal nonlinear media. We also investigate HG and LG solitons clusters analytically and numerically. In particular, we obtain bifurcated solutions of HG and LG soliton clusters with variational approach. Dynamics of these HG and LG solitons clusters are demonstrated with split-step Fourier transform.