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Existence and Multiplicity of Wave Trains in a 2D Diatomic Face-Centered Lattice

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Abstract

We investigate the existence and branching patterns of wave trains (also called periodic traveling waves) in a 2D face-centered square lattice consisting of alternating light and heavy atoms, with linear coupling between nearest particles and a nonlinear substrate potential. In contrast to monatomic chains, we consider two different periodic waveform functions corresponding to light and heavy particles, respectively. As a result, we have to solve coupled advanced-delay differential equations, which are reduced to a finite-dimensional bifurcation equation with an inherited Hamiltonian structure by applying a Lyapunov–Schmidt reduction. Then we obtain the small-amplitude solutions in the Hamiltonian system near equilibria in nonresonance and p : q resonance, respectively. In particular, the results can be applied to some one-dimensional diatomic lattices.

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Correspondence to Shangjiang Guo.

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Communicated by Alan R. Champney.

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This work was supported in part by the National Natural Science Foundation of P.R. China (Grant Nos. 11701532 and 12071446) and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (Grant No. CUGST2).

Appendix: Proof of generic nondegeneracy conditions

Appendix: Proof of generic nondegeneracy conditions

We shall prove Theorem 4.5 in more details by computing a series of derivatives of the reduced Hamiltonian function h.

Proof of Theorem 4.5(i)

At first, we expand \((u_1,u_2,u_3,u_4)\in {\mathscr {X}}^l ={\mathcal {K}}\oplus {\mathcal {M}}^*\) as same as in formula (25). Then the variables \(\{z_{k_1^*},z_{-k_1^*},z_{k_2^*},z_{-k_2^*}\}\) are used to describe the elements of \({\mathcal {K}}\), while the others describe the elements of \({\mathcal {M}}^*\). Recall that \(z_k\ (k\ne \pm k^*_1,\pm k^*_2)\) and \(y_{i,k}\) can all be viewed as functions of the seven independent coordinates \(z_k\ (k=\pm k_1^*,\pm k_2^*),\theta ,v,\omega \) that satisfy \(z_k={\mathcal {O}}(\Vert (z_{k_1^*},z_{-k_1^*},z_{k_2^*},z_{-k_2^*},\theta ^*-\theta ,v^*-v,\omega ^*-\omega )\Vert ^2)\) for \(k\ne \pm k^*_1,\pm k^*_2\) and \(y_{i,k}={\mathcal {O}}(\Vert (z_{k_1^*},z_{-k_1^*},z_{k_2^*},z_{-k_2^*},\theta ^*-\theta ,v^*-v,\omega ^*-\omega )\Vert ^2)\) for all k and \(i\in \{1,2,3\}\). Then one obtains from (26) that

$$\begin{aligned} \begin{aligned}&h(z_{k_1^*},z_{-k_1^*},z_{k_2^*},z_{-k_2^*},\theta ,v,\omega )\\=&\Big [2k_1^{*2}M_1\omega ^*\omega v^*v+2k_1^{*2}M_2\sigma _{k_1^*}^{2}\omega ^*\omega v^*v -k_1^{*2}M_1(v^*\omega ^*)^{2}-k_1^{*2}M_2\sigma _{k_1^*}^{2}(v^*\omega ^*)^{2} \\ {}&+2\sigma _{k_1^*}\rho (\theta ,\omega ,k_1^*) -(4\lambda +\gamma )(1+\sigma _{k_1^*}^{2})\Big ]z_{k_1^*}z_{-k_1^*} \\&+\,\Big [2k_2^{*2}M_1\omega ^*\omega v^*v+2k_2^{*2}M_2\sigma _{k_2^*}^{2}\omega ^*\omega v^*v-k_2^{*2}M_1(v^*\omega ^*)^{2}-k_2^{*2}M_2\sigma _{k_2^*}^{2}(v^*\omega ^*)^{2} \\ {}&+2\sigma _{k_2^*}\rho (\theta ,\omega ,k_2^*)-(4\lambda +\gamma )(1+\sigma _{k_2^*}^{2})\Big ]z_{k_2^*}z_{-k_2^*}\\ {}&+ {\mathcal {O}}(\Vert (z_{k_1^*},z_{-k_1^*},z_{k_2^*},z_{-k_2^*})\Vert ^3)+{\mathcal {O}}(\Vert (z_{k_1^*},z_{-k_1^*},z_{k_2^*},z_{-k_2^*},\theta ^*-\theta ,v^*-v,\omega ^*-\omega )\Vert ^4). \end{aligned} \end{aligned}$$

It is easy to check that all the second-order subdeterminants of the matrix (50) are nonzero exactly when the normal vectors of surface \(v^2= g_{-}(\theta ,\omega ,k^*_1)\) and \(v^2= g_{+}(\theta ,\omega ,k^*_2)\) at \((\theta ^*,v^*,\omega ^*)\) are not collinear. \(\square \)

Proof of Theorem 4.5(ii)

In fact, it suffices to prove the theorem under the assumption that \(f(x)=\gamma x+\frac{\delta }{(p+q-1)!}x^{p+q-1}\), where \(p+q\) is even. Firstly, equating all inner products of \(F(U,\theta ^*,v^*,\omega ^*)\) with basis vectors for \({\mathcal {M}}\) to zero yields that for \(k\ne \pm k_1^*,\pm k_2^*\)

$$\begin{aligned} \left\{ \begin{aligned} ((k\omega ^*)^2v^{*2}M_1+1)y_{1,k}-k\omega ^*v^*\sqrt{M_1}\mathrm {i}\sigma _ky_{3,k}&=0, \\ ((k\omega ^*)^2v^{*2}M_2+1)y_{2,k}+k\omega ^*v^*\sqrt{M_2}\mathrm {i}y_{3,k}&=0,\\ \big [(k\omega ^*)^2v^{*2}M_1+\sigma _k\rho _k-(4\lambda +\gamma )\big ]z_k-k\omega ^*v^*\sqrt{M_1}\mathrm {i}(1+4\lambda +\gamma )y_{1,k}&\\ +k\omega ^*v^*\sqrt{M_2}\mathrm {i}\rho _ky_{2,k} -(\rho _k+\sigma _k(4\lambda +\gamma ))y_{3,k}&=\delta D_{k,1},\\ \big [(k\omega ^*)^2v^{*2}M_2\sigma _k+\rho _k-\sigma _k(4\lambda +\gamma )\big ]z_k+k\omega ^*v^*\sqrt{M_1}\mathrm {i}\rho _ky_{1,k}&\\ -k\omega ^*v^*\sqrt{M_2}\mathrm {i}(1+4\lambda +\gamma )y_{2,k} +(\sigma _k\rho _k+(4\lambda +\gamma ))y_{3,k}&=\delta D_{k,2} \end{aligned} \right. \end{aligned}$$
(66)

and for \(k= \pm {k}_{1}^{*},\pm {k}_{2}^{*}\),

$$\begin{aligned} \left\{ \begin{aligned} \big [(k\omega ^*)^2v^{*2}M_1(2+4\lambda +\gamma )+1\big ]y_{1,k}-(k\omega ^*)^2v^{*2}\sqrt{M_1M_2}\rho _ky_{2,k}&\\ -k\omega ^*v^*\sqrt{M_1}\mathrm {i}(\sigma _k(1+4\lambda +\gamma )+\rho _k)y_{3,k}=kv^*\omega ^*\sqrt{M_1}&\mathrm {i}\delta D_{k,1},\\ -(k\omega ^*)^2v^{*2}\sqrt{M_1M_2}\rho _ky_{1,k}+\big [(k\omega ^*)^2v^{*2}M_2(2+4\lambda +\gamma )+1\big ]y_{2,k}&\\ +k\omega ^*v^*\sqrt{M_2}\mathrm {i}(1+\sigma _k\rho _k+4\lambda +\gamma )y_{3,k}=kv^*\omega ^*\sqrt{M_2}\mathrm {i}&\delta D_{k,2},\\ -k\omega ^*v^*\sqrt{M_1}\mathrm {i}[\sigma _k(1+4\lambda +\gamma )+\rho _k]y_{1,k}+k\omega ^*v^*\sqrt{M_2}\mathrm {i}\big [1+ \sigma _k\rho _k+4\lambda +\gamma \big ]y_{2,k}&\\ -\big [(\sigma _{k}^2+1)(4\lambda +\gamma ) +2\sigma _k\rho _k\big ]y_{3,k}=\sigma _k\delta D_{k,1}&-\delta {D}_{k,2}, \end{aligned}\right. \end{aligned}$$
(67)

where

$$\begin{aligned} D_{k,1}:&=\frac{1}{2\pi (p+q-1)!}\int _0^{2\pi }\mathrm {e}^{-\mathrm {i}ks}u_1^{p+q-1}(s)\mathrm{d}s\\&=\frac{1}{(p+q-1)!}\sum _{\begin{array}{c}n\in {\mathbb {Z}}^{p+q-1}\\ \sum _{j=1}^{p+q-1}n_j=k\end{array}} \prod _{j=1}^{p+q-1}\Big (z_{n_j}+n_j\omega ^*v^*\sqrt{M_1}\mathrm {i}y_{1,n_j}+\sigma _{n_j}y_{3,n_j}\Big ),\\ D_{k,2}:&=\frac{1}{2\pi (p+q-1)!}\int _0^{2\pi }\mathrm {e}^{-\mathrm {i}ks}u_2^{p+q-1}(s)\mathrm{d}s\\&=\frac{1}{(p+q-1)!}\sum _{\begin{array}{c}n\in {\mathbb {Z}}^{p+q-1}\\ \sum _{j=1}^{p+q-1}n_j=k\end{array}} \prod _{j=1}^{p+q-1}\Big (\sigma _{n_j}z_{n_j}+n_j\omega ^*v^*\sqrt{M_2}\mathrm {i}y_{2,n_j}-y_{3,n_j}\Big ), \end{aligned}$$

and \(\rho _k=\rho (\theta ^*,\omega ^*,k)\). One now observes from these equations that

$$\begin{aligned} \frac{\partial y_{i,n}}{\partial z_k}(0,\theta ^*,v^*,\omega ^*)=0,\ \ \frac{\partial z_{n}}{\partial z_k}(0,\theta ^*,v^*,\omega ^*)=\delta _n^k(\mathrm{the\ Kronecker\ delta}) \end{aligned}$$

for all n, \(k=\pm k_1^*,\pm k_2^*\) and \(i\in \{1,2,3\}\), which implies that \(D_{n,j}={\mathcal {O}} (\parallel (z_{k_1^*},z_{-k_1^*},z_{k_2^*},z_{-k_2^*})\parallel ^{p+q-1})\) for \(j=1,2\). Hence, \(z_k={\mathcal {O}}(\Vert (z_{k_1^*},z_{-k_1^*},z_{k_2^*},z_{-k_2^*})\Vert ^{p+q-1})\) for \(k\ne \pm k_1^*,\pm k_2^*\) and \(y_{i,k}={\mathcal {O}}(\Vert (z_{k_1^*},z_{-k_1^*},z_{k_2^*},z_{-k_2^*})\Vert ^{p+q-1})\) for all k and \(i\in \{1,2,3\}\). In order to compute the derivative of h with respect to d, we should compute the reduced function h:

$$\begin{aligned} h&(z_{k_1^*},z_{-k_1^*},z_{k_2^*},z_{-k_2^*},\theta ^*,v^*,\omega ^*)\\ =&\sum _{k\in {\mathbb {Z}}_{>0}}\Big [ (k\omega ^*v^*)^2M_1+(k\omega ^*v^*)^2M_2\sigma _{k}^2 +2\sigma _k\rho _k-(4\lambda +\gamma )(1+\sigma _{k}^2)\Big ]z_kz_{-k}\\&+\sum _{k\in {\mathbb {Z}}_{>0}}\Big [-(k\omega ^*v^*\sqrt{M_1})^3\mathrm {i}-k\omega ^*v^*\sqrt{M_1}\mathrm {i}\sigma _k\rho _k+k\omega ^*v^*\sqrt{M_1}\mathrm {i}(4\lambda +\gamma )\Big ]\\&\qquad (z_ky_{1,-k}-z_{-k}y_{1,k})\\&+\sum _{k\in {\mathbb {Z}}_{>0}}\Big [-(k\omega ^*v^*\sqrt{M_2})^3\sigma _k\mathrm {i}-k\omega ^*v^*\sqrt{M_2}\mathrm {i}\rho _k +k\omega ^*v^*\sqrt{M_2}\mathrm {i}\sigma _k(4\lambda +\gamma )\Big ]\\&\qquad (z_ky_{2,-k}-z_{-k}y_{2,k})\\&+\sum _{k\in {\mathbb {Z}}_{>0}}\Big [(k\omega ^*v^*)^2M_1\sigma _k-(k\omega ^*v^*)^2M_2\sigma _k+\sigma _{k}^2\rho _k-\rho _k\Big ](z_ky_{3,-k}-z_{-k}y_{3,k})\\&+\frac{\delta }{(p+q)!}\sum _{\begin{array}{c}n\in {\mathbb {Z}}^{p+q}\\ \sum _{j=1}^{p+q}n_j=0\end{array}} \prod _{j=1}^{p+q}(z_{n_j}+n_jv^*\sqrt{M_1}\mathrm {i}y_{1,n_j}+\sigma _{n_j}y_{3,n_j})\\&+\frac{\delta }{(p+q)!}\sum _{\begin{array}{c}n\in {\mathbb {Z}}^{p+q}\\ \sum _{j=1}^{p+q}n_j=0\end{array}} \prod _{j=1}^{p+q}(\sigma _{n_j}z_{n_j}+n_jv^*\sqrt{M_2}\mathrm {i}y_{2,n_j}-y_{3,n_j})\\&+{\mathcal {O}}(\Vert (z_{k_1^*},z_{-k_1^*},z_{k_2^*},z_{-k_2^*})\Vert ^{2(p+q-1)})\\ =&\,r(a,b)+\frac{\delta }{p!q!}(1+\sigma _{k_1^*}^{q}\sigma _{k_2^*}^{p})d+{\mathcal {O}}(\Vert (z_{k_1^*},z_{-k_1^*},z_{k_2^*},z_{-k_2^*})\Vert ^{2(p+q-1)}), \end{aligned}$$

where the function r(ab) appears only when \(p+q\) is even. Thus, if \(1+\sigma _{k_1^*}^{q}\sigma _{k_2^*}^{p}\ne 0\), then we show that \(\tau \ne 0\). This concludes the proof of (ii). \(\square \)

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Zhang, L., Guo, S. Existence and Multiplicity of Wave Trains in a 2D Diatomic Face-Centered Lattice. J Nonlinear Sci 32, 54 (2022). https://doi.org/10.1007/s00332-022-09813-w

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