Trapped solitary waves over an uneven bottom

Abstract

Steady two-dimensional surface waves on an ideal irrotational fluid over a complex multi-bumped topography are studied analytically in the case when the far upstream flow is slightly supercritical. Fully nonlinear equations are formulated via the von Mises variables that parametrize the bundle of streamlines in the flow over obstacles. For a given small-height topography, we construct approximate two-parametric solution sets which approach the branches of solitary waves as the typical height of the obstacle vanishes.

Appendices

Appendix A

Here, we consider the Green’s operator $\varGamma$ defined by Formula (3.13)

$$\begin{aligned} \varGamma b(x;c_0)= -3\tau _0'(x;c_0)\int \limits _{c_0}^{x}r_0(s;c_0)b(s)\mathrm{d}s -3r_0(x;c_0)\int \limits _{x}^{\infty }\tau _0'(s;c_0)b(s)\mathrm{d}s \end{aligned}$$

where analytic functions $\tau _0$ and $r_0$ have exponential behavior at infinity such that

$$\begin{aligned} |\tau _0'(x;c_0)|\leqslant C\,\mathrm{e}^{-|x-c_0|}, \qquad |r_0(x;c_0)|\leqslant C\,\mathrm{e}^{|x-c_0|}. \end{aligned}$$

We suppose that the function b belongs to the class $C_{\alpha }({\mathbb {R}})$ of continuous functions having an exponential decay $|b(x)|\leqslant C\exp (-\alpha |x|)$ as $|x|\rightarrow \infty$ with $0<\alpha <1$. In this case, the function $\varGamma b(x;c_0)$ has continuous derivatives $D_x\varGamma b$ and $D^2_x\varGamma b$, vanishes exponentially as $x\rightarrow +\infty$, but can be unbounded since

$$\begin{aligned} \varGamma b(x;c_0)\sim -3\mu (c_0; b)\,r_0(x;c_0)\qquad (x\rightarrow -\infty ) \end{aligned}$$

where $\mu$ is the Melnikov integral from Formula (3.11). This unbounded term can be removed if the necessary condition (3.11) is valid, so the routine technique provides the estimate

$$\begin{aligned} |\varGamma b(x;c_0)|+|D_x\varGamma b(x;c_0)|+|D^2_x\varGamma b(x;c_0)|\leqslant C\,\mathrm{e}^{-\alpha |x|} \end{aligned}$$

where the constant C does not depend on x. It means that the linear operator $\varGamma :\,C_{\alpha }({\mathbb {R}})\rightarrow C_{\alpha }^2({\mathbb {R}})$ is bounded under the condition (3.11) for the class $C^2_{\alpha }({\mathbb {R}})$ which consists of twice continuously differentiable functions having exponential decay rate $0<\alpha <1$ at infinity.

Appendix B

Here, we prove the lemma formulated in Sect. 3. Let us consider the orthogonality condition prescribed with the function $\tau _0'$ to the right-hand term from Eq. (3.14):

$$\begin{aligned} \int \limits _{-\infty }^{\infty } \big (\varGamma b(s;c_0)+k_1\tau _0'(s;c_0)\big )^2 \tau _0'(s;c_0)\mathrm{d}s=0. \end{aligned}$$

Note that the term with $k_1^2$ actually disappears since the eigenfunction $\tau _0'(s;c_0)$ is odd with respect to the point $s=c_0$, so the integral with a cubic term $\tau _0'^3$ vanishes. Therefore, we find immediately that the constant $k_1$ should be taken as follows:

$$\begin{aligned} k_1=-\,\frac{\displaystyle {\int \limits _{-\infty }^{\infty } \tau _0'(s;c_0)\bigg (\varGamma b(s;c_0)\bigg )^2 \mathrm{d}s}}{\displaystyle {2\int \limits _{-\infty }^{\infty }\tau _0'^2(s;c_0)\,\varGamma b(s;c_0)\, \mathrm{d}s}} \end{aligned}$$

(B.1)

if the denominator is not zero here. Now, we demonstrate that this condition is equivalent to the criterion formulated in the lemma. For that reason, we consider the linear differential operator

$$\begin{aligned} \mathcal{A}(c_0)u:= u_{xx}+\big (9\tau _0(x;c_0)-1\big )u \end{aligned}$$

defined by the left-side part of Eq. (3.9) for the class $C^2_{\alpha }({\mathbb {R}})$ (see Appendix A). It is clear that the operator $\mathcal A$ is symmetric with respect to scalar product of the Hilbert space $L_2({\mathbb {R}})$,

$$\begin{aligned} \int \limits _{-\infty }^{\infty }\mathcal{A}u_1\!(s)\, u_2(s) \mathrm{d}s = \int \limits _{-\infty }^{\infty } u_1\!(s)\, \mathcal{A}u_2(s) \mathrm{d}s, \qquad u_1,u_2\in C^2_{\alpha }({\mathbb {R}}). \end{aligned}$$

Further, we introduce modified Green’s operator $\mathcal K$ acting by the rule

$$\begin{aligned} \mathcal{K} b(x;c_0):= m_{\delta }\tau _0'(x;c_0)+\varGamma b(x;c_0) \end{aligned}$$

(B.2)

where the constant $m_{\delta }$ is defined by Formula (3.18) such that the element $\mathcal{K} b$ occurs to be orthogonal to $\tau _0'$ automatically in $L_2({\mathbb {R}})$ for any given function $b\in C_{\alpha }({\mathbb {R}})$. In other words, the solution $\tau _1$, taken from (3.12) with the constant $k_1=m_{\delta }+c_{\delta }$, splits into the orthogonal sum $\tau _1=c_{\delta }\tau _0'+\mathcal{K}b$ where $\mu (c_0;\mathcal{K}b)=0$. Note that the operator $\mathcal A$, defined above, also satisfies the orthogonality condition $\mu (c_0;\mathcal{A}u)=0$ for any functions $u\in C^2_{\alpha }({\mathbb {R}})$. With these facts, we obtain from definition of the operator $\mathcal{K}$ the identity

$$\begin{aligned} \mathcal{A}\mathcal{K} u = \mathcal{K} \mathcal{A} u = u \end{aligned}$$

(B.3)

which holds for any $u\in C^2_{\alpha }({\mathbb {R}})$ such that $\mu (c_0;u)=0$. The relation (B.3) means that $\mathcal{K}$ is the resolvent operator which is inverse to $\mathcal{A}$ defined on the subspace in $C^2_{\alpha }({\mathbb {R}})$ being orthogonal to $\tau _0'$. In that sense, we are following here to standard Lyapunov–Schmidt projection method [31] applied to the Fredholm operator $\mathcal{A}: C^2_{\alpha }({\mathbb {R}})\rightarrow C_{\alpha }({\mathbb {R}})$ which has one-dimensional kernel appearing due to non-symmetry of the flow. Returning back to Formula (B.1), we can rewrite the denominator to the form announced in the lemma:

$$\begin{aligned} \int \limits _{-\infty }^{\infty }\tau _0'^2(s;c_0)\,\varGamma b (s;c_0)\,\mathrm{d}s= & {} \int \limits _{-\infty }^{\infty }\tau _0'^2(s;c_0)\,\mathcal{K}b (s;c_0)\,\mathrm{d}s\\= & {} \int \limits _{-\infty }^{\infty }\bigg (\mathcal{K}\tau _0'^2(s;c_0)\bigg )b(s)\,\mathrm{d}s= -\frac{1}{9}\int \limits _{-\infty }^{\infty }\tau _0''(s;c_0) b(s)\,\mathrm{d}s\\= & {} \frac{1}{9}\left( \frac{\partial \mu (c;b)}{\partial c}\right) _{\big |c=c_0}. \end{aligned}$$

Indeed, the first step uses here the fact that the elements $\varGamma b$ and $\mathcal{K}b$ differ in (B.2) only by the term $m_{\delta }\tau _0'$ being odd function. Further, we involve the symmetry of resolvent operator $\mathcal{K}$ transposed to the element $\tau _0'^2$ via scalar product. Finally, it remains to be noticed that the function $u=\tau _0''$ is clearly a solution of linear ODE equation $\mathcal{A}u=-9\tau _0'^2$ where the right-hand term satisfies the solvability condition $\mu (c_0;\tau _0'^2)=0$.

Concluding remark concerns with transformation of approximate solution $w=\varepsilon ^2(w_{10}+\delta w_{11})$ to the form (3.17). This property follows immediately from asymptotic decomposition

$$\begin{aligned} \tau _0(x;c_0)+\delta \tau _1(x;c_0)=\tau _0(x;c_0+\delta c_{\delta })+\delta \mathcal{K} b(x;c_0) + h.o.t. \end{aligned}$$

which takes into account the definition (B.2) of the operator $\mathcal{K}$ and Formula (3.12) with $k_1=m_{\delta }+c_{\delta }$.

Hence, the lemma is proven.